3.1664 \(\int \frac{1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=295 \[ \frac{9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}-\frac{9009 b^2 e^5}{128 \sqrt{d+e x} (b d-a e)^8}-\frac{3003 b e^5}{128 (d+e x)^{3/2} (b d-a e)^7}-\frac{9009 e^5}{640 (d+e x)^{5/2} (b d-a e)^6}-\frac{1287 e^4}{128 (a+b x) (d+e x)^{5/2} (b d-a e)^5}+\frac{143 e^3}{64 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}-\frac{13 e^2}{16 (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}+\frac{3 e}{8 (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

[Out]

(-9009*e^5)/(640*(b*d - a*e)^6*(d + e*x)^(5/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(
d + e*x)^(5/2)) + (3*e)/(8*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) - (13*e^2)
/(16*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) + (143*e^3)/(64*(b*d - a*e)^4*(a
+ b*x)^2*(d + e*x)^(5/2)) - (1287*e^4)/(128*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(
5/2)) - (3003*b*e^5)/(128*(b*d - a*e)^7*(d + e*x)^(3/2)) - (9009*b^2*e^5)/(128*(
b*d - a*e)^8*Sqrt[d + e*x]) + (9009*b^(5/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/
Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))

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Rubi [A]  time = 0.790176, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}-\frac{9009 b^2 e^5}{128 \sqrt{d+e x} (b d-a e)^8}-\frac{3003 b e^5}{128 (d+e x)^{3/2} (b d-a e)^7}-\frac{9009 e^5}{640 (d+e x)^{5/2} (b d-a e)^6}-\frac{1287 e^4}{128 (a+b x) (d+e x)^{5/2} (b d-a e)^5}+\frac{143 e^3}{64 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}-\frac{13 e^2}{16 (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}+\frac{3 e}{8 (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-9009*e^5)/(640*(b*d - a*e)^6*(d + e*x)^(5/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(
d + e*x)^(5/2)) + (3*e)/(8*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) - (13*e^2)
/(16*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) + (143*e^3)/(64*(b*d - a*e)^4*(a
+ b*x)^2*(d + e*x)^(5/2)) - (1287*e^4)/(128*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(
5/2)) - (3003*b*e^5)/(128*(b*d - a*e)^7*(d + e*x)^(3/2)) - (9009*b^2*e^5)/(128*(
b*d - a*e)^8*Sqrt[d + e*x]) + (9009*b^(5/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/
Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{9009 b^{4} e^{5} \int \frac{1}{2 b \left (a + b x\right ) \sqrt{d + e x}}\, dx}{128 \left (a e - b d\right )^{8}} - \frac{9009 b^{3} e^{4} \sqrt{d + e x}}{128 \left (a + b x\right ) \left (a e - b d\right )^{8}} - \frac{3003 b^{3} e^{3} \sqrt{d + e x}}{64 \left (a + b x\right )^{2} \left (a e - b d\right )^{7}} - \frac{3003 b^{3} e^{2} \sqrt{d + e x}}{80 \left (a + b x\right )^{3} \left (a e - b d\right )^{6}} - \frac{1287 b^{3} e \sqrt{d + e x}}{40 \left (a + b x\right )^{4} \left (a e - b d\right )^{5}} - \frac{143 b^{3} \sqrt{d + e x}}{5 \left (a + b x\right )^{5} \left (a e - b d\right )^{4}} - \frac{26 b^{2}}{\left (a + b x\right )^{5} \sqrt{d + e x} \left (a e - b d\right )^{3}} + \frac{2 b}{\left (a + b x\right )^{5} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} - \frac{2}{5 \left (a + b x\right )^{5} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-9009*b**4*e**5*Integral(1/(2*b*(a + b*x)*sqrt(d + e*x)), x)/(128*(a*e - b*d)**8
) - 9009*b**3*e**4*sqrt(d + e*x)/(128*(a + b*x)*(a*e - b*d)**8) - 3003*b**3*e**3
*sqrt(d + e*x)/(64*(a + b*x)**2*(a*e - b*d)**7) - 3003*b**3*e**2*sqrt(d + e*x)/(
80*(a + b*x)**3*(a*e - b*d)**6) - 1287*b**3*e*sqrt(d + e*x)/(40*(a + b*x)**4*(a*
e - b*d)**5) - 143*b**3*sqrt(d + e*x)/(5*(a + b*x)**5*(a*e - b*d)**4) - 26*b**2/
((a + b*x)**5*sqrt(d + e*x)*(a*e - b*d)**3) + 2*b/((a + b*x)**5*(d + e*x)**(3/2)
*(a*e - b*d)**2) - 2/(5*(a + b*x)**5*(d + e*x)**(5/2)*(a*e - b*d))

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Mathematica [A]  time = 1.80312, size = 243, normalized size = 0.82 \[ \frac{1}{640} \left (\frac{45045 b^{5/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{17/2}}-\frac{\sqrt{d+e x} \left (-\frac{5710 b^3 e^3 (b d-a e)}{(a+b x)^2}+\frac{2008 b^3 e^2 (b d-a e)^2}{(a+b x)^3}-\frac{624 b^3 e (b d-a e)^3}{(a+b x)^4}+\frac{128 b^3 (b d-a e)^4}{(a+b x)^5}+\frac{18165 b^3 e^4}{a+b x}+\frac{2560 b e^5 (b d-a e)}{(d+e x)^2}+\frac{256 e^5 (b d-a e)^2}{(d+e x)^3}+\frac{26880 b^2 e^5}{d+e x}\right )}{(b d-a e)^8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-((Sqrt[d + e*x]*((128*b^3*(b*d - a*e)^4)/(a + b*x)^5 - (624*b^3*e*(b*d - a*e)^
3)/(a + b*x)^4 + (2008*b^3*e^2*(b*d - a*e)^2)/(a + b*x)^3 - (5710*b^3*e^3*(b*d -
a*e))/(a + b*x)^2 + (18165*b^3*e^4)/(a + b*x) + (256*e^5*(b*d - a*e)^2)/(d + e*
x)^3 + (2560*b*e^5*(b*d - a*e))/(d + e*x)^2 + (26880*b^2*e^5)/(d + e*x)))/(b*d -
a*e)^8) + (45045*b^(5/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/
(b*d - a*e)^(17/2))/640

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Maple [B]  time = 0.045, size = 693, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2/5*e^5/(a*e-b*d)^6/(e*x+d)^(5/2)-42*e^5/(a*e-b*d)^8*b^2/(e*x+d)^(1/2)+4*e^5/(a
*e-b*d)^7*b/(e*x+d)^(3/2)-3633/128*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/
2)-7837/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a+7837/64*e^5/(a*e-b*
d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d-1001/5*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*
(e*x+d)^(5/2)*a^2+2002/5*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d-100
1/5*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^2-9443/64*e^8/(a*e-b*d)^8*
b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3+28329/64*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(
e*x+d)^(3/2)*a^2*d-28329/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^
2+9443/64*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^3-5327/128*e^9/(a*e-
b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4+5327/32*e^8/(a*e-b*d)^8*b^4/(b*e*x+a*
e)^5*(e*x+d)^(1/2)*a^3*d-15981/64*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*a^2*d^2+5327/32*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^3-5327/128
*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^4-9009/128*e^5/(a*e-b*d)^8*b^
3/(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)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313747, 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)^(7/2)),x, algorithm="fricas")

[Out]

[-1/1280*(90090*b^7*e^7*x^7 + 256*b^7*d^7 - 2272*a*b^6*d^6*e + 9296*a^2*b^5*d^5*
e^2 - 24220*a^3*b^4*d^4*e^3 + 53270*a^4*b^3*d^3*e^4 + 59392*a^5*b^2*d^2*e^5 - 61
44*a^6*b*d*e^6 + 512*a^7*e^7 + 210210*(b^7*d*e^6 + 2*a*b^6*e^7)*x^6 + 6006*(23*b
^7*d^2*e^5 + 164*a*b^6*d*e^6 + 128*a^2*b^5*e^7)*x^5 + 4290*(3*b^7*d^3*e^4 + 152*
a*b^6*d^2*e^5 + 422*a^2*b^5*d*e^6 + 158*a^3*b^4*e^7)*x^4 - 1430*(2*b^7*d^4*e^3 -
44*a*b^6*d^3*e^4 - 846*a^2*b^5*d^2*e^5 - 1124*a^3*b^4*d*e^6 - 193*a^4*b^3*e^7)*
x^3 + 130*(8*b^7*d^5*e^2 - 106*a*b^6*d^4*e^3 + 938*a^2*b^5*d^3*e^4 + 8368*a^3*b^
4*d^2*e^5 + 5089*a^4*b^3*d*e^6 + 256*a^5*b^2*e^7)*x^2 - 45045*(b^7*e^7*x^7 + a^5
*b^2*d^2*e^5 + (2*b^7*d*e^6 + 5*a*b^6*e^7)*x^6 + (b^7*d^2*e^5 + 10*a*b^6*d*e^6 +
10*a^2*b^5*e^7)*x^5 + 5*(a*b^6*d^2*e^5 + 4*a^2*b^5*d*e^6 + 2*a^3*b^4*e^7)*x^4 +
5*(2*a^2*b^5*d^2*e^5 + 4*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + (10*a^3*b^4*d^2*e^5
+ 10*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^2 + (5*a^4*b^3*d^2*e^5 + 2*a^5*b^2*d*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)*sqr
t(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 10*(48*b^7*d^6*e - 496*a*b^6*d^5*e^
2 + 2618*a^2*b^5*d^4*e^3 - 11620*a^3*b^4*d^3*e^4 - 45677*a^4*b^3*d^2*e^5 - 8192*
a^5*b^2*d*e^6 + 256*a^6*b*e^7)*x)/((a^5*b^8*d^10 - 8*a^6*b^7*d^9*e + 28*a^7*b^6*
d^8*e^2 - 56*a^8*b^5*d^7*e^3 + 70*a^9*b^4*d^6*e^4 - 56*a^10*b^3*d^5*e^5 + 28*a^1
1*b^2*d^4*e^6 - 8*a^12*b*d^3*e^7 + a^13*d^2*e^8 + (b^13*d^8*e^2 - 8*a*b^12*d^7*e
^3 + 28*a^2*b^11*d^6*e^4 - 56*a^3*b^10*d^5*e^5 + 70*a^4*b^9*d^4*e^6 - 56*a^5*b^8
*d^3*e^7 + 28*a^6*b^7*d^2*e^8 - 8*a^7*b^6*d*e^9 + a^8*b^5*e^10)*x^7 + (2*b^13*d^
9*e - 11*a*b^12*d^8*e^2 + 16*a^2*b^11*d^7*e^3 + 28*a^3*b^10*d^6*e^4 - 140*a^4*b^
9*d^5*e^5 + 238*a^5*b^8*d^4*e^6 - 224*a^6*b^7*d^3*e^7 + 124*a^7*b^6*d^2*e^8 - 38
*a^8*b^5*d*e^9 + 5*a^9*b^4*e^10)*x^6 + (b^13*d^10 + 2*a*b^12*d^9*e - 42*a^2*b^11
*d^8*e^2 + 144*a^3*b^10*d^7*e^3 - 210*a^4*b^9*d^6*e^4 + 84*a^5*b^8*d^5*e^5 + 168
*a^6*b^7*d^4*e^6 - 288*a^7*b^6*d^3*e^7 + 201*a^8*b^5*d^2*e^8 - 70*a^9*b^4*d*e^9
+ 10*a^10*b^3*e^10)*x^5 + 5*(a*b^12*d^10 - 4*a^2*b^11*d^9*e - 2*a^3*b^10*d^8*e^2
+ 40*a^4*b^9*d^7*e^3 - 98*a^5*b^8*d^6*e^4 + 112*a^6*b^7*d^5*e^5 - 56*a^7*b^6*d^
4*e^6 - 8*a^8*b^5*d^3*e^7 + 25*a^9*b^4*d^2*e^8 - 12*a^10*b^3*d*e^9 + 2*a^11*b^2*
e^10)*x^4 + 5*(2*a^2*b^11*d^10 - 12*a^3*b^10*d^9*e + 25*a^4*b^9*d^8*e^2 - 8*a^5*
b^8*d^7*e^3 - 56*a^6*b^7*d^6*e^4 + 112*a^7*b^6*d^5*e^5 - 98*a^8*b^5*d^4*e^6 + 40
*a^9*b^4*d^3*e^7 - 2*a^10*b^3*d^2*e^8 - 4*a^11*b^2*d*e^9 + a^12*b*e^10)*x^3 + (1
0*a^3*b^10*d^10 - 70*a^4*b^9*d^9*e + 201*a^5*b^8*d^8*e^2 - 288*a^6*b^7*d^7*e^3 +
168*a^7*b^6*d^6*e^4 + 84*a^8*b^5*d^5*e^5 - 210*a^9*b^4*d^4*e^6 + 144*a^10*b^3*d
^3*e^7 - 42*a^11*b^2*d^2*e^8 + 2*a^12*b*d*e^9 + a^13*e^10)*x^2 + (5*a^4*b^9*d^10
- 38*a^5*b^8*d^9*e + 124*a^6*b^7*d^8*e^2 - 224*a^7*b^6*d^7*e^3 + 238*a^8*b^5*d^
6*e^4 - 140*a^9*b^4*d^5*e^5 + 28*a^10*b^3*d^4*e^6 + 16*a^11*b^2*d^3*e^7 - 11*a^1
2*b*d^2*e^8 + 2*a^13*d*e^9)*x)*sqrt(e*x + d)), -1/640*(45045*b^7*e^7*x^7 + 128*b
^7*d^7 - 1136*a*b^6*d^6*e + 4648*a^2*b^5*d^5*e^2 - 12110*a^3*b^4*d^4*e^3 + 26635
*a^4*b^3*d^3*e^4 + 29696*a^5*b^2*d^2*e^5 - 3072*a^6*b*d*e^6 + 256*a^7*e^7 + 1051
05*(b^7*d*e^6 + 2*a*b^6*e^7)*x^6 + 3003*(23*b^7*d^2*e^5 + 164*a*b^6*d*e^6 + 128*
a^2*b^5*e^7)*x^5 + 2145*(3*b^7*d^3*e^4 + 152*a*b^6*d^2*e^5 + 422*a^2*b^5*d*e^6 +
158*a^3*b^4*e^7)*x^4 - 715*(2*b^7*d^4*e^3 - 44*a*b^6*d^3*e^4 - 846*a^2*b^5*d^2*
e^5 - 1124*a^3*b^4*d*e^6 - 193*a^4*b^3*e^7)*x^3 + 65*(8*b^7*d^5*e^2 - 106*a*b^6*
d^4*e^3 + 938*a^2*b^5*d^3*e^4 + 8368*a^3*b^4*d^2*e^5 + 5089*a^4*b^3*d*e^6 + 256*
a^5*b^2*e^7)*x^2 - 45045*(b^7*e^7*x^7 + a^5*b^2*d^2*e^5 + (2*b^7*d*e^6 + 5*a*b^6
*e^7)*x^6 + (b^7*d^2*e^5 + 10*a*b^6*d*e^6 + 10*a^2*b^5*e^7)*x^5 + 5*(a*b^6*d^2*e
^5 + 4*a^2*b^5*d*e^6 + 2*a^3*b^4*e^7)*x^4 + 5*(2*a^2*b^5*d^2*e^5 + 4*a^3*b^4*d*e
^6 + a^4*b^3*e^7)*x^3 + (10*a^3*b^4*d^2*e^5 + 10*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^
2 + (5*a^4*b^3*d^2*e^5 + 2*a^5*b^2*d*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)) - 5*(48*b^7*d^6*e -
496*a*b^6*d^5*e^2 + 2618*a^2*b^5*d^4*e^3 - 11620*a^3*b^4*d^3*e^4 - 45677*a^4*b^3
*d^2*e^5 - 8192*a^5*b^2*d*e^6 + 256*a^6*b*e^7)*x)/((a^5*b^8*d^10 - 8*a^6*b^7*d^9
*e + 28*a^7*b^6*d^8*e^2 - 56*a^8*b^5*d^7*e^3 + 70*a^9*b^4*d^6*e^4 - 56*a^10*b^3*
d^5*e^5 + 28*a^11*b^2*d^4*e^6 - 8*a^12*b*d^3*e^7 + a^13*d^2*e^8 + (b^13*d^8*e^2
- 8*a*b^12*d^7*e^3 + 28*a^2*b^11*d^6*e^4 - 56*a^3*b^10*d^5*e^5 + 70*a^4*b^9*d^4*
e^6 - 56*a^5*b^8*d^3*e^7 + 28*a^6*b^7*d^2*e^8 - 8*a^7*b^6*d*e^9 + a^8*b^5*e^10)*
x^7 + (2*b^13*d^9*e - 11*a*b^12*d^8*e^2 + 16*a^2*b^11*d^7*e^3 + 28*a^3*b^10*d^6*
e^4 - 140*a^4*b^9*d^5*e^5 + 238*a^5*b^8*d^4*e^6 - 224*a^6*b^7*d^3*e^7 + 124*a^7*
b^6*d^2*e^8 - 38*a^8*b^5*d*e^9 + 5*a^9*b^4*e^10)*x^6 + (b^13*d^10 + 2*a*b^12*d^9
*e - 42*a^2*b^11*d^8*e^2 + 144*a^3*b^10*d^7*e^3 - 210*a^4*b^9*d^6*e^4 + 84*a^5*b
^8*d^5*e^5 + 168*a^6*b^7*d^4*e^6 - 288*a^7*b^6*d^3*e^7 + 201*a^8*b^5*d^2*e^8 - 7
0*a^9*b^4*d*e^9 + 10*a^10*b^3*e^10)*x^5 + 5*(a*b^12*d^10 - 4*a^2*b^11*d^9*e - 2*
a^3*b^10*d^8*e^2 + 40*a^4*b^9*d^7*e^3 - 98*a^5*b^8*d^6*e^4 + 112*a^6*b^7*d^5*e^5
- 56*a^7*b^6*d^4*e^6 - 8*a^8*b^5*d^3*e^7 + 25*a^9*b^4*d^2*e^8 - 12*a^10*b^3*d*e
^9 + 2*a^11*b^2*e^10)*x^4 + 5*(2*a^2*b^11*d^10 - 12*a^3*b^10*d^9*e + 25*a^4*b^9*
d^8*e^2 - 8*a^5*b^8*d^7*e^3 - 56*a^6*b^7*d^6*e^4 + 112*a^7*b^6*d^5*e^5 - 98*a^8*
b^5*d^4*e^6 + 40*a^9*b^4*d^3*e^7 - 2*a^10*b^3*d^2*e^8 - 4*a^11*b^2*d*e^9 + a^12*
b*e^10)*x^3 + (10*a^3*b^10*d^10 - 70*a^4*b^9*d^9*e + 201*a^5*b^8*d^8*e^2 - 288*a
^6*b^7*d^7*e^3 + 168*a^7*b^6*d^6*e^4 + 84*a^8*b^5*d^5*e^5 - 210*a^9*b^4*d^4*e^6
+ 144*a^10*b^3*d^3*e^7 - 42*a^11*b^2*d^2*e^8 + 2*a^12*b*d*e^9 + a^13*e^10)*x^2 +
(5*a^4*b^9*d^10 - 38*a^5*b^8*d^9*e + 124*a^6*b^7*d^8*e^2 - 224*a^7*b^6*d^7*e^3
+ 238*a^8*b^5*d^6*e^4 - 140*a^9*b^4*d^5*e^5 + 28*a^10*b^3*d^4*e^6 + 16*a^11*b^2*
d^3*e^7 - 11*a^12*b*d^2*e^8 + 2*a^13*d*e^9)*x)*sqrt(e*x + d))]

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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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231578, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^(7/2)),x, algorithm="giac")

[Out]

Done