3.2068 \(\int \frac{1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=457 \[ \frac{1155 c^4 d^4 e \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1155 c^4 d^4 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 \left (c d^2-a e^2\right )^{13/2}}-\frac{385 c^3 d^3 e}{64 \sqrt{d+e x} \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{33 c^2 d^2}{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{11 c d}{24 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

1/(4*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2)) + (11*c*d)/(24*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2)) + (33*c^2*d^2)/(32*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (77*c^3*d^3*Sqrt[d + e*x])/(32*(c*d^2 -
a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (385*c^3*d^3*e)/(64*(c
*d^2 - a*e^2)^5*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (11
55*c^4*d^4*e*Sqrt[d + e*x])/(64*(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2]) + (1155*c^4*d^4*e^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(64*(c*d^2 - a*e^2)^(1
3/2))

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Rubi [A]  time = 1.32216, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{1155 c^4 d^4 e \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1155 c^4 d^4 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 \left (c d^2-a e^2\right )^{13/2}}-\frac{385 c^3 d^3 e}{64 \sqrt{d+e x} \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{33 c^2 d^2}{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{11 c d}{24 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

1/(4*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2)) + (11*c*d)/(24*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2)) + (33*c^2*d^2)/(32*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (77*c^3*d^3*Sqrt[d + e*x])/(32*(c*d^2 -
a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (385*c^3*d^3*e)/(64*(c
*d^2 - a*e^2)^5*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (11
55*c^4*d^4*e*Sqrt[d + e*x])/(64*(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2]) + (1155*c^4*d^4*e^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(64*(c*d^2 - a*e^2)^(1
3/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 2.28806, size = 286, normalized size = 0.63 \[ \frac{(d+e x)^{5/2} \left (\frac{(a e+c d x)^3 \left (\frac{1920 c^4 d^4 e}{a e+c d x}+\frac{128 c^4 d^4 \left (a e^2-c d^2\right )}{(a e+c d x)^2}+\frac{518 c^2 d^2 e^2 \left (c d^2-a e^2\right )}{(d+e x)^2}+\frac{184 c d \left (c d^2 e-a e^3\right )^2}{(d+e x)^3}-\frac{48 e^2 \left (a e^2-c d^2\right )^3}{(d+e x)^4}+\frac{1545 c^3 d^3 e^2}{d+e x}\right )}{3 \left (c d^2-a e^2\right )^6}-\frac{1155 c^4 d^4 e^{3/2} (a e+c d x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\left (a e^2-c d^2\right )^{13/2}}\right )}{64 ((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

((d + e*x)^(5/2)*(((a*e + c*d*x)^3*((128*c^4*d^4*(-(c*d^2) + a*e^2))/(a*e + c*d*
x)^2 + (1920*c^4*d^4*e)/(a*e + c*d*x) - (48*e^2*(-(c*d^2) + a*e^2)^3)/(d + e*x)^
4 + (184*c*d*(c*d^2*e - a*e^3)^2)/(d + e*x)^3 + (518*c^2*d^2*e^2*(c*d^2 - a*e^2)
)/(d + e*x)^2 + (1545*c^3*d^3*e^2)/(d + e*x)))/(3*(c*d^2 - a*e^2)^6) - (1155*c^4
*d^4*e^(3/2)*(a*e + c*d*x)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^
2) + a*e^2]])/(-(c*d^2) + a*e^2)^(13/2)))/(64*((a*e + c*d*x)*(d + e*x))^(5/2))

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Maple [B]  time = 0.055, size = 1225, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-1/192*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3465*arctanh(e*(c*d*x+a*e)^(1/2)
/((a*e^2-c*d^2)*e)^(1/2))*x^5*c^5*d^5*e^6*(c*d*x+a*e)^(1/2)+13860*arctanh(e*(c*d
*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^4*c^5*d^6*e^5*(c*d*x+a*e)^(1/2)+20790*a
rctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^3*c^5*d^7*e^4*(c*d*x+a*e)^
(1/2)+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^2*c^5*d^8*e^3
*(c*d*x+a*e)^(1/2)+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*c
^5*d^9*e^2*(c*d*x+a*e)^(1/2)+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^
(1/2))*a*c^4*d^8*e^3*(c*d*x+a*e)^(1/2)-4620*((a*e^2-c*d^2)*e)^(1/2)*x^4*a*c^4*d^
4*e^6-17094*((a*e^2-c*d^2)*e)^(1/2)*x^3*a*c^4*d^5*e^5-22968*((a*e^2-c*d^2)*e)^(1
/2)*x^2*a*c^4*d^6*e^4-12782*((a*e^2-c*d^2)*e)^(1/2)*x*a*c^4*d^7*e^3-693*((a*e^2-
c*d^2)*e)^(1/2)*x^3*a^2*c^3*d^3*e^7+198*((a*e^2-c*d^2)*e)^(1/2)*x^2*a^3*c^2*d^2*
e^8-2673*((a*e^2-c*d^2)*e)^(1/2)*x^2*a^2*c^3*d^4*e^6-88*((a*e^2-c*d^2)*e)^(1/2)*
x*a^4*c*d*e^9+748*((a*e^2-c*d^2)*e)^(1/2)*x*a^3*c^2*d^3*e^7-3795*((a*e^2-c*d^2)*
e)^(1/2)*x*a^2*c^3*d^5*e^5+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(
1/2))*x^3*a*c^4*d^5*e^6*(c*d*x+a*e)^(1/2)+20790*arctanh(e*(c*d*x+a*e)^(1/2)/((a*
e^2-c*d^2)*e)^(1/2))*x^2*a*c^4*d^6*e^5*(c*d*x+a*e)^(1/2)+13860*arctanh(e*(c*d*x+
a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*a*c^4*d^7*e^4*(c*d*x+a*e)^(1/2)+3465*arcta
nh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^4*a*c^4*d^4*e^7*(c*d*x+a*e)^(1
/2)-9207*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^5*d^8*e^2-1408*((a*e^2-c*d^2)*e)^(1/2)*x*
c^5*d^9*e-3465*((a*e^2-c*d^2)*e)^(1/2)*x^5*c^5*d^5*e^5-12705*((a*e^2-c*d^2)*e)^(
1/2)*x^4*c^5*d^6*e^4-16863*((a*e^2-c*d^2)*e)^(1/2)*x^3*c^5*d^7*e^3-2048*((a*e^2-
c*d^2)*e)^(1/2)*a*c^4*d^8*e^2-328*((a*e^2-c*d^2)*e)^(1/2)*a^4*c*d^2*e^8+1030*((a
*e^2-c*d^2)*e)^(1/2)*a^3*c^2*d^4*e^6-2295*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^3*d^6*e^
4+128*((a*e^2-c*d^2)*e)^(1/2)*c^5*d^10+48*((a*e^2-c*d^2)*e)^(1/2)*a^5*e^10)/(e*x
+d)^(9/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^6/((a*e^2-c*d^2)*e)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264173, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/384*(3465*(c^6*d^6*e^6*x^7 + a^2*c^4*d^9*e^3 + (5*c^6*d^7*e^5 + 2*a*c^5*d^5*e
^7)*x^6 + (10*c^6*d^8*e^4 + 10*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^5 + 5*(2*c^6*d
^9*e^3 + 4*a*c^5*d^7*e^5 + a^2*c^4*d^5*e^7)*x^4 + 5*(c^6*d^10*e^2 + 4*a*c^5*d^8*
e^4 + 2*a^2*c^4*d^6*e^6)*x^3 + (c^6*d^11*e + 10*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e
^5)*x^2 + (2*a*c^5*d^10*e^2 + 5*a^2*c^4*d^8*e^4)*x)*sqrt(-e/(c*d^2 - a*e^2))*log
(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d
^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2
+ 2*d*e*x + d^2)) + 2*(3465*c^5*d^5*e^5*x^5 - 128*c^5*d^10 + 2048*a*c^4*d^8*e^2
+ 2295*a^2*c^3*d^6*e^4 - 1030*a^3*c^2*d^4*e^6 + 328*a^4*c*d^2*e^8 - 48*a^5*e^10
+ 1155*(11*c^5*d^6*e^4 + 4*a*c^4*d^4*e^6)*x^4 + 231*(73*c^5*d^7*e^3 + 74*a*c^4*d
^5*e^5 + 3*a^2*c^3*d^3*e^7)*x^3 + 99*(93*c^5*d^8*e^2 + 232*a*c^4*d^6*e^4 + 27*a^
2*c^3*d^4*e^6 - 2*a^3*c^2*d^2*e^8)*x^2 + 11*(128*c^5*d^9*e + 1162*a*c^4*d^7*e^3
+ 345*a^2*c^3*d^5*e^5 - 68*a^3*c^2*d^3*e^7 + 8*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^6*d^17*e^2 - 6*a^3*c^5*d^15*e^4
+ 15*a^4*c^4*d^13*e^6 - 20*a^5*c^3*d^11*e^8 + 15*a^6*c^2*d^9*e^10 - 6*a^7*c*d^7
*e^12 + a^8*d^5*e^14 + (c^8*d^14*e^5 - 6*a*c^7*d^12*e^7 + 15*a^2*c^6*d^10*e^9 -
20*a^3*c^5*d^8*e^11 + 15*a^4*c^4*d^6*e^13 - 6*a^5*c^3*d^4*e^15 + a^6*c^2*d^2*e^1
7)*x^7 + (5*c^8*d^15*e^4 - 28*a*c^7*d^13*e^6 + 63*a^2*c^6*d^11*e^8 - 70*a^3*c^5*
d^9*e^10 + 35*a^4*c^4*d^7*e^12 - 7*a^6*c^2*d^3*e^16 + 2*a^7*c*d*e^18)*x^6 + (10*
c^8*d^16*e^3 - 50*a*c^7*d^14*e^5 + 91*a^2*c^6*d^12*e^7 - 56*a^3*c^5*d^10*e^9 - 3
5*a^4*c^4*d^8*e^11 + 70*a^5*c^3*d^6*e^13 - 35*a^6*c^2*d^4*e^15 + 4*a^7*c*d^2*e^1
7 + a^8*e^19)*x^5 + 5*(2*c^8*d^17*e^2 - 8*a*c^7*d^15*e^4 + 7*a^2*c^6*d^13*e^6 +
14*a^3*c^5*d^11*e^8 - 35*a^4*c^4*d^9*e^10 + 28*a^5*c^3*d^7*e^12 - 7*a^6*c^2*d^5*
e^14 - 2*a^7*c*d^3*e^16 + a^8*d*e^18)*x^4 + 5*(c^8*d^18*e - 2*a*c^7*d^16*e^3 - 7
*a^2*c^6*d^14*e^5 + 28*a^3*c^5*d^12*e^7 - 35*a^4*c^4*d^10*e^9 + 14*a^5*c^3*d^8*e
^11 + 7*a^6*c^2*d^6*e^13 - 8*a^7*c*d^4*e^15 + 2*a^8*d^2*e^17)*x^3 + (c^8*d^19 +
4*a*c^7*d^17*e^2 - 35*a^2*c^6*d^15*e^4 + 70*a^3*c^5*d^13*e^6 - 35*a^4*c^4*d^11*e
^8 - 56*a^5*c^3*d^9*e^10 + 91*a^6*c^2*d^7*e^12 - 50*a^7*c*d^5*e^14 + 10*a^8*d^3*
e^16)*x^2 + (2*a*c^7*d^18*e - 7*a^2*c^6*d^16*e^3 + 35*a^4*c^4*d^12*e^7 - 70*a^5*
c^3*d^10*e^9 + 63*a^6*c^2*d^8*e^11 - 28*a^7*c*d^6*e^13 + 5*a^8*d^4*e^15)*x), -1/
192*(3465*(c^6*d^6*e^6*x^7 + a^2*c^4*d^9*e^3 + (5*c^6*d^7*e^5 + 2*a*c^5*d^5*e^7)
*x^6 + (10*c^6*d^8*e^4 + 10*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^5 + 5*(2*c^6*d^9*
e^3 + 4*a*c^5*d^7*e^5 + a^2*c^4*d^5*e^7)*x^4 + 5*(c^6*d^10*e^2 + 4*a*c^5*d^8*e^4
+ 2*a^2*c^4*d^6*e^6)*x^3 + (c^6*d^11*e + 10*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5)
*x^2 + (2*a*c^5*d^10*e^2 + 5*a^2*c^4*d^8*e^4)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(
sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d^2 - a*e^2
)))) - (3465*c^5*d^5*e^5*x^5 - 128*c^5*d^10 + 2048*a*c^4*d^8*e^2 + 2295*a^2*c^3*
d^6*e^4 - 1030*a^3*c^2*d^4*e^6 + 328*a^4*c*d^2*e^8 - 48*a^5*e^10 + 1155*(11*c^5*
d^6*e^4 + 4*a*c^4*d^4*e^6)*x^4 + 231*(73*c^5*d^7*e^3 + 74*a*c^4*d^5*e^5 + 3*a^2*
c^3*d^3*e^7)*x^3 + 99*(93*c^5*d^8*e^2 + 232*a*c^4*d^6*e^4 + 27*a^2*c^3*d^4*e^6 -
2*a^3*c^2*d^2*e^8)*x^2 + 11*(128*c^5*d^9*e + 1162*a*c^4*d^7*e^3 + 345*a^2*c^3*d
^5*e^5 - 68*a^3*c^2*d^3*e^7 + 8*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*sqrt(e*x + d))/(a^2*c^6*d^17*e^2 - 6*a^3*c^5*d^15*e^4 + 15*a^4*c^4*d
^13*e^6 - 20*a^5*c^3*d^11*e^8 + 15*a^6*c^2*d^9*e^10 - 6*a^7*c*d^7*e^12 + a^8*d^5
*e^14 + (c^8*d^14*e^5 - 6*a*c^7*d^12*e^7 + 15*a^2*c^6*d^10*e^9 - 20*a^3*c^5*d^8*
e^11 + 15*a^4*c^4*d^6*e^13 - 6*a^5*c^3*d^4*e^15 + a^6*c^2*d^2*e^17)*x^7 + (5*c^8
*d^15*e^4 - 28*a*c^7*d^13*e^6 + 63*a^2*c^6*d^11*e^8 - 70*a^3*c^5*d^9*e^10 + 35*a
^4*c^4*d^7*e^12 - 7*a^6*c^2*d^3*e^16 + 2*a^7*c*d*e^18)*x^6 + (10*c^8*d^16*e^3 -
50*a*c^7*d^14*e^5 + 91*a^2*c^6*d^12*e^7 - 56*a^3*c^5*d^10*e^9 - 35*a^4*c^4*d^8*e
^11 + 70*a^5*c^3*d^6*e^13 - 35*a^6*c^2*d^4*e^15 + 4*a^7*c*d^2*e^17 + a^8*e^19)*x
^5 + 5*(2*c^8*d^17*e^2 - 8*a*c^7*d^15*e^4 + 7*a^2*c^6*d^13*e^6 + 14*a^3*c^5*d^11
*e^8 - 35*a^4*c^4*d^9*e^10 + 28*a^5*c^3*d^7*e^12 - 7*a^6*c^2*d^5*e^14 - 2*a^7*c*
d^3*e^16 + a^8*d*e^18)*x^4 + 5*(c^8*d^18*e - 2*a*c^7*d^16*e^3 - 7*a^2*c^6*d^14*e
^5 + 28*a^3*c^5*d^12*e^7 - 35*a^4*c^4*d^10*e^9 + 14*a^5*c^3*d^8*e^11 + 7*a^6*c^2
*d^6*e^13 - 8*a^7*c*d^4*e^15 + 2*a^8*d^2*e^17)*x^3 + (c^8*d^19 + 4*a*c^7*d^17*e^
2 - 35*a^2*c^6*d^15*e^4 + 70*a^3*c^5*d^13*e^6 - 35*a^4*c^4*d^11*e^8 - 56*a^5*c^3
*d^9*e^10 + 91*a^6*c^2*d^7*e^12 - 50*a^7*c*d^5*e^14 + 10*a^8*d^3*e^16)*x^2 + (2*
a*c^7*d^18*e - 7*a^2*c^6*d^16*e^3 + 35*a^4*c^4*d^12*e^7 - 70*a^5*c^3*d^10*e^9 +
63*a^6*c^2*d^8*e^11 - 28*a^7*c*d^6*e^13 + 5*a^8*d^4*e^15)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, 2]