3.1663 \(\int \frac{1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=266 \[ \frac{3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}-\frac{3003 b e^5}{128 \sqrt{d+e x} (b d-a e)^7}-\frac{1001 e^5}{128 (d+e x)^{3/2} (b d-a e)^6}-\frac{3003 e^4}{640 (a+b x) (d+e x)^{3/2} (b d-a e)^5}+\frac{429 e^3}{320 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac{143 e^2}{240 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}+\frac{13 e}{40 (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]

[Out]

(-1001*e^5)/(128*(b*d - a*e)^6*(d + e*x)^(3/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(
d + e*x)^(3/2)) + (13*e)/(40*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)) - (143*e
^2)/(240*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) + (429*e^3)/(320*(b*d - a*e)
^4*(a + b*x)^2*(d + e*x)^(3/2)) - (3003*e^4)/(640*(b*d - a*e)^5*(a + b*x)*(d + e
*x)^(3/2)) - (3003*b*e^5)/(128*(b*d - a*e)^7*Sqrt[d + e*x]) + (3003*b^(3/2)*e^5*
ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(15/2))

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Rubi [A]  time = 0.699661, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}-\frac{3003 b e^5}{128 \sqrt{d+e x} (b d-a e)^7}-\frac{1001 e^5}{128 (d+e x)^{3/2} (b d-a e)^6}-\frac{3003 e^4}{640 (a+b x) (d+e x)^{3/2} (b d-a e)^5}+\frac{429 e^3}{320 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac{143 e^2}{240 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}+\frac{13 e}{40 (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-1001*e^5)/(128*(b*d - a*e)^6*(d + e*x)^(3/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(
d + e*x)^(3/2)) + (13*e)/(40*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)) - (143*e
^2)/(240*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) + (429*e^3)/(320*(b*d - a*e)
^4*(a + b*x)^2*(d + e*x)^(3/2)) - (3003*e^4)/(640*(b*d - a*e)^5*(a + b*x)*(d + e
*x)^(3/2)) - (3003*b*e^5)/(128*(b*d - a*e)^7*Sqrt[d + e*x]) + (3003*b^(3/2)*e^5*
ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(15/2))

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Rubi in Sympy [A]  time = 170.366, size = 264, normalized size = 0.99 \[ \frac{3003 b^{\frac{3}{2}} e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 \left (a e - b d\right )^{\frac{15}{2}}} + \frac{3003 b^{2} e^{4} \sqrt{d + e x}}{128 \left (a + b x\right ) \left (a e - b d\right )^{7}} + \frac{1001 b^{2} e^{3} \sqrt{d + e x}}{64 \left (a + b x\right )^{2} \left (a e - b d\right )^{6}} + \frac{1001 b^{2} e^{2} \sqrt{d + e x}}{80 \left (a + b x\right )^{3} \left (a e - b d\right )^{5}} + \frac{429 b^{2} e \sqrt{d + e x}}{40 \left (a + b x\right )^{4} \left (a e - b d\right )^{4}} + \frac{143 b^{2} \sqrt{d + e x}}{15 \left (a + b x\right )^{5} \left (a e - b d\right )^{3}} + \frac{26 b}{3 \left (a + b x\right )^{5} \sqrt{d + e x} \left (a e - b d\right )^{2}} - \frac{2}{3 \left (a + b x\right )^{5} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

3003*b**(3/2)*e**5*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*(a*e - b*d)*
*(15/2)) + 3003*b**2*e**4*sqrt(d + e*x)/(128*(a + b*x)*(a*e - b*d)**7) + 1001*b*
*2*e**3*sqrt(d + e*x)/(64*(a + b*x)**2*(a*e - b*d)**6) + 1001*b**2*e**2*sqrt(d +
e*x)/(80*(a + b*x)**3*(a*e - b*d)**5) + 429*b**2*e*sqrt(d + e*x)/(40*(a + b*x)*
*4*(a*e - b*d)**4) + 143*b**2*sqrt(d + e*x)/(15*(a + b*x)**5*(a*e - b*d)**3) + 2
6*b/(3*(a + b*x)**5*sqrt(d + e*x)*(a*e - b*d)**2) - 2/(3*(a + b*x)**5*(d + e*x)*
*(3/2)*(a*e - b*d))

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Mathematica [A]  time = 1.36763, size = 217, normalized size = 0.82 \[ \frac{\frac{45045 b^{3/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{15/2}}+\frac{\sqrt{d+e x} \left (\frac{8270 b^2 e^3 (b d-a e)}{(a+b x)^2}-\frac{3544 b^2 e^2 (b d-a e)^2}{(a+b x)^3}+\frac{1392 b^2 e (b d-a e)^3}{(a+b x)^4}-\frac{384 b^2 (b d-a e)^4}{(a+b x)^5}-\frac{22005 b^2 e^4}{a+b x}+\frac{1280 e^5 (a e-b d)}{(d+e x)^2}-\frac{23040 b e^5}{d+e x}\right )}{(b d-a e)^7}}{1920} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((Sqrt[d + e*x]*((-384*b^2*(b*d - a*e)^4)/(a + b*x)^5 + (1392*b^2*e*(b*d - a*e)^
3)/(a + b*x)^4 - (3544*b^2*e^2*(b*d - a*e)^2)/(a + b*x)^3 + (8270*b^2*e^3*(b*d -
a*e))/(a + b*x)^2 - (22005*b^2*e^4)/(a + b*x) + (1280*e^5*(-(b*d) + a*e))/(d +
e*x)^2 - (23040*b*e^5)/(d + e*x)))/(b*d - a*e)^7 + (45045*b^(3/2)*e^5*ArcTanh[(S
qrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(15/2))/1920

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Maple [B]  time = 0.044, size = 668, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2/3*e^5/(a*e-b*d)^6/(e*x+d)^(3/2)+12*e^5/(a*e-b*d)^7*b/(e*x+d)^(1/2)+1467/128*e
^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)+9629/192*e^6/(a*e-b*d)^7*b^5/(b*e
*x+a*e)^5*(e*x+d)^(7/2)*a-9629/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(7/
2)*d+1253/15*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^2-2506/15*e^6/(a*
e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d+1253/15*e^5/(a*e-b*d)^7*b^6/(b*e*x+
a*e)^5*(e*x+d)^(5/2)*d^2+12131/192*e^8/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/
2)*a^3-12131/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d+12131/64*e
^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^2-12131/192*e^5/(a*e-b*d)^7*b
^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^3+2373/128*e^9/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5*(e
*x+d)^(1/2)*a^4-2373/32*e^8/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d+71
19/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^2*d^2-2373/32*e^6/(a*e-b
*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^3+2373/128*e^5/(a*e-b*d)^7*b^6/(b*e*x+
a*e)^5*(e*x+d)^(1/2)*d^4+3003/128*e^5/(a*e-b*d)^7*b^2/(b*(a*e-b*d))^(1/2)*arctan
((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(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/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302465, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[-1/3840*(90090*b^6*e^6*x^6 + 768*b^6*d^6 - 5856*a*b^5*d^5*e + 20048*a^2*b^4*d^4
*e^2 - 42140*a^3*b^3*d^3*e^3 + 71190*a^4*b^2*d^2*e^4 + 48640*a^5*b*d*e^5 - 2560*
a^6*e^6 + 60060*(2*b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 6006*(3*b^6*d^2*e^4 + 94*a*b^5
*d*e^5 + 128*a^2*b^4*e^6)*x^4 - 1716*(3*b^6*d^3*e^3 - 51*a*b^5*d^2*e^4 - 607*a^2
*b^4*d*e^5 - 395*a^3*b^3*e^6)*x^3 + 286*(8*b^6*d^4*e^2 - 86*a*b^5*d^3*e^3 + 588*
a^2*b^4*d^2*e^4 + 3250*a^3*b^3*d*e^5 + 965*a^4*b^2*e^6)*x^2 + 45045*(b^6*e^6*x^6
+ a^5*b*d*e^5 + (b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 + 2*a^2*b^4*e^6)
*x^4 + 10*(a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 5*(2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*
x^2 + (5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((b*
e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a))
- 52*(24*b^6*d^5*e - 208*a*b^5*d^4*e^2 + 889*a^2*b^4*d^3*e^3 - 3045*a^3*b^3*d^2*
e^4 - 7415*a^4*b^2*d*e^5 - 640*a^5*b*e^6)*x)/((a^5*b^7*d^8 - 7*a^6*b^6*d^7*e + 2
1*a^7*b^5*d^6*e^2 - 35*a^8*b^4*d^5*e^3 + 35*a^9*b^3*d^4*e^4 - 21*a^10*b^2*d^3*e^
5 + 7*a^11*b*d^2*e^6 - a^12*d*e^7 + (b^12*d^7*e - 7*a*b^11*d^6*e^2 + 21*a^2*b^10
*d^5*e^3 - 35*a^3*b^9*d^4*e^4 + 35*a^4*b^8*d^3*e^5 - 21*a^5*b^7*d^2*e^6 + 7*a^6*
b^6*d*e^7 - a^7*b^5*e^8)*x^6 + (b^12*d^8 - 2*a*b^11*d^7*e - 14*a^2*b^10*d^6*e^2
+ 70*a^3*b^9*d^5*e^3 - 140*a^4*b^8*d^4*e^4 + 154*a^5*b^7*d^3*e^5 - 98*a^6*b^6*d^
2*e^6 + 34*a^7*b^5*d*e^7 - 5*a^8*b^4*e^8)*x^5 + 5*(a*b^11*d^8 - 5*a^2*b^10*d^7*e
+ 7*a^3*b^9*d^6*e^2 + 7*a^4*b^8*d^5*e^3 - 35*a^5*b^7*d^4*e^4 + 49*a^6*b^6*d^3*e
^5 - 35*a^7*b^5*d^2*e^6 + 13*a^8*b^4*d*e^7 - 2*a^9*b^3*e^8)*x^4 + 10*(a^2*b^10*d
^8 - 6*a^3*b^9*d^7*e + 14*a^4*b^8*d^6*e^2 - 14*a^5*b^7*d^5*e^3 + 14*a^7*b^5*d^3*
e^5 - 14*a^8*b^4*d^2*e^6 + 6*a^9*b^3*d*e^7 - a^10*b^2*e^8)*x^3 + 5*(2*a^3*b^9*d^
8 - 13*a^4*b^8*d^7*e + 35*a^5*b^7*d^6*e^2 - 49*a^6*b^6*d^5*e^3 + 35*a^7*b^5*d^4*
e^4 - 7*a^8*b^4*d^3*e^5 - 7*a^9*b^3*d^2*e^6 + 5*a^10*b^2*d*e^7 - a^11*b*e^8)*x^2
+ (5*a^4*b^8*d^8 - 34*a^5*b^7*d^7*e + 98*a^6*b^6*d^6*e^2 - 154*a^7*b^5*d^5*e^3
+ 140*a^8*b^4*d^4*e^4 - 70*a^9*b^3*d^3*e^5 + 14*a^10*b^2*d^2*e^6 + 2*a^11*b*d*e^
7 - a^12*e^8)*x)*sqrt(e*x + d)), -1/1920*(45045*b^6*e^6*x^6 + 384*b^6*d^6 - 2928
*a*b^5*d^5*e + 10024*a^2*b^4*d^4*e^2 - 21070*a^3*b^3*d^3*e^3 + 35595*a^4*b^2*d^2
*e^4 + 24320*a^5*b*d*e^5 - 1280*a^6*e^6 + 30030*(2*b^6*d*e^5 + 7*a*b^5*e^6)*x^5
+ 3003*(3*b^6*d^2*e^4 + 94*a*b^5*d*e^5 + 128*a^2*b^4*e^6)*x^4 - 858*(3*b^6*d^3*e
^3 - 51*a*b^5*d^2*e^4 - 607*a^2*b^4*d*e^5 - 395*a^3*b^3*e^6)*x^3 + 143*(8*b^6*d^
4*e^2 - 86*a*b^5*d^3*e^3 + 588*a^2*b^4*d^2*e^4 + 3250*a^3*b^3*d*e^5 + 965*a^4*b^
2*e^6)*x^2 - 45045*(b^6*e^6*x^6 + a^5*b*d*e^5 + (b^6*d*e^5 + 5*a*b^5*e^6)*x^5 +
5*(a*b^5*d*e^5 + 2*a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 5*(
2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + (5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*sqrt(e*x +
d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)
*b)) - 26*(24*b^6*d^5*e - 208*a*b^5*d^4*e^2 + 889*a^2*b^4*d^3*e^3 - 3045*a^3*b^3
*d^2*e^4 - 7415*a^4*b^2*d*e^5 - 640*a^5*b*e^6)*x)/((a^5*b^7*d^8 - 7*a^6*b^6*d^7*
e + 21*a^7*b^5*d^6*e^2 - 35*a^8*b^4*d^5*e^3 + 35*a^9*b^3*d^4*e^4 - 21*a^10*b^2*d
^3*e^5 + 7*a^11*b*d^2*e^6 - a^12*d*e^7 + (b^12*d^7*e - 7*a*b^11*d^6*e^2 + 21*a^2
*b^10*d^5*e^3 - 35*a^3*b^9*d^4*e^4 + 35*a^4*b^8*d^3*e^5 - 21*a^5*b^7*d^2*e^6 + 7
*a^6*b^6*d*e^7 - a^7*b^5*e^8)*x^6 + (b^12*d^8 - 2*a*b^11*d^7*e - 14*a^2*b^10*d^6
*e^2 + 70*a^3*b^9*d^5*e^3 - 140*a^4*b^8*d^4*e^4 + 154*a^5*b^7*d^3*e^5 - 98*a^6*b
^6*d^2*e^6 + 34*a^7*b^5*d*e^7 - 5*a^8*b^4*e^8)*x^5 + 5*(a*b^11*d^8 - 5*a^2*b^10*
d^7*e + 7*a^3*b^9*d^6*e^2 + 7*a^4*b^8*d^5*e^3 - 35*a^5*b^7*d^4*e^4 + 49*a^6*b^6*
d^3*e^5 - 35*a^7*b^5*d^2*e^6 + 13*a^8*b^4*d*e^7 - 2*a^9*b^3*e^8)*x^4 + 10*(a^2*b
^10*d^8 - 6*a^3*b^9*d^7*e + 14*a^4*b^8*d^6*e^2 - 14*a^5*b^7*d^5*e^3 + 14*a^7*b^5
*d^3*e^5 - 14*a^8*b^4*d^2*e^6 + 6*a^9*b^3*d*e^7 - a^10*b^2*e^8)*x^3 + 5*(2*a^3*b
^9*d^8 - 13*a^4*b^8*d^7*e + 35*a^5*b^7*d^6*e^2 - 49*a^6*b^6*d^5*e^3 + 35*a^7*b^5
*d^4*e^4 - 7*a^8*b^4*d^3*e^5 - 7*a^9*b^3*d^2*e^6 + 5*a^10*b^2*d*e^7 - a^11*b*e^8
)*x^2 + (5*a^4*b^8*d^8 - 34*a^5*b^7*d^7*e + 98*a^6*b^6*d^6*e^2 - 154*a^7*b^5*d^5
*e^3 + 140*a^8*b^4*d^4*e^4 - 70*a^9*b^3*d^3*e^5 + 14*a^10*b^2*d^2*e^6 + 2*a^11*b
*d*e^7 - a^12*e^8)*x)*sqrt(e*x + d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{6} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Integral(1/((a + b*x)**6*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.23129, size = 860, normalized size = 3.23 \[ -\frac{3003 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (18 \,{\left (x e + d\right )} b e^{5} + b d e^{5} - a e^{6}\right )}}{3 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} - \frac{22005 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{6} e^{5} - 96290 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} d e^{5} + 160384 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d^{2} e^{5} - 121310 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{3} e^{5} + 35595 \, \sqrt{x e + d} b^{6} d^{4} e^{5} + 96290 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{5} e^{6} - 320768 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} d e^{6} + 363930 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d^{2} e^{6} - 142380 \, \sqrt{x e + d} a b^{5} d^{3} e^{6} + 160384 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{4} e^{7} - 363930 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} d e^{7} + 213570 \, \sqrt{x e + d} a^{2} b^{4} d^{2} e^{7} + 121310 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{3} e^{8} - 142380 \, \sqrt{x e + d} a^{3} b^{3} d e^{8} + 35595 \, \sqrt{x e + d} a^{4} b^{2} e^{9}}{1920 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

-3003/128*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^7*d^7 - 7*a*b
^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5
*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*sqrt(-b^2*d + a*b*e)) - 2/3*(18*(x*e + d
)*b*e^5 + b*d*e^5 - a*e^6)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a
^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e
^7)*(x*e + d)^(3/2)) - 1/1920*(22005*(x*e + d)^(9/2)*b^6*e^5 - 96290*(x*e + d)^(
7/2)*b^6*d*e^5 + 160384*(x*e + d)^(5/2)*b^6*d^2*e^5 - 121310*(x*e + d)^(3/2)*b^6
*d^3*e^5 + 35595*sqrt(x*e + d)*b^6*d^4*e^5 + 96290*(x*e + d)^(7/2)*a*b^5*e^6 - 3
20768*(x*e + d)^(5/2)*a*b^5*d*e^6 + 363930*(x*e + d)^(3/2)*a*b^5*d^2*e^6 - 14238
0*sqrt(x*e + d)*a*b^5*d^3*e^6 + 160384*(x*e + d)^(5/2)*a^2*b^4*e^7 - 363930*(x*e
+ d)^(3/2)*a^2*b^4*d*e^7 + 213570*sqrt(x*e + d)*a^2*b^4*d^2*e^7 + 121310*(x*e +
d)^(3/2)*a^3*b^3*e^8 - 142380*sqrt(x*e + d)*a^3*b^3*d*e^8 + 35595*sqrt(x*e + d)
*a^4*b^2*e^9)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^
3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*((x*e + d
)*b - b*d + a*e)^5)