3.298 \(\int (f+g x^3)^2 \log ^3(c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=1124 \[ \text{result too large to display} \]

[Out]

-48*f^2*p^3*x + (351136*d^3*g^2*p^3*x)/(25725*e^3) + (6*d*f*g*p^3*x^2)/e - (55456*d^2*g^2*p^3*x^3)/(77175*e^2)
+ (5232*d*g^2*p^3*x^5)/(42875*e) - (48*g^2*p^3*x^7)/2401 - (3*f*g*p^3*(d + e*x^2)^2)/(8*e^2) + (48*Sqrt[d]*f^
2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (351136*d^(7/2)*g^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(25725*e^(7/
2)) - ((24*I)*Sqrt[d]*f^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] + (((1408*I)/245)*d^(7/2)*g^2*p^3*ArcTan[
(Sqrt[e]*x)/Sqrt[d]]^2)/e^(7/2) - (48*Sqrt[d]*f^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I
*Sqrt[e]*x)])/Sqrt[e] + (2816*d^(7/2)*g^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]
*x)])/(245*e^(7/2)) + 24*f^2*p^2*x*Log[c*(d + e*x^2)^p] - (1408*d^3*g^2*p^2*x*Log[c*(d + e*x^2)^p])/(245*e^3)
+ (568*d^2*g^2*p^2*x^3*Log[c*(d + e*x^2)^p])/(735*e^2) - (288*d*g^2*p^2*x^5*Log[c*(d + e*x^2)^p])/(1225*e) + (
24*g^2*p^2*x^7*Log[c*(d + e*x^2)^p])/343 - (6*d*f*g*p^2*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 + (3*f*g*p^2*(d
+ e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) - (24*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p
])/Sqrt[e] + (1408*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(245*e^(7/2)) - 6*f^2*p*x
*Log[c*(d + e*x^2)^p]^2 + (6*d^3*g^2*p*x*Log[c*(d + e*x^2)^p]^2)/(7*e^3) - (2*d^2*g^2*p*x^3*Log[c*(d + e*x^2)^
p]^2)/(7*e^2) + (6*d*g^2*p*x^5*Log[c*(d + e*x^2)^p]^2)/(35*e) - (6*g^2*p*x^7*Log[c*(d + e*x^2)^p]^2)/49 + (3*d
*f*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/e^2 - (3*f*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + f^2*
x*Log[c*(d + e*x^2)^p]^3 + (g^2*x^7*Log[c*(d + e*x^2)^p]^3)/7 - (d*f*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^3)/e^2
+ (f*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^3)/(2*e^2) - ((24*I)*Sqrt[d]*f^2*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sq
rt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (((1408*I)/245)*d^(7/2)*g^2*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[
e]*x)])/e^(7/2) + 6*d*f^2*p*Unintegrable[Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x] - (6*d^4*g^2*p*Unintegrable[Lo
g[c*(d + e*x^2)^p]^2/(d + e*x^2), x])/(7*e^3)

________________________________________________________________________________________

Rubi [A]  time = 2.64937, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(f + g*x^3)^2*Log[c*(d + e*x^2)^p]^3,x]

[Out]

-48*f^2*p^3*x + (351136*d^3*g^2*p^3*x)/(25725*e^3) + (6*d*f*g*p^3*x^2)/e - (55456*d^2*g^2*p^3*x^3)/(77175*e^2)
+ (5232*d*g^2*p^3*x^5)/(42875*e) - (48*g^2*p^3*x^7)/2401 - (3*f*g*p^3*(d + e*x^2)^2)/(8*e^2) + (48*Sqrt[d]*f^
2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (351136*d^(7/2)*g^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(25725*e^(7/
2)) - ((24*I)*Sqrt[d]*f^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] + (((1408*I)/245)*d^(7/2)*g^2*p^3*ArcTan[
(Sqrt[e]*x)/Sqrt[d]]^2)/e^(7/2) - (48*Sqrt[d]*f^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I
*Sqrt[e]*x)])/Sqrt[e] + (2816*d^(7/2)*g^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]
*x)])/(245*e^(7/2)) + 24*f^2*p^2*x*Log[c*(d + e*x^2)^p] - (1408*d^3*g^2*p^2*x*Log[c*(d + e*x^2)^p])/(245*e^3)
+ (568*d^2*g^2*p^2*x^3*Log[c*(d + e*x^2)^p])/(735*e^2) - (288*d*g^2*p^2*x^5*Log[c*(d + e*x^2)^p])/(1225*e) + (
24*g^2*p^2*x^7*Log[c*(d + e*x^2)^p])/343 - (6*d*f*g*p^2*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 + (3*f*g*p^2*(d
+ e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) - (24*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p
])/Sqrt[e] + (1408*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(245*e^(7/2)) - 6*f^2*p*x
*Log[c*(d + e*x^2)^p]^2 + (6*d^3*g^2*p*x*Log[c*(d + e*x^2)^p]^2)/(7*e^3) - (2*d^2*g^2*p*x^3*Log[c*(d + e*x^2)^
p]^2)/(7*e^2) + (6*d*g^2*p*x^5*Log[c*(d + e*x^2)^p]^2)/(35*e) - (6*g^2*p*x^7*Log[c*(d + e*x^2)^p]^2)/49 + (3*d
*f*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/e^2 - (3*f*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + f^2*
x*Log[c*(d + e*x^2)^p]^3 + (g^2*x^7*Log[c*(d + e*x^2)^p]^3)/7 - (d*f*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^3)/e^2
+ (f*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^3)/(2*e^2) - ((24*I)*Sqrt[d]*f^2*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sq
rt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (((1408*I)/245)*d^(7/2)*g^2*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[
e]*x)])/e^(7/2) + 6*d*f^2*p*Defer[Int][Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x] - (6*d^4*g^2*p*Defer[Int][Log[c*
(d + e*x^2)^p]^2/(d + e*x^2), x])/(7*e^3)

Rubi steps

\begin{align*} \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^2 \log ^3\left (c \left (d+e x^2\right )^p\right )+2 f g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log ^3\left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^3 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )+(f g) \operatorname{Subst}\left (\int x \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (6 e f^2 p\right ) \int \frac{x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{1}{7} \left (6 e g^2 p\right ) \int \frac{x^8 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )+(f g) \operatorname{Subst}\left (\int \left (-\frac{d \log ^3\left (c (d+e x)^p\right )}{e}+\frac{(d+e x) \log ^3\left (c (d+e x)^p\right )}{e}\right ) \, dx,x,x^2\right )-\left (6 e f^2 p\right ) \int \left (\frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{d \log ^2\left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx-\frac{1}{7} \left (6 e g^2 p\right ) \int \left (-\frac{d^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^4}+\frac{d^2 x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac{d x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^6 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac{d^4 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{(f g) \operatorname{Subst}\left (\int (d+e x) \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{e}-\frac{(d f g) \operatorname{Subst}\left (\int \log ^3\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{e}-\left (6 f^2 p\right ) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{1}{7} \left (6 g^2 p\right ) \int x^6 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\frac{\left (6 d^3 g^2 p\right ) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}-\frac{\left (6 d^2 g^2 p\right ) \int x^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}+\frac{\left (6 d g^2 p\right ) \int x^4 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e}\\ &=-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{(f g) \operatorname{Subst}\left (\int x \log ^3\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\frac{(d f g) \operatorname{Subst}\left (\int \log ^3\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\left (24 e f^2 p^2\right ) \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{1}{35} \left (24 d g^2 p^2\right ) \int \frac{x^6 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (24 d^3 g^2 p^2\right ) \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^2}+\frac{\left (8 d^2 g^2 p^2\right ) \int \frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e}+\frac{1}{49} \left (24 e g^2 p^2\right ) \int \frac{x^8 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{(3 f g p) \operatorname{Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\frac{(3 d f g p) \operatorname{Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\left (24 e f^2 p^2\right ) \int \left (\frac{\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac{d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx-\frac{1}{35} \left (24 d g^2 p^2\right ) \int \left (\frac{d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac{d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^3 \left (d+e x^2\right )}\right ) \, dx-\frac{\left (24 d^3 g^2 p^2\right ) \int \left (\frac{\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac{d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx}{7 e^2}+\frac{\left (8 d^2 g^2 p^2\right ) \int \left (-\frac{d \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx}{7 e}+\frac{1}{49} \left (24 e g^2 p^2\right ) \int \left (-\frac{d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^4}+\frac{d^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac{d x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^6 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{d^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\left (24 f^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\left (24 d f^2 p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\frac{\left (3 f g p^2\right ) \operatorname{Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}-\frac{\left (6 d f g p^2\right ) \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}+\frac{1}{49} \left (24 g^2 p^2\right ) \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\frac{\left (24 d^3 g^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{49 e^3}-\frac{\left (24 d^3 g^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{35 e^3}-\frac{\left (8 d^3 g^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac{\left (24 d^3 g^2 p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}+\frac{\left (24 d^4 g^2 p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{49 e^3}+\frac{\left (24 d^4 g^2 p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{35 e^3}+\frac{\left (8 d^4 g^2 p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\frac{\left (24 d^4 g^2 p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\frac{\left (24 d^2 g^2 p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{49 e^2}+\frac{\left (24 d^2 g^2 p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{35 e^2}+\frac{\left (8 d^2 g^2 p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}-\frac{\left (24 d g^2 p^2\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{49 e}-\frac{\left (24 d g^2 p^2\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{35 e}\\ &=\frac{6 d f g p^3 x^2}{e}-\frac{3 f g p^3 \left (d+e x^2\right )^2}{8 e^2}+24 f^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{1408 d^3 g^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{245 e^3}+\frac{568 d^2 g^2 p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )}{735 e^2}-\frac{288 d g^2 p^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )}{1225 e}+\frac{24}{343} g^2 p^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{6 d f g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 f g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{24 \sqrt{d} f^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{245 e^{7/2}}-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}-\left (48 e f^2 p^3\right ) \int \frac{x^2}{d+e x^2} \, dx+\left (48 d e f^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx+\frac{1}{245} \left (48 d g^2 p^3\right ) \int \frac{x^6}{d+e x^2} \, dx+\frac{1}{175} \left (48 d g^2 p^3\right ) \int \frac{x^6}{d+e x^2} \, dx+\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{x^2}{d+e x^2} \, dx}{49 e^2}+\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{x^2}{d+e x^2} \, dx}{35 e^2}+\frac{\left (16 d^3 g^2 p^3\right ) \int \frac{x^2}{d+e x^2} \, dx}{7 e^2}+\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{x^2}{d+e x^2} \, dx}{7 e^2}-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx}{49 e^2}-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx}{35 e^2}-\frac{\left (16 d^4 g^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx}{7 e^2}-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx}{7 e^2}-\frac{\left (16 d^2 g^2 p^3\right ) \int \frac{x^4}{d+e x^2} \, dx}{49 e}-\frac{\left (16 d^2 g^2 p^3\right ) \int \frac{x^4}{d+e x^2} \, dx}{35 e}-\frac{\left (16 d^2 g^2 p^3\right ) \int \frac{x^4}{d+e x^2} \, dx}{21 e}-\frac{1}{343} \left (48 e g^2 p^3\right ) \int \frac{x^8}{d+e x^2} \, dx\\ &=-48 f^2 p^3 x+\frac{2816 d^3 g^2 p^3 x}{245 e^3}+\frac{6 d f g p^3 x^2}{e}-\frac{3 f g p^3 \left (d+e x^2\right )^2}{8 e^2}+24 f^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{1408 d^3 g^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{245 e^3}+\frac{568 d^2 g^2 p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )}{735 e^2}-\frac{288 d g^2 p^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )}{1225 e}+\frac{24}{343} g^2 p^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{6 d f g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 f g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{24 \sqrt{d} f^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{245 e^{7/2}}-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\left (48 d f^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx+\left (48 \sqrt{d} \sqrt{e} f^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx+\frac{1}{245} \left (48 d g^2 p^3\right ) \int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx+\frac{1}{175} \left (48 d g^2 p^3\right ) \int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{49 e^3}-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{35 e^3}-\frac{\left (16 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{7 e^3}-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{7 e^3}-\frac{\left (48 d^{7/2} g^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{49 e^{5/2}}-\frac{\left (48 d^{7/2} g^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{35 e^{5/2}}-\frac{\left (16 d^{7/2} g^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{7 e^{5/2}}-\frac{\left (48 d^{7/2} g^2 p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{7 e^{5/2}}-\frac{\left (16 d^2 g^2 p^3\right ) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx}{49 e}-\frac{\left (16 d^2 g^2 p^3\right ) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx}{35 e}-\frac{\left (16 d^2 g^2 p^3\right ) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx}{21 e}-\frac{1}{343} \left (48 e g^2 p^3\right ) \int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-48 f^2 p^3 x+\frac{351136 d^3 g^2 p^3 x}{25725 e^3}+\frac{6 d f g p^3 x^2}{e}-\frac{55456 d^2 g^2 p^3 x^3}{77175 e^2}+\frac{5232 d g^2 p^3 x^5}{42875 e}-\frac{48 g^2 p^3 x^7}{2401}-\frac{3 f g p^3 \left (d+e x^2\right )^2}{8 e^2}+\frac{48 \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{2816 d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{245 e^{7/2}}-\frac{24 i \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{1408 i d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{245 e^{7/2}}+24 f^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{1408 d^3 g^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{245 e^3}+\frac{568 d^2 g^2 p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )}{735 e^2}-\frac{288 d g^2 p^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )}{1225 e}+\frac{24}{343} g^2 p^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{6 d f g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 f g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{24 \sqrt{d} f^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{245 e^{7/2}}-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}-\left (48 f^2 p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx+\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{49 e^3}+\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{35 e^3}+\frac{\left (16 d^3 g^2 p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{7 e^3}+\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{7 e^3}-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{343 e^3}-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{245 e^3}-\frac{\left (48 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{175 e^3}-\frac{\left (16 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{49 e^3}-\frac{\left (16 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{35 e^3}-\frac{\left (16 d^4 g^2 p^3\right ) \int \frac{1}{d+e x^2} \, dx}{21 e^3}\\ &=-48 f^2 p^3 x+\frac{351136 d^3 g^2 p^3 x}{25725 e^3}+\frac{6 d f g p^3 x^2}{e}-\frac{55456 d^2 g^2 p^3 x^3}{77175 e^2}+\frac{5232 d g^2 p^3 x^5}{42875 e}-\frac{48 g^2 p^3 x^7}{2401}-\frac{3 f g p^3 \left (d+e x^2\right )^2}{8 e^2}+\frac{48 \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{351136 d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{25725 e^{7/2}}-\frac{24 i \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{1408 i d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{245 e^{7/2}}-\frac{48 \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{2816 d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{245 e^{7/2}}+24 f^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{1408 d^3 g^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{245 e^3}+\frac{568 d^2 g^2 p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )}{735 e^2}-\frac{288 d g^2 p^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )}{1225 e}+\frac{24}{343} g^2 p^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{6 d f g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 f g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{24 \sqrt{d} f^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{245 e^{7/2}}-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}+\left (48 f^2 p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx-\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx}{49 e^3}-\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx}{35 e^3}-\frac{\left (16 d^3 g^2 p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx}{7 e^3}-\frac{\left (48 d^3 g^2 p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx}{7 e^3}\\ &=-48 f^2 p^3 x+\frac{351136 d^3 g^2 p^3 x}{25725 e^3}+\frac{6 d f g p^3 x^2}{e}-\frac{55456 d^2 g^2 p^3 x^3}{77175 e^2}+\frac{5232 d g^2 p^3 x^5}{42875 e}-\frac{48 g^2 p^3 x^7}{2401}-\frac{3 f g p^3 \left (d+e x^2\right )^2}{8 e^2}+\frac{48 \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{351136 d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{25725 e^{7/2}}-\frac{24 i \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{1408 i d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{245 e^{7/2}}-\frac{48 \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{2816 d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{245 e^{7/2}}+24 f^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{1408 d^3 g^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{245 e^3}+\frac{568 d^2 g^2 p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )}{735 e^2}-\frac{288 d g^2 p^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )}{1225 e}+\frac{24}{343} g^2 p^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{6 d f g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 f g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{24 \sqrt{d} f^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{245 e^{7/2}}-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}-\frac{\left (48 i \sqrt{d} f^2 p^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{\sqrt{e}}+\frac{\left (48 i d^{7/2} g^2 p^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{49 e^{7/2}}+\frac{\left (48 i d^{7/2} g^2 p^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{35 e^{7/2}}+\frac{\left (16 i d^{7/2} g^2 p^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{7 e^{7/2}}+\frac{\left (48 i d^{7/2} g^2 p^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{7 e^{7/2}}\\ &=-48 f^2 p^3 x+\frac{351136 d^3 g^2 p^3 x}{25725 e^3}+\frac{6 d f g p^3 x^2}{e}-\frac{55456 d^2 g^2 p^3 x^3}{77175 e^2}+\frac{5232 d g^2 p^3 x^5}{42875 e}-\frac{48 g^2 p^3 x^7}{2401}-\frac{3 f g p^3 \left (d+e x^2\right )^2}{8 e^2}+\frac{48 \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{351136 d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{25725 e^{7/2}}-\frac{24 i \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{1408 i d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{245 e^{7/2}}-\frac{48 \sqrt{d} f^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{2816 d^{7/2} g^2 p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{245 e^{7/2}}+24 f^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{1408 d^3 g^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{245 e^3}+\frac{568 d^2 g^2 p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )}{735 e^2}-\frac{288 d g^2 p^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )}{1225 e}+\frac{24}{343} g^2 p^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac{6 d f g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{3 f g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{24 \sqrt{d} f^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{245 e^{7/2}}-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{24 i \sqrt{d} f^2 p^3 \text{Li}_2\left (1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{1408 i d^{7/2} g^2 p^3 \text{Li}_2\left (1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{245 e^{7/2}}+\left (6 d f^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (6 d^4 g^2 p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}\\ \end{align*}

Mathematica [A]  time = 9.21294, size = 2539, normalized size = 2.26 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(f + g*x^3)^2*Log[c*(d + e*x^2)^p]^3,x]

[Out]

(f*g*p^3*(d + e*x^2)*(45*d - 3*e*x^2 + (-42*d + 6*e*x^2)*Log[d + e*x^2] + 6*(3*d - e*x^2)*Log[d + e*x^2]^2 - 4
*(d - e*x^2)*Log[d + e*x^2]^3))/(8*e^2) + (g^2*p^3*x*(-280*d^3*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1
+ (e*x^2)/d] - 280*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d] - 112*d^3*Hypergeom
etricPFQ[{-5/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^2)/d] - 112*d^2*e*x^2*HypergeometricPFQ[{-5/2, 1, 1, 1, 1}
, {2, 2, 2, 2}, 1 + (e*x^2)/d] + 280*d^3*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^2)/d] +
280*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^2)/d] - 210*d^3*HypergeometricPFQ[{
-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^2)/d] - 210*d^2*e*x^2*HypergeometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2
, 2}, 1 + (e*x^2)/d] + 16*d^3*Log[d + e*x^2] + 16*e^3*x^6*Sqrt[-((e*x^2)/d)]*Log[d + e*x^2] + 280*d^3*Hypergeo
metricPFQ[{-3/2, 1, 1}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + 280*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1},
{2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] - 280*d^3*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d]*L
og[d + e*x^2] - 280*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + 21
0*d^3*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] + 210*d^2*e*x^2*Hypergeometr
icPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2] - 32*d^3*Log[d + e*x^2]^2 + 28*d*e^2*x^4*Sqrt[
-((e*x^2)/d)]*Log[d + e*x^2]^2 - 4*e^3*x^6*Sqrt[-((e*x^2)/d)]*Log[d + e*x^2]^2 + 140*d^3*HypergeometricPFQ[{-3
/2, 1, 1}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2]^2 + 140*d^2*e*x^2*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, 1 +
(e*x^2)/d]*Log[d + e*x^2]^2 - 105*d^3*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2]^2
- 105*d^2*e*x^2*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + (e*x^2)/d]*Log[d + e*x^2]^2 + 10*d^3*Log[d + e*x^
2]^3 + 10*e^3*x^6*Sqrt[-((e*x^2)/d)]*Log[d + e*x^2]^3 + 56*d^2*(d + e*x^2)*HypergeometricPFQ[{-5/2, 1, 1, 1},
{2, 2, 2}, 1 + (e*x^2)/d]*(3 + 2*Log[d + e*x^2]) - 56*d^2*(d + e*x^2)*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2},
1 + (e*x^2)/d]*(1 + 3*Log[d + e*x^2] + Log[d + e*x^2]^2)))/(70*e^3*Sqrt[-((e*x^2)/d)]) - (3*f*g*p^2*(e*x^2*(-6
*d + e*x^2) + (6*d^2 + 4*d*e*x^2 - 2*e^2*x^4)*Log[d + e*x^2] - 2*(d^2 - e^2*x^4)*Log[d + e*x^2]^2)*(p*Log[d +
e*x^2] - Log[c*(d + e*x^2)^p]))/(4*e^2) + (3*d*f*g*p*x^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(2*e)
- (2*d^2*g^2*p*x^3*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(7*e^2) + (6*d*g^2*p*x^5*(-(p*Log[d + e*x^
2]) + Log[c*(d + e*x^2)^p])^2)/(35*e) - (6*Sqrt[d]*(-7*e^3*f^2 + d^3*g^2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*L
og[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(7*e^(7/2)) - (3*d^2*f*g*p*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log
[c*(d + e*x^2)^p])^2)/(2*e^2) + (3*p*x*(14*f^2 + 7*f*g*x^3 + 2*g^2*x^6)*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) +
Log[c*(d + e*x^2)^p])^2)/14 - (g^2*x^7*(6*p + 7*p*Log[d + e*x^2] - 7*Log[c*(d + e*x^2)^p])*(-(p*Log[d + e*x^2]
) + Log[c*(d + e*x^2)^p])^2)/49 - (f*g*x^4*(3*p + 2*p*Log[d + e*x^2] - 2*Log[c*(d + e*x^2)^p])*(-(p*Log[d + e*
x^2]) + Log[c*(d + e*x^2)^p])^2)/4 + (x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-42*e^3*f^2*p + 6*d^3*
g^2*p + 7*e^3*f^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/(7*e^3) - (3*f^2*p^2*(p*Log[d + e*x^2] - Log[
c*(d + e*x^2)^p])*((4*I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + 4*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 2
*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[d + e*x^2]) + Sqrt[e]*x*(8 - 4*Log[d + e*x^2] + Log[d + e*x^2]
^2) + (4*I)*Sqrt[d]*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)]))/Sqrt[e] + 3*g^2*p^2*(-(p*
Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*((x^7*Log[d + e*x^2]^2)/7 - (4*((11025*I)*d^(7/2)*ArcTan[(Sqrt[e]*x)/S
qrt[d]]^2 + 105*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-352 + 210*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + 105
*Log[d + e*x^2]) + Sqrt[e]*x*(36960*d^3 - 4970*d^2*e*x^2 + 1512*d*e^2*x^4 - 450*e^3*x^6 - 105*(105*d^3 - 35*d^
2*e*x^2 + 21*d*e^2*x^4 - 15*e^3*x^6)*Log[d + e*x^2]) + (11025*I)*d^(7/2)*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((
-I)*Sqrt[d] + Sqrt[e]*x)]))/(77175*e^(7/2))) + (f^2*p^3*(-48*Sqrt[-d^2]*Sqrt[(e*x^2)/(d + e*x^2)]*Sqrt[d + e*x
^2]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] + Sqrt[-d]*e*x^2*(-48 + 24*Log[d + e*x^2] - 6*Log[d + e*x^2]^2 + Log[d + e
*x^2]^3) - 6*Sqrt[-d^2]*Sqrt[(e*x^2)/(d + e*x^2)]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2
, 3/2}, d/(d + e*x^2)] + Log[d + e*x^2]*(4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^2
)] + Sqrt[d + e*x^2]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2])) + 24*d*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/S
qrt[-d]]*(Log[d + e*x^2] - Log[1 + (e*x^2)/d]) + 6*(-d)^(3/2)*Sqrt[-((e*x^2)/d)]*(Log[1 + (e*x^2)/d]^2 - 4*Log
[1 + (e*x^2)/d]*Log[(1 + Sqrt[-((e*x^2)/d)])/2] + 2*Log[(1 + Sqrt[-((e*x^2)/d)])/2]^2 - 4*PolyLog[2, 1/2 - Sqr
t[-((e*x^2)/d)]/2])))/(Sqrt[-d]*e*x)

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Maple [A]  time = 105., size = 0, normalized size = 0. \begin{align*} \int \left ( g{x}^{3}+f \right ) ^{2} \left ( \ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^3+f)^2*ln(c*(e*x^2+d)^p)^3,x)

[Out]

int((g*x^3+f)^2*ln(c*(e*x^2+d)^p)^3,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)^2*log(c*(e*x^2+d)^p)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (g^{2} x^{6} + 2 \, f g x^{3} + f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)^2*log(c*(e*x^2+d)^p)^3,x, algorithm="fricas")

[Out]

integral((g^2*x^6 + 2*f*g*x^3 + f^2)*log((e*x^2 + d)^p*c)^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**3+f)**2*ln(c*(e*x**2+d)**p)**3,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)^2*log(c*(e*x^2+d)^p)^3,x, algorithm="giac")

[Out]

integrate((g*x^3 + f)^2*log((e*x^2 + d)^p*c)^3, x)