3.127 \(\int \frac{x^5}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=128 \[ \frac{x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{(16 d-15 e x) \sqrt{d^2-e^2 x^2}}{6 e^6}-\frac{5 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6}-\frac{x^2 (4 d-5 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}} \]

[Out]

(x^4*(d - e*x))/(3*e^2*(d^2 - e^2*x^2)^(3/2)) - (x^2*(4*d - 5*e*x))/(3*e^4*Sqrt[
d^2 - e^2*x^2]) - ((16*d - 15*e*x)*Sqrt[d^2 - e^2*x^2])/(6*e^6) - (5*d^2*ArcTan[
(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^6)

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Rubi [A]  time = 0.367588, antiderivative size = 128, 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 \[ \frac{x^4 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{(16 d-15 e x) \sqrt{d^2-e^2 x^2}}{6 e^6}-\frac{5 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6}-\frac{x^2 (4 d-5 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]

[Out]

(x^4*(d - e*x))/(3*e^2*(d^2 - e^2*x^2)^(3/2)) - (x^2*(4*d - 5*e*x))/(3*e^4*Sqrt[
d^2 - e^2*x^2]) - ((16*d - 15*e*x)*Sqrt[d^2 - e^2*x^2])/(6*e^6) - (5*d^2*ArcTan[
(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^6)

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Rubi in Sympy [A]  time = 59.4011, size = 163, normalized size = 1.27 \[ \frac{d^{4}}{3 e^{6} \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} - \frac{2 d^{3}}{e^{6} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{4 d^{2} x}{3 e^{5} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{5 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{6}} - \frac{d \sqrt{d^{2} - e^{2} x^{2}}}{e^{6}} + \frac{x^{3}}{e^{3} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{3 x \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(e*x+d)/(-e**2*x**2+d**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.11664, size = 106, normalized size = 0.83 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (16 d^4+d^3 e x-23 d^2 e^2 x^2-3 d e^3 x^3+3 e^4 x^4\right )}{(e x-d) (d+e x)^2}-15 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{6 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(16*d^4 + d^3*e*x - 23*d^2*e^2*x^2 - 3*d*e^3*x^3 + 3*e^4*x
^4))/((-d + e*x)*(d + e*x)^2) - 15*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e^6
)

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Maple [A]  time = 0.026, size = 208, normalized size = 1.6 \[{\frac{7\,{d}^{2}x}{2\,{e}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{{x}^{3}}{2\,{e}^{3}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{5\,{d}^{2}}{2\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d{x}^{2}}{{e}^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-3\,{\frac{{d}^{3}}{{e}^{6}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}+{\frac{{d}^{4}}{3\,{e}^{7}} \left ( x+{\frac{d}{e}} \right ) ^{-1}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{2\,{d}^{2}x}{3\,{e}^{5}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282576, size = 605, normalized size = 4.73 \[ \frac{3 \, e^{8} x^{8} - 3 \, d e^{7} x^{7} - 47 \, d^{2} e^{6} x^{6} - 39 \, d^{3} e^{5} x^{5} + 160 \, d^{4} e^{4} x^{4} + 160 \, d^{5} e^{3} x^{3} - 120 \, d^{6} e^{2} x^{2} - 120 \, d^{7} e x + 30 \,{\left (4 \, d^{3} e^{5} x^{5} + 4 \, d^{4} e^{4} x^{4} - 12 \, d^{5} e^{3} x^{3} - 12 \, d^{6} e^{2} x^{2} + 8 \, d^{7} e x + 8 \, d^{8} -{\left (d^{2} e^{5} x^{5} + d^{3} e^{4} x^{4} - 8 \, d^{4} e^{3} x^{3} - 8 \, d^{5} e^{2} x^{2} + 8 \, d^{6} e x + 8 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 4 \,{\left (3 \, d e^{6} x^{6} + d^{2} e^{5} x^{5} - 25 \, d^{3} e^{4} x^{4} - 25 \, d^{4} e^{3} x^{3} + 30 \, d^{5} e^{2} x^{2} + 30 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (4 \, d e^{11} x^{5} + 4 \, d^{2} e^{10} x^{4} - 12 \, d^{3} e^{9} x^{3} - 12 \, d^{4} e^{8} x^{2} + 8 \, d^{5} e^{7} x + 8 \, d^{6} e^{6} -{\left (e^{11} x^{5} + d e^{10} x^{4} - 8 \, d^{2} e^{9} x^{3} - 8 \, d^{3} e^{8} x^{2} + 8 \, d^{4} e^{7} x + 8 \, d^{5} e^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)),x, algorithm="fricas")

[Out]

1/6*(3*e^8*x^8 - 3*d*e^7*x^7 - 47*d^2*e^6*x^6 - 39*d^3*e^5*x^5 + 160*d^4*e^4*x^4
+ 160*d^5*e^3*x^3 - 120*d^6*e^2*x^2 - 120*d^7*e*x + 30*(4*d^3*e^5*x^5 + 4*d^4*e
^4*x^4 - 12*d^5*e^3*x^3 - 12*d^6*e^2*x^2 + 8*d^7*e*x + 8*d^8 - (d^2*e^5*x^5 + d^
3*e^4*x^4 - 8*d^4*e^3*x^3 - 8*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*(3*d*e^6*x^6 + d^2*e^5*x^5 -
25*d^3*e^4*x^4 - 25*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^11*x^5 + 4*d^2*e^10*x^4 - 12*d^3*e^9*x^3 - 12*d^4*e^8*x^2 + 8*d^5*e^
7*x + 8*d^6*e^6 - (e^11*x^5 + d*e^10*x^4 - 8*d^2*e^9*x^3 - 8*d^3*e^8*x^2 + 8*d^4
*e^7*x + 8*d^5*e^6)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(e*x+d)/(-e**2*x**2+d**2)**(3/2),x)

[Out]

Integral(x**5/((-(-d + e*x)*(d + e*x))**(3/2)*(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)^(3/2)*(e*x + d)),x, algorithm="giac")

[Out]

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