\(\int (f+g x^3)^2 \log ^3(c (d+e x^2)^p) \, dx\) [298]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\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 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {351136 d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{25725 e^{7/2}}-\frac {24 i \sqrt {d} f^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {1408 i d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{245 e^{7/2}}-\frac {48 \sqrt {d} f^2 p^3 \arctan \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 \arctan \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 \arctan \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 \arctan \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 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {1408 i d^{7/2} g^2 p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{245 e^{7/2}}+6 d f^2 p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )-\frac {6 d^4 g^2 p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{7 e^3} \]

[Out]

6*d*f*g*p^3*x^2/e-1408/245*d^3*g^2*p^2*x*ln(c*(e*x^2+d)^p)/e^3+568/735*d^2*g^2*p^2*x^3*ln(c*(e*x^2+d)^p)/e^2-2
88/1225*d*g^2*p^2*x^5*ln(c*(e*x^2+d)^p)/e+3/4*f*g*p^2*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)/e^2+1408/245*d^(7/2)*g^2*p
^2*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(7/2)+6/7*d^3*g^2*p*x*ln(c*(e*x^2+d)^p)^2/e^3-2/7*d^2*g^2*p*x
^3*ln(c*(e*x^2+d)^p)^2/e^2+6/35*d*g^2*p*x^5*ln(c*(e*x^2+d)^p)^2/e-3/4*f*g*p*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^2/e^
2+2816/245*d^(7/2)*g^2*p^3*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))/e^(7/2)-24*f^2*p^2*ar
ctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)*d^(1/2)/e^(1/2)-48*f^2*p^3*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d
^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)-24*I*f^2*p^3*arctan(x*e^(1/2)/d^(1/2))^2*d^(1/2)/e^(1/2)-24*I*f^2*p^3*pol
ylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)-d*f*g*(e*x^2+d)*ln(c*(e*x^2+d)^p)^3/e^2-48*f^2*p^3*x
-48/2401*g^2*p^3*x^7+1/7*g^2*x^7*ln(c*(e*x^2+d)^p)^3+1408/245*I*d^(7/2)*g^2*p^3*polylog(2,1-2*d^(1/2)/(d^(1/2)
+I*x*e^(1/2)))/e^(7/2)+1408/245*I*d^(7/2)*g^2*p^3*arctan(x*e^(1/2)/d^(1/2))^2/e^(7/2)+24*f^2*p^2*x*ln(c*(e*x^2
+d)^p)+24/343*g^2*p^2*x^7*ln(c*(e*x^2+d)^p)-6*f^2*p*x*ln(c*(e*x^2+d)^p)^2-6/49*g^2*p*x^7*ln(c*(e*x^2+d)^p)^2+6
*d*f^2*p*Unintegrable(ln(c*(e*x^2+d)^p)^2/(e*x^2+d),x)+f^2*x*ln(c*(e*x^2+d)^p)^3-6*d*f*g*p^2*(e*x^2+d)*ln(c*(e
*x^2+d)^p)/e^2+3*d*f*g*p*(e*x^2+d)*ln(c*(e*x^2+d)^p)^2/e^2-351136/25725*d^(7/2)*g^2*p^3*arctan(x*e^(1/2)/d^(1/
2))/e^(7/2)+1/2*f*g*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^3/e^2+48*f^2*p^3*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(1/2)-6
/7*d^4*g^2*p*Unintegrable(ln(c*(e*x^2+d)^p)^2/(e*x^2+d),x)/e^3+351136/25725*d^3*g^2*p^3*x/e^3-55456/77175*d^2*
g^2*p^3*x^3/e^2+5232/42875*d*g^2*p^3*x^5/e-3/8*f*g*p^3*(e*x^2+d)^2/e^2

Rubi [N/A]

Not integrable

Time = 1.66 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]

[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*} \text {integral}& = \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) \text {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) \text {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) \text {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) \text {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) \text {Subst}\left (\int x \log ^3\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\frac {(d f g) \text {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) \text {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) \text {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 ) \text {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 ) \text {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} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2385\) vs. \(2(1126)=2252\).

Time = 8.70 (sec) , antiderivative size = 2385, normalized size of antiderivative = 99.38 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Result too large to show} \]

[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)*(-8*d*(-6 + 6*Log[d + e*x^2] - 3*Log[d + e*x^2]^2 + Log[d + e*x^2]^3) + (d + e*x^2)*(-3 +
6*Log[d + e*x^2] - 6*Log[d + e*x^2]^2 + 4*Log[d + e*x^2]^3)))/(8*e^2) + 6*f*g*p^2*((x^4*Log[d + e*x^2]^2)/4 -
e*((3*d*x^2)/(4*e^2) - x^4/(8*e) - (3*d^2*Log[d + e*x^2])/(4*e^3) - (d*x^2*Log[d + e*x^2])/(2*e^2) + (x^4*Log
[d + e*x^2])/(4*e) + (d^2*Log[d + e*x^2]^2)/(4*e^3)))*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]) + (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) - (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 + (f*g*x^4*(-(p*Log[d + e*x^2]) + Log[c
*(d + e*x^2)^p])^2*(-3*p + 2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/4 + (g^2*x^7*(-(p*Log[d + e*x^2])
+ Log[c*(d + e*x^2)^p])^2*(-6*p + 7*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/49 + (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) - (6*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-7*d*e^3*f^2*p*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2
+ d^4*g^2*p*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(7*Sqrt[d]*e^(7/2)) + 3*f^2*p^2*(-(p*Log[d + e*x^
2]) + Log[c*(d + e*x^2)^p])*(x*Log[d + e*x^2]^2 - (4*((-I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*(
-2 + Log[d + e*x^2]) - 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]) - 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[(Sqr
t[e]*x)/Sqrt[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))) + (g^2*p^3*(702272*Sqrt[-d]*d^(7/2)*Sqrt[d + e*x^2]*Sqrt
[1 - d/(d + e*x^2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] + 44100*Sqrt[-d]*d^(7/2)*Sqrt[1 - d/(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)] + 4*Sqrt[d]*HypergeometricPFQ[{1/2,
1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^2)]*Log[d + e*x^2] + Sqrt[d + e*x^2]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d +
e*x^2]^2) - (2*Sqrt[-d]*e*x^2*(-1125*(d + e*x^2)^3*(-48 + 168*Log[d + e*x^2] - 294*Log[d + e*x^2]^2 + 343*Log
[d + e*x^2]^3) + 27*d*(d + e*x^2)^2*(-18208 + 44520*Log[d + e*x^2] - 53900*Log[d + e*x^2]^2 + 42875*Log[d + e*
x^2]^3) + d^3*(-39193856 + 18434640*Log[d + e*x^2] - 3880800*Log[d + e*x^2]^2 + 385875*Log[d + e*x^2]^3) - d^2
*(d + e*x^2)*(-2762192 + 3924480*Log[d + e*x^2] - 2690100*Log[d + e*x^2]^2 + 1157625*Log[d + e*x^2]^3)))/105 -
73920*d^4*(4*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] - Log[(d + e*x^2)/d]) - Sqrt[-d]*Sqrt[
1 - (d + e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*Log[(d + e*x^2)/d]*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2] + 2*Log[(
1 + Sqrt[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^2)/d]/2]))))/(51450*Sqrt[-d]*e^4*x) +
(f^2*p^3*(-48*Sqrt[-d^2]*Sqrt[d + e*x^2]*Sqrt[1 - d/(d + e*x^2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] - 6*Sqrt[-d^
2]*Sqrt[1 - d/(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)]
+ 4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^2)]*Log[d + e*x^2] + Sqrt[d + e*x^2]*Arc
Sin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2]^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) + 24*d*Sqrt[e*x^2]*ArcTanh[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] - Log[(d + e*x^2)/d]) + 6
*(-d)^(3/2)*Sqrt[1 - (d + e*x^2)/d]*(Log[(d + e*x^2)/d]^2 - 4*Log[(d + e*x^2)/d]*Log[(1 + Sqrt[1 - (d + e*x^2)
/d])/2] + 2*Log[(1 + Sqrt[1 - (d + e*x^2)/d])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (d + e*x^2)/d]/2])))/(Sqrt[-d
]*e*x)

Maple [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \left (g \,x^{3}+f \right )^{2} {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}^{3}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 32.81 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{3}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,{\left (g\,x^3+f\right )}^2 \,d x \]

[In]

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

[Out]

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