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

Optimal. Leaf size=123 \[ -\frac{x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

-x^2/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (2*(d + 2*e*x))/(35*d*e^3*(d^2 -
e^2*x^2)^(5/2)) - (4*x)/(105*d^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*e^2
*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.201322, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{x^2}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-x^2/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (2*(d + 2*e*x))/(35*d*e^3*(d^2 -
e^2*x^2)^(5/2)) - (4*x)/(105*d^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*e^2
*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 17.1857, size = 105, normalized size = 0.85 \[ - \frac{x^{2} \left (d - e x\right )}{7 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}} + \frac{2 d + 4 e x}{35 d e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{4 x}{105 d^{3} e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{8 x}{105 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2*(d - e*x)/(7*d*e*(d**2 - e**2*x**2)**(7/2)) + (2*d + 4*e*x)/(35*d*e**3*(d*
*2 - e**2*x**2)**(5/2)) - 4*x/(105*d**3*e**2*(d**2 - e**2*x**2)**(3/2)) - 8*x/(1
05*d**5*e**2*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0699624, size = 104, normalized size = 0.85 \[ \frac{\sqrt{d^2-e^2 x^2} \left (6 d^6+6 d^5 e x-15 d^4 e^2 x^2+20 d^3 e^3 x^3+20 d^2 e^4 x^4-8 d e^5 x^5-8 e^6 x^6\right )}{105 d^5 e^3 (d-e x)^3 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(6*d^6 + 6*d^5*e*x - 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 20*d
^2*e^4*x^4 - 8*d*e^5*x^5 - 8*e^6*x^6))/(105*d^5*e^3*(d - e*x)^3*(d + e*x)^4)

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Maple [A]  time = 0.012, size = 92, normalized size = 0.8 \[{\frac{ \left ( -ex+d \right ) \left ( -8\,{e}^{6}{x}^{6}-8\,{e}^{5}{x}^{5}d+20\,{e}^{4}{x}^{4}{d}^{2}+20\,{x}^{3}{d}^{3}{e}^{3}-15\,{x}^{2}{d}^{4}{e}^{2}+6\,x{d}^{5}e+6\,{d}^{6} \right ) }{105\,{d}^{5}{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(-e*x+d)*(-8*e^6*x^6-8*d*e^5*x^5+20*d^2*e^4*x^4+20*d^3*e^3*x^3-15*d^4*e^2*
x^2+6*d^5*e*x+6*d^6)/d^5/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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.344249, size = 636, normalized size = 5.17 \[ \frac{8 \, e^{9} x^{12} + 44 \, d e^{8} x^{11} - 128 \, d^{2} e^{7} x^{10} - 464 \, d^{3} e^{6} x^{9} + 459 \, d^{4} e^{5} x^{8} + 1614 \, d^{5} e^{4} x^{7} - 616 \, d^{6} e^{3} x^{6} - 2296 \, d^{7} e^{2} x^{5} + 280 \, d^{8} e x^{4} + 1120 \, d^{9} x^{3} - 2 \,{\left (3 \, e^{8} x^{11} - 21 \, d e^{7} x^{10} - 84 \, d^{2} e^{6} x^{9} + 128 \, d^{3} e^{5} x^{8} + 443 \, d^{4} e^{4} x^{7} - 238 \, d^{5} e^{3} x^{6} - 868 \, d^{6} e^{2} x^{5} + 140 \, d^{7} e x^{4} + 560 \, d^{8} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{105 \,{\left (6 \, d^{6} e^{11} x^{11} + 6 \, d^{7} e^{10} x^{10} - 50 \, d^{8} e^{9} x^{9} - 50 \, d^{9} e^{8} x^{8} + 146 \, d^{10} e^{7} x^{7} + 146 \, d^{11} e^{6} x^{6} - 198 \, d^{12} e^{5} x^{5} - 198 \, d^{13} e^{4} x^{4} + 128 \, d^{14} e^{3} x^{3} + 128 \, d^{15} e^{2} x^{2} - 32 \, d^{16} e x - 32 \, d^{17} -{\left (d^{5} e^{11} x^{11} + d^{6} e^{10} x^{10} - 20 \, d^{7} e^{9} x^{9} - 20 \, d^{8} e^{8} x^{8} + 85 \, d^{9} e^{7} x^{7} + 85 \, d^{10} e^{6} x^{6} - 146 \, d^{11} e^{5} x^{5} - 146 \, d^{12} e^{4} x^{4} + 112 \, d^{13} e^{3} x^{3} + 112 \, d^{14} e^{2} x^{2} - 32 \, d^{15} e x - 32 \, d^{16}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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)),x, algorithm="fricas")

[Out]

1/105*(8*e^9*x^12 + 44*d*e^8*x^11 - 128*d^2*e^7*x^10 - 464*d^3*e^6*x^9 + 459*d^4
*e^5*x^8 + 1614*d^5*e^4*x^7 - 616*d^6*e^3*x^6 - 2296*d^7*e^2*x^5 + 280*d^8*e*x^4
+ 1120*d^9*x^3 - 2*(3*e^8*x^11 - 21*d*e^7*x^10 - 84*d^2*e^6*x^9 + 128*d^3*e^5*x
^8 + 443*d^4*e^4*x^7 - 238*d^5*e^3*x^6 - 868*d^6*e^2*x^5 + 140*d^7*e*x^4 + 560*d
^8*x^3)*sqrt(-e^2*x^2 + d^2))/(6*d^6*e^11*x^11 + 6*d^7*e^10*x^10 - 50*d^8*e^9*x^
9 - 50*d^9*e^8*x^8 + 146*d^10*e^7*x^7 + 146*d^11*e^6*x^6 - 198*d^12*e^5*x^5 - 19
8*d^13*e^4*x^4 + 128*d^14*e^3*x^3 + 128*d^15*e^2*x^2 - 32*d^16*e*x - 32*d^17 - (
d^5*e^11*x^11 + d^6*e^10*x^10 - 20*d^7*e^9*x^9 - 20*d^8*e^8*x^8 + 85*d^9*e^7*x^7
+ 85*d^10*e^6*x^6 - 146*d^11*e^5*x^5 - 146*d^12*e^4*x^4 + 112*d^13*e^3*x^3 + 11
2*d^14*e^2*x^2 - 32*d^15*e*x - 32*d^16)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)),x, algorithm="giac")

[Out]

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