3.138 \(\int \frac{x^5}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=122 \[ \frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 d-15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.350774, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 d-15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 58.691, size = 160, normalized size = 1.31 \[ \frac{d^{4}}{5 e^{6} \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2 d^{3}}{3 e^{6} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{d^{2} x}{15 e^{5} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{d}{e^{6} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 x^{3}}{3 e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{13 x}{15 e^{5} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

d**4/(5*e**6*(d + e*x)*(d**2 - e**2*x**2)**(3/2)) - 2*d**3/(3*e**6*(d**2 - e**2*
x**2)**(3/2)) + d**2*x/(15*e**5*(d**2 - e**2*x**2)**(3/2)) + d/(e**6*sqrt(d**2 -
e**2*x**2)) + 2*x**3/(3*e**3*(d**2 - e**2*x**2)**(3/2)) - 13*x/(15*e**5*sqrt(d*
*2 - e**2*x**2)) + atan(e*x/sqrt(d**2 - e**2*x**2))/e**6

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Mathematica [A]  time = 0.117251, size = 103, normalized size = 0.84 \[ \frac{15 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (8 d^4-7 d^3 e x-27 d^2 e^2 x^2+8 d e^3 x^3+23 e^4 x^4\right )}{(d-e x)^2 (d+e x)^3}}{15 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(8*d^4 - 7*d^3*e*x - 27*d^2*e^2*x^2 + 8*d*e^3*x^3 + 23*e^4
*x^4))/((d - e*x)^2*(d + e*x)^3) + 15*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^6
)

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Maple [B]  time = 0.015, size = 259, normalized size = 2.1 \[{\frac{2\,{d}^{2}x}{3\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,x}{3\,{e}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{{x}^{3}}{3\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d{x}^{2}}{{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{{d}^{3}}{3\,{e}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{{d}^{4}}{5\,{e}^{7}} \left ( x+{\frac{d}{e}} \right ) ^{-1} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{d}^{2}x}{15\,{e}^{5}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,x}{15\,{e}^{5}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

2/3*d^2/e^5*x/(-e^2*x^2+d^2)^(3/2)-2/3/e^5*x/(-e^2*x^2+d^2)^(1/2)+1/3/e^3*x^3/(-
e^2*x^2+d^2)^(3/2)+1/e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-
d/e^4*x^2/(-e^2*x^2+d^2)^(3/2)+1/3*d^3/e^6/(-e^2*x^2+d^2)^(3/2)+1/5*d^4/e^7/(x+d
/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-4/15*d^2/e^5/(-(x+d/e)^2*e^2+2*d*e*(x+d
/e))^(3/2)*x-8/15/e^5/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x

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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^5/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306031, size = 726, normalized size = 5.95 \[ \frac{23 \, e^{8} x^{8} + 40 \, d e^{7} x^{7} - 179 \, d^{2} e^{6} x^{6} - 199 \, d^{3} e^{5} x^{5} + 280 \, d^{4} e^{4} x^{4} + 280 \, d^{5} e^{3} x^{3} - 120 \, d^{6} e^{2} x^{2} - 120 \, d^{7} e x - 30 \,{\left (4 \, d e^{7} x^{7} + 4 \, d^{2} e^{6} x^{6} - 16 \, d^{3} e^{5} x^{5} - 16 \, d^{4} e^{4} x^{4} + 20 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} - 8 \, d^{7} e x - 8 \, d^{8} -{\left (e^{7} x^{7} + d e^{6} x^{6} - 9 \, d^{2} e^{5} x^{5} - 9 \, d^{3} e^{4} x^{4} + 16 \, d^{4} e^{3} x^{3} + 16 \, d^{5} e^{2} x^{2} - 8 \, d^{6} e x - 8 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 4 \,{\left (2 \, e^{7} x^{7} - 21 \, d e^{6} x^{6} - 26 \, d^{2} e^{5} x^{5} + 55 \, d^{3} e^{4} x^{4} + 55 \, d^{4} e^{3} x^{3} - 30 \, d^{5} e^{2} x^{2} - 30 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d e^{13} x^{7} + 4 \, d^{2} e^{12} x^{6} - 16 \, d^{3} e^{11} x^{5} - 16 \, d^{4} e^{10} x^{4} + 20 \, d^{5} e^{9} x^{3} + 20 \, d^{6} e^{8} x^{2} - 8 \, d^{7} e^{7} x - 8 \, d^{8} e^{6} -{\left (e^{13} x^{7} + d e^{12} x^{6} - 9 \, d^{2} e^{11} x^{5} - 9 \, d^{3} e^{10} x^{4} + 16 \, d^{4} e^{9} x^{3} + 16 \, d^{5} e^{8} x^{2} - 8 \, d^{6} e^{7} x - 8 \, d^{7} e^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(23*e^8*x^8 + 40*d*e^7*x^7 - 179*d^2*e^6*x^6 - 199*d^3*e^5*x^5 + 280*d^4*e^
4*x^4 + 280*d^5*e^3*x^3 - 120*d^6*e^2*x^2 - 120*d^7*e*x - 30*(4*d*e^7*x^7 + 4*d^
2*e^6*x^6 - 16*d^3*e^5*x^5 - 16*d^4*e^4*x^4 + 20*d^5*e^3*x^3 + 20*d^6*e^2*x^2 -
8*d^7*e*x - 8*d^8 - (e^7*x^7 + d*e^6*x^6 - 9*d^2*e^5*x^5 - 9*d^3*e^4*x^4 + 16*d^
4*e^3*x^3 + 16*d^5*e^2*x^2 - 8*d^6*e*x - 8*d^7)*sqrt(-e^2*x^2 + d^2))*arctan(-(d
- sqrt(-e^2*x^2 + d^2))/(e*x)) - 4*(2*e^7*x^7 - 21*d*e^6*x^6 - 26*d^2*e^5*x^5 +
55*d^3*e^4*x^4 + 55*d^4*e^3*x^3 - 30*d^5*e^2*x^2 - 30*d^6*e*x)*sqrt(-e^2*x^2 +
d^2))/(4*d*e^13*x^7 + 4*d^2*e^12*x^6 - 16*d^3*e^11*x^5 - 16*d^4*e^10*x^4 + 20*d^
5*e^9*x^3 + 20*d^6*e^8*x^2 - 8*d^7*e^7*x - 8*d^8*e^6 - (e^13*x^7 + d*e^12*x^6 -
9*d^2*e^11*x^5 - 9*d^3*e^10*x^4 + 16*d^4*e^9*x^3 + 16*d^5*e^8*x^2 - 8*d^6*e^7*x
- 8*d^7*e^6)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(x**5/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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