3.171 \(\int \frac{x^4}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac{17 d^2 (d-e x)}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (15 d-13 e x)}{15 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}-\frac{d^3 (d-e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

-(d^3*(d - e*x)^2)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) + (17*d^2*(d - e*x))/(15*e^5*(d
^2 - e^2*x^2)^(3/2)) - (2*(15*d - 13*e*x))/(15*e^5*Sqrt[d^2 - e^2*x^2]) - ArcTan
[(e*x)/Sqrt[d^2 - e^2*x^2]]/e^5

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Rubi [A]  time = 0.402498, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{17 d^2 (d-e x)}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (15 d-13 e x)}{15 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}-\frac{d^3 (d-e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]

[Out]

-(d^3*(d - e*x)^2)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) + (17*d^2*(d - e*x))/(15*e^5*(d
^2 - e^2*x^2)^(3/2)) - (2*(15*d - 13*e*x))/(15*e^5*Sqrt[d^2 - e^2*x^2]) - ArcTan
[(e*x)/Sqrt[d^2 - e^2*x^2]]/e^5

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Rubi in Sympy [A]  time = 43.0961, size = 121, normalized size = 0.98 \[ - \frac{d^{3}}{5 e^{5} \left (d + e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{17 d^{2}}{15 e^{5} \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} - \frac{2 d}{e^{5} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{26 x}{15 e^{4} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(e*x+d)**2/(-e**2*x**2+d**2)**(3/2),x)

[Out]

-d**3/(5*e**5*(d + e*x)**2*sqrt(d**2 - e**2*x**2)) + 17*d**2/(15*e**5*(d + e*x)*
sqrt(d**2 - e**2*x**2)) - 2*d/(e**5*sqrt(d**2 - e**2*x**2)) + 26*x/(15*e**4*sqrt
(d**2 - e**2*x**2)) - atan(e*x/sqrt(d**2 - e**2*x**2))/e**5

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Mathematica [A]  time = 0.115957, size = 106, normalized size = 0.86 \[ \sqrt{d^2-e^2 x^2} \left (-\frac{d^2}{10 e^5 (d+e x)^3}+\frac{31 d}{60 e^5 (d+e x)^2}-\frac{1}{8 e^5 (e x-d)}-\frac{193}{120 e^5 (d+e x)}\right )-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]

[Out]

Sqrt[d^2 - e^2*x^2]*(-1/(8*e^5*(-d + e*x)) - d^2/(10*e^5*(d + e*x)^3) + (31*d)/(
60*e^5*(d + e*x)^2) - 193/(120*e^5*(d + e*x))) - ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2
]]/e^5

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Maple [A]  time = 0.02, size = 198, normalized size = 1.6 \[ 4\,{\frac{x}{{e}^{4}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}-{\frac{1}{{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{d}^{3}}{5\,{e}^{7}} \left ( x+{\frac{d}{e}} \right ) ^{-2}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}+{\frac{17\,{d}^{2}}{15\,{e}^{6}} \left ( x+{\frac{d}{e}} \right ) ^{-1}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{34\,x}{15\,{e}^{4}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-2\,{\frac{d}{{e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x)

[Out]

4/e^4*x/(-e^2*x^2+d^2)^(1/2)-1/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^
2)^(1/2))-1/5*d^3/e^7/(x+d/e)^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+17/15*d^2/e
^6/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-34/15/e^4/(-(x+d/e)^2*e^2+2*d*e*
(x+d/e))^(1/2)*x-2*d/e^5/(-e^2*x^2+d^2)^(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(x^4/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292889, size = 547, normalized size = 4.45 \[ -\frac{16 \, e^{6} x^{6} - 46 \, d e^{5} x^{5} - 130 \, d^{2} e^{4} x^{4} - 5 \, d^{3} e^{3} x^{3} + 120 \, d^{4} e^{2} x^{2} + 60 \, d^{5} e x - 30 \,{\left (e^{6} x^{6} + 2 \, d e^{5} x^{5} - 4 \, d^{2} e^{4} x^{4} - 10 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} + 8 \, d^{5} e x + 4 \, d^{6} +{\left (3 \, d e^{4} x^{4} + 6 \, d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2} - 8 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (26 \, e^{5} x^{5} + 70 \, d e^{4} x^{4} - 25 \, d^{2} e^{3} x^{3} - 120 \, d^{3} e^{2} x^{2} - 60 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{11} x^{6} + 2 \, d e^{10} x^{5} - 4 \, d^{2} e^{9} x^{4} - 10 \, d^{3} e^{8} x^{3} - d^{4} e^{7} x^{2} + 8 \, d^{5} e^{6} x + 4 \, d^{6} e^{5} +{\left (3 \, d e^{9} x^{4} + 6 \, d^{2} e^{8} x^{3} - d^{3} e^{7} x^{2} - 8 \, d^{4} e^{6} x - 4 \, d^{5} e^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

-1/15*(16*e^6*x^6 - 46*d*e^5*x^5 - 130*d^2*e^4*x^4 - 5*d^3*e^3*x^3 + 120*d^4*e^2
*x^2 + 60*d^5*e*x - 30*(e^6*x^6 + 2*d*e^5*x^5 - 4*d^2*e^4*x^4 - 10*d^3*e^3*x^3 -
d^4*e^2*x^2 + 8*d^5*e*x + 4*d^6 + (3*d*e^4*x^4 + 6*d^2*e^3*x^3 - d^3*e^2*x^2 -
8*d^4*e*x - 4*d^5)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x
)) + (26*e^5*x^5 + 70*d*e^4*x^4 - 25*d^2*e^3*x^3 - 120*d^3*e^2*x^2 - 60*d^4*e*x)
*sqrt(-e^2*x^2 + d^2))/(e^11*x^6 + 2*d*e^10*x^5 - 4*d^2*e^9*x^4 - 10*d^3*e^8*x^3
- d^4*e^7*x^2 + 8*d^5*e^6*x + 4*d^6*e^5 + (3*d*e^9*x^4 + 6*d^2*e^8*x^3 - d^3*e^
7*x^2 - 8*d^4*e^6*x - 4*d^5*e^5)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(e*x+d)**2/(-e**2*x**2+d**2)**(3/2),x)

[Out]

Integral(x**4/((-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)**2), x)

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GIAC/XCAS [A]  time = 0.67318, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

sage0*x