3.467 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^7 (d+e x)} \, dx\)

Optimal. Leaf size=386 \[ -\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}+\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 x^5}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6} \]

[Out]

-((c*d^2 - a*e^2)^3*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2])/(512*a^3*d^4*e^3*x^2) + ((c*d^2 - a*e^2)*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*a^2*d^3*e^2*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(6*d*x^
6) - (((5*c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*x^5) + ((c*d^2 - a*e^2)^5*(
5*c*d^2 + 7*a*e^2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2])])/(1024*a^(7/2)*d^(9/2)*e^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.49468, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {849, 834, 806, 720, 724, 206} \[ -\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}+\frac{\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 x^5}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)),x]

[Out]

-((c*d^2 - a*e^2)^3*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2])/(512*a^3*d^4*e^3*x^2) + ((c*d^2 - a*e^2)*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*a^2*d^3*e^2*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(6*d*x^
6) - (((5*c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*x^5) + ((c*d^2 - a*e^2)^5*(
5*c*d^2 + 7*a*e^2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2])])/(1024*a^(7/2)*d^(9/2)*e^(7/2))

Rule 849

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx &=\int \frac{(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7} \, dx\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac{\int \frac{\left (-\frac{1}{2} a e \left (5 c d^2-7 a e^2\right )+a c d e^2 x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6} \, dx}{6 a d e}\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}-\frac{\left (\frac{5 c^2 d^2}{a}+2 c e^2-\frac{7 a e^4}{d^2}\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5} \, dx}{24 e}\\ &=\frac{\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac{\left (\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right )\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{128 a^2 d^3 e^2}\\ &=-\frac{\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac{\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}-\frac{\left (\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right )\right ) \int \frac{1}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 a^3 d^4 e^3}\\ &=-\frac{\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac{\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac{\left (\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a d e-x^2} \, dx,x,\frac{2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 a^3 d^4 e^3}\\ &=-\frac{\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac{\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac{\left (\frac{5 c}{a e}-\frac{7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac{\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right ) \tanh ^{-1}\left (\frac{2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.02725, size = 344, normalized size = 0.89 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (-\frac{\left (7 a e^2+5 c d^2\right ) \left (5 x \left (c d^2-a e^2\right ) \left (\frac{x \left (c d^2-a e^2\right ) \left (\frac{x \left (a e^2-c d^2\right ) \left (3 x^2 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )+\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a e (2 d+5 e x)-3 c d^2 x\right )\right )}{a^{5/2} \sqrt{d} e^{5/2}}-8 (d+e x)^{5/2} \sqrt{a e+c d x}\right )}{d}-16 (d+e x)^{5/2} (a e+c d x)^{3/2}\right )-128 d (d+e x)^{5/2} (a e+c d x)^{5/2}\right )}{1280 d^2 x^5 (d+e x)^{3/2} (a e+c d x)^{3/2}}-\frac{(d+e x) (a e+c d x)^2}{x^6}\right )}{6 a d e} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*(-(((a*e + c*d*x)^2*(d + e*x))/x^6) - ((5*c*d^2 + 7*a*e^2)*(-128*d*(a*e + c*d
*x)^(5/2)*(d + e*x)^(5/2) + 5*(c*d^2 - a*e^2)*x*(-16*(a*e + c*d*x)^(3/2)*(d + e*x)^(5/2) + ((c*d^2 - a*e^2)*x*
(-8*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2) + ((-(c*d^2) + a*e^2)*x*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[
d + e*x]*(-3*c*d^2*x + a*e*(2*d + 5*e*x)) + 3*(c*d^2 - a*e^2)^2*x^2*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[
a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(5/2)*Sqrt[d]*e^(5/2))))/d)))/(1280*d^2*x^5*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2
))))/(6*a*d*e)

________________________________________________________________________________________

Maple [B]  time = 0.121, size = 4735, normalized size = 12.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x)

[Out]

-1/5*e^6/d^7*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(5/2)-101/512*e^6/d^7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
5/2)-59/320/d/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3-35/384*e^5/d^4*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(3/2)+25/512*e^4/d*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+7/1536*e^7/d^6*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)+7/512*e^8/d^5*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/16*e^5/d^4*c*(c*d*e*(d/e+x)^2+(a*e^2-
c*d^2)*(d/e+x))^(3/2)-3/128*e^4/d*c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+17/60/d^3/a/x^5*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+1/512*d^3/a^2*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-2681/7680*e^2/d^3/a^2*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2-1543/3840*e^3/d^6/a/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-35
/1536*e^3/d^2/a*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1017/2560*e^4/d^7/a/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(7/2)-1/64*e^2*d/a*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-397/960*e^4/d^5/a*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(5/2)*c+377/960*e^2/d^5/a/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+5/256*e^3*d^2/(a*d*e)^(1/2)*l
n((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^3-5/256*e^3/a*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3-45/1024*e^5*a/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^2+381/1280/d^3/a^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^
2+9/64*e^9/d^6*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/128*e^12/d^9*a^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)+15/128*e^10/d^5*a^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2))/(d*e*c)^(1/2)+15/512*e^5/d^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/256*e^4*d*c^3*ln((1/2
*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-185/1536*e^6/d^
5*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-7/1024*e^9/d^4*a^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(
a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)+3/64*e^10/d^7*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)-5/64*e^6/d^3*a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*e^8/d^7*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(3/2)*x+1/16*e^9/d^8*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-57/160*e/d^4/a/x^4*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(7/2)-221/7680/e^2*d/a^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^4+1/120/e^4*d^3/a^5*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^5+49/1536/e*d^2/a^3*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+7/384/e^3*d^4/a
^4*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/64/e^2*d^5/a^3*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+89
/320/d^3/a^2/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c+35/768*d/a^3*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(3/2)*x-81/1280/e/a^4*c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x-11/480/e^3/a^4/x^2*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(7/2)*c^3-1/16*e^9/d^8*a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)-9/64*e^9/d^6*a^2*(c*d*e*
(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+3/128*e^12/d^9*a^4/c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+
3/64*e^6/d^3*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-15/128*e^10/d^5*a^3*ln((1/2*a*e^2-1/2*c*d^2+(d/
e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-3/64*e^5/d^2*c^2*(c*d*e
*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-1/8*e^8/d^7*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x-3/64*e
^10/d^7*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+1/8*e^6/d^5*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+
x))^(3/2)*x+3/256*e^4*d*c^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2
)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-5/1536*d^6/a^5/e^5*c^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-5/512*d^7/a^4/e
^4*c^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/6/d^2/a/e/x^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-1/512*d
^5/a^6/e^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^6-1/32/a^3/e^3/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2
)*c^2-65/512*e^7/d^4*a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+3/256*e^14/d^9*a^5/c^2*ln((1/2*a*e^2+1/2*c*
d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-15/128*e^8/d^3*a^2*c*ln((1/2
*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-89/7680/e^4*d/a
^5/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^4+43/1536/e^2*d^3/a^4*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)*x-113/640/e/d^2/a^3/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^2+7/256*e*d^2/a^2*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)*x*c^4+3/512/e*d^4/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^5+15/1024*e*d^4/a/(a*d*e)^(
1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^4-9/512/e*d^6/a
^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^5+2
9/320/e^2/d/a^3/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^2+89/7680/e^3*d^2/a^5*c^5*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(5/2)*x+81/1280/e^2/d/a^4/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^3-43/240/e/d^2/a^2/x^4*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c+15/512*e^7/d^2*a^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1
/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c-11/30*e/d^4/a^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*
c-65/1536*e^4/d^3/a*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1017/2560*e^5/d^6/a*c*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(5/2)*x-3211/7680*e^3/d^4/a^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x-381/1280*e/d^2/a^3*c^3
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+3211/7680*e^2/d^5/a^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c-2
5/768*e^2/d/a^2*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+1/12/d/a^2/e^2/x^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(7/2)*c+1/512*d^3/a^6/e^6/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^5-5/1536*d^5/a^5/e^4*c^6*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-5/512*d^6/a^4/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^6+5/1024*d^8/a^3
/e^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^6
+1/192*d/a^4/e^4/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^3+1/768*d^2/a^5/e^5/x^2*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(7/2)*c^4-1/512*d^4/a^6/e^5*c^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+15/256*e^12/d^7*a^4/c*ln(
(1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)
+3/64*e^11/d^8*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+9/64*e^7/d^4*a*c*(c*d*e*(d/e+x)^2+(a*e^2-
c*d^2)*(d/e+x))^(1/2)*x-3/256*e^14/d^9*a^5/c^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/
e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+15/128*e^8/d^3*a^2*c*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)
/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-15/256*e^6/d*a*c^2*ln((1/2*a*e^2-1
/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+15/256*e^6/
d*a*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-
15/256*e^12/d^7*a^4/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/
(d*e*c)^(1/2)-3/64*e^11/d^8*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )} x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^7), x)

________________________________________________________________________________________

Fricas [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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**7/(e*x+d),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x, algorithm="giac")

[Out]

Timed out