3.213 \(\int \frac{x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=209 \[ -\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(14*x)/(2145*d^4*e^2*(d^2 - e^2*x^2)^(5/2)) - d/(13*e^3*(d + e*x)^4*(d^2 - e^2*x
^2)^(5/2)) + 17/(143*e^3*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d*e^3*(d +
e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d^2*e^3*(d + e*x)*(d^2 - e^2*x^2)^(5/2)
) + (56*x)/(6435*d^6*e^2*(d^2 - e^2*x^2)^(3/2)) + (112*x)/(6435*d^8*e^2*Sqrt[d^2
- e^2*x^2])

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Rubi [A]  time = 0.415857, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(14*x)/(2145*d^4*e^2*(d^2 - e^2*x^2)^(5/2)) - d/(13*e^3*(d + e*x)^4*(d^2 - e^2*x
^2)^(5/2)) + 17/(143*e^3*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d*e^3*(d +
e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d^2*e^3*(d + e*x)*(d^2 - e^2*x^2)^(5/2)
) + (56*x)/(6435*d^6*e^2*(d^2 - e^2*x^2)^(3/2)) + (112*x)/(6435*d^8*e^2*Sqrt[d^2
- e^2*x^2])

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Rubi in Sympy [A]  time = 62.1787, size = 187, normalized size = 0.89 \[ - \frac{d}{13 e^{3} \left (d + e x\right )^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{17}{143 e^{3} \left (d + e x\right )^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{7}{1287 d e^{3} \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{7}{1287 d^{2} e^{3} \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{14 x}{2145 d^{4} e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{56 x}{6435 d^{6} e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{112 x}{6435 d^{8} e^{2} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-d/(13*e**3*(d + e*x)**4*(d**2 - e**2*x**2)**(5/2)) + 17/(143*e**3*(d + e*x)**3*
(d**2 - e**2*x**2)**(5/2)) - 7/(1287*d*e**3*(d + e*x)**2*(d**2 - e**2*x**2)**(5/
2)) - 7/(1287*d**2*e**3*(d + e*x)*(d**2 - e**2*x**2)**(5/2)) + 14*x/(2145*d**4*e
**2*(d**2 - e**2*x**2)**(5/2)) + 56*x/(6435*d**6*e**2*(d**2 - e**2*x**2)**(3/2))
+ 112*x/(6435*d**8*e**2*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0974611, size = 137, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (200 d^9+800 d^8 e x+700 d^7 e^2 x^2+945 d^6 e^3 x^3-280 d^5 e^4 x^4-1358 d^4 e^5 x^5-672 d^3 e^6 x^6+392 d^2 e^7 x^7+448 d e^8 x^8+112 e^9 x^9\right )}{6435 d^8 e^3 (d-e x)^3 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(200*d^9 + 800*d^8*e*x + 700*d^7*e^2*x^2 + 945*d^6*e^3*x^3
- 280*d^5*e^4*x^4 - 1358*d^4*e^5*x^5 - 672*d^3*e^6*x^6 + 392*d^2*e^7*x^7 + 448*d
*e^8*x^8 + 112*e^9*x^9))/(6435*d^8*e^3*(d - e*x)^3*(d + e*x)^7)

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Maple [A]  time = 0.015, size = 132, normalized size = 0.6 \[{\frac{ \left ( -ex+d \right ) \left ( 112\,{e}^{9}{x}^{9}+448\,{e}^{8}{x}^{8}d+392\,{e}^{7}{x}^{7}{d}^{2}-672\,{e}^{6}{x}^{6}{d}^{3}-1358\,{e}^{5}{x}^{5}{d}^{4}-280\,{e}^{4}{x}^{4}{d}^{5}+945\,{x}^{3}{d}^{6}{e}^{3}+700\,{x}^{2}{d}^{7}{e}^{2}+800\,x{d}^{8}e+200\,{d}^{9} \right ) }{6435\,{e}^{3}{d}^{8} \left ( ex+d \right ) ^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/6435*(-e*x+d)*(112*e^9*x^9+448*d*e^8*x^8+392*d^2*e^7*x^7-672*d^3*e^6*x^6-1358*
d^4*e^5*x^5-280*d^5*e^4*x^4+945*d^6*e^3*x^3+700*d^7*e^2*x^2+800*d^8*e*x+200*d^9)
/(e*x+d)^3/d^8/e^3/(-e^2*x^2+d^2)^(7/2)

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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(x^2/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.51384, size = 992, normalized size = 4.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

1/6435*(200*e^15*x^18 + 1808*d*e^14*x^17 - 3368*d^2*e^13*x^16 - 43512*d^3*e^12*x
^15 - 38608*d^4*e^11*x^14 + 245122*d^5*e^10*x^13 + 440856*d^6*e^9*x^12 - 464503*
d^7*e^8*x^11 - 1510652*d^8*e^7*x^10 - 33176*d^9*e^6*x^9 + 2402400*d^10*e^5*x^8 +
1201200*d^11*e^4*x^7 - 1839552*d^12*e^3*x^6 - 1455168*d^13*e^2*x^5 + 549120*d^1
4*e*x^4 + 549120*d^15*x^3 - (112*e^14*x^17 - 1352*d*e^13*x^16 - 11288*d^2*e^12*x
^15 - 1792*d^3*e^11*x^14 + 113042*d^4*e^10*x^13 + 161720*d^5*e^9*x^12 - 335231*d
^6*e^8*x^11 - 827684*d^7*e^7*x^10 + 193336*d^8*e^6*x^9 + 1688544*d^9*e^5*x^8 + 6
79536*d^10*e^4*x^7 - 1564992*d^11*e^3*x^6 - 1180608*d^12*e^2*x^5 + 549120*d^13*e
*x^4 + 549120*d^14*x^3)*sqrt(-e^2*x^2 + d^2))/(d^8*e^18*x^18 + 4*d^9*e^17*x^17 -
37*d^10*e^16*x^16 - 168*d^11*e^15*x^15 + 106*d^12*e^14*x^14 + 1280*d^13*e^13*x^
13 + 846*d^14*e^12*x^12 - 3704*d^15*e^11*x^11 - 5011*d^16*e^10*x^10 + 4284*d^17*
e^9*x^9 + 10519*d^18*e^8*x^8 + 32*d^19*e^7*x^7 - 10536*d^20*e^6*x^6 - 4544*d^21*
e^5*x^5 + 4688*d^22*e^4*x^4 + 3840*d^23*e^3*x^3 - 320*d^24*e^2*x^2 - 1024*d^25*e
*x - 256*d^26 + (9*d^9*e^16*x^16 + 36*d^10*e^15*x^15 - 84*d^11*e^14*x^14 - 516*d
^12*e^13*x^13 - 138*d^13*e^12*x^12 + 2172*d^14*e^11*x^11 + 2388*d^15*e^10*x^10 -
3516*d^16*e^9*x^9 - 6839*d^17*e^8*x^8 + 1120*d^18*e^7*x^7 + 8392*d^19*e^6*x^6 +
3008*d^20*e^5*x^5 - 4432*d^21*e^4*x^4 - 3328*d^22*e^3*x^3 + 448*d^23*e^2*x^2 +
1024*d^24*e*x + 256*d^25)*sqrt(-e^2*x^2 + d^2))

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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(x**2/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]