3.148 \(\int \frac{x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=118 \[ \frac{x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{35 d^4 e^3 \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.268329, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{35 d^4 e^3 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.1506, size = 116, normalized size = 0.98 \[ - \frac{x^{3} \left (d - e x\right )}{7 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{7}{2}}} + \frac{x^{2} \left (3 d + 3 e x\right )}{35 d^{2} e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{6 d - 6 e x}{105 d^{2} e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2 x}{35 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**3*(d - e*x)/(7*d*e*(d**2 - e**2*x**2)**(7/2)) + x**2*(3*d + 3*e*x)/(35*d**2*
e**2*(d**2 - e**2*x**2)**(5/2)) - (6*d - 6*e*x)/(105*d**2*e**4*(d**2 - e**2*x**2
)**(3/2)) - 2*x/(35*d**4*e**3*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0834439, size = 104, normalized size = 0.88 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (2 d^6+2 d^5 e x-5 d^4 e^2 x^2-5 d^3 e^3 x^3-5 d^2 e^4 x^4+2 d e^5 x^5+2 e^6 x^6\right )}{35 d^4 e^4 (d-e x)^3 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d^2 - e^2*x^2]*(2*d^6 + 2*d^5*e*x - 5*d^4*e^2*x^2 - 5*d^3*e^3*x^3 - 5*d^2
*e^4*x^4 + 2*d*e^5*x^5 + 2*e^6*x^6))/(35*d^4*e^4*(d - e*x)^3*(d + e*x)^4)

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Maple [A]  time = 0.012, size = 92, normalized size = 0.8 \[ -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{6}{x}^{6}+2\,{e}^{5}{x}^{5}d-5\,{x}^{4}{d}^{2}{e}^{4}-5\,{x}^{3}{d}^{3}{e}^{3}-5\,{x}^{2}{d}^{4}{e}^{2}+2\,{d}^{5}xe+2\,{d}^{6} \right ) }{35\,{d}^{4}{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(-e*x+d)*(2*e^6*x^6+2*d*e^5*x^5-5*d^2*e^4*x^4-5*d^3*e^3*x^3-5*d^4*e^2*x^2+
2*d^5*e*x+2*d^6)/d^4/e^4/(-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^3/((-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 = 0.341498, size = 605, normalized size = 5.13 \[ \frac{2 \, e^{8} x^{12} - 10 \, d e^{7} x^{11} - 53 \, d^{2} e^{6} x^{10} + 59 \, d^{3} e^{5} x^{9} + 281 \, d^{4} e^{4} x^{8} - 104 \, d^{5} e^{3} x^{7} - 504 \, d^{6} e^{2} x^{6} + 56 \, d^{7} e x^{5} + 280 \, d^{8} x^{4} + 2 \,{\left (e^{7} x^{11} + 7 \, d e^{6} x^{10} - 14 \, d^{2} e^{5} x^{9} - 67 \, d^{3} e^{4} x^{8} + 38 \, d^{4} e^{3} x^{7} + 182 \, d^{5} e^{2} x^{6} - 28 \, d^{6} e x^{5} - 140 \, d^{7} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{35 \,{\left (6 \, d^{5} e^{11} x^{11} + 6 \, d^{6} e^{10} x^{10} - 50 \, d^{7} e^{9} x^{9} - 50 \, d^{8} e^{8} x^{8} + 146 \, d^{9} e^{7} x^{7} + 146 \, d^{10} e^{6} x^{6} - 198 \, d^{11} e^{5} x^{5} - 198 \, d^{12} e^{4} x^{4} + 128 \, d^{13} e^{3} x^{3} + 128 \, d^{14} e^{2} x^{2} - 32 \, d^{15} e x - 32 \, d^{16} -{\left (d^{4} e^{11} x^{11} + d^{5} e^{10} x^{10} - 20 \, d^{6} e^{9} x^{9} - 20 \, d^{7} e^{8} x^{8} + 85 \, d^{8} e^{7} x^{7} + 85 \, d^{9} e^{6} x^{6} - 146 \, d^{10} e^{5} x^{5} - 146 \, d^{11} e^{4} x^{4} + 112 \, d^{12} e^{3} x^{3} + 112 \, d^{13} e^{2} x^{2} - 32 \, d^{14} e x - 32 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)),x, algorithm="fricas")

[Out]

1/35*(2*e^8*x^12 - 10*d*e^7*x^11 - 53*d^2*e^6*x^10 + 59*d^3*e^5*x^9 + 281*d^4*e^
4*x^8 - 104*d^5*e^3*x^7 - 504*d^6*e^2*x^6 + 56*d^7*e*x^5 + 280*d^8*x^4 + 2*(e^7*
x^11 + 7*d*e^6*x^10 - 14*d^2*e^5*x^9 - 67*d^3*e^4*x^8 + 38*d^4*e^3*x^7 + 182*d^5
*e^2*x^6 - 28*d^6*e*x^5 - 140*d^7*x^4)*sqrt(-e^2*x^2 + d^2))/(6*d^5*e^11*x^11 +
6*d^6*e^10*x^10 - 50*d^7*e^9*x^9 - 50*d^8*e^8*x^8 + 146*d^9*e^7*x^7 + 146*d^10*e
^6*x^6 - 198*d^11*e^5*x^5 - 198*d^12*e^4*x^4 + 128*d^13*e^3*x^3 + 128*d^14*e^2*x
^2 - 32*d^15*e*x - 32*d^16 - (d^4*e^11*x^11 + d^5*e^10*x^10 - 20*d^6*e^9*x^9 - 2
0*d^7*e^8*x^8 + 85*d^8*e^7*x^7 + 85*d^9*e^6*x^6 - 146*d^10*e^5*x^5 - 146*d^11*e^
4*x^4 + 112*d^12*e^3*x^3 + 112*d^13*e^2*x^2 - 32*d^14*e*x - 32*d^15)*sqrt(-e^2*x
^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(x**3/((-(-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^3/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)),x, algorithm="giac")

[Out]

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