3.180 \(\int \frac{x^4}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=146 \[ \frac{6 d^2 (d-e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{24 d (d-e x)}{5 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{e^5}-\frac{3 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}-\frac{d^3 (d-e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

-(d^3*(d - e*x)^3)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) + (6*d^2*(d - e*x)^2)/(5*e^5*(d
^2 - e^2*x^2)^(3/2)) - (24*d*(d - e*x))/(5*e^5*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 -
e^2*x^2]/e^5 - (3*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^5

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Rubi [A]  time = 0.594183, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{6 d^2 (d-e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{24 d (d-e x)}{5 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{e^5}-\frac{3 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}-\frac{d^3 (d-e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(d^3*(d - e*x)^3)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) + (6*d^2*(d - e*x)^2)/(5*e^5*(d
^2 - e^2*x^2)^(3/2)) - (24*d*(d - e*x))/(5*e^5*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 -
e^2*x^2]/e^5 - (3*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^5

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Rubi in Sympy [A]  time = 45.0295, size = 128, normalized size = 0.88 \[ - \frac{d^{3} \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{5} \left (d + e x\right )^{3}} + \frac{6 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{5} \left (d + e x\right )^{2}} - \frac{3 d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{5}} - \frac{24 d \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{5} \left (d + e x\right )} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)

[Out]

-d**3*sqrt(d**2 - e**2*x**2)/(5*e**5*(d + e*x)**3) + 6*d**2*sqrt(d**2 - e**2*x**
2)/(5*e**5*(d + e*x)**2) - 3*d*atan(e*x/sqrt(d**2 - e**2*x**2))/e**5 - 24*d*sqrt
(d**2 - e**2*x**2)/(5*e**5*(d + e*x)) - sqrt(d**2 - e**2*x**2)/e**5

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Mathematica [A]  time = 0.164934, size = 85, normalized size = 0.58 \[ -\frac{15 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (24 d^3+57 d^2 e x+39 d e^2 x^2+5 e^3 x^3\right )}{(d+e x)^3}}{5 e^5} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[d^2 - e^2*x^2]*(24*d^3 + 57*d^2*e*x + 39*d*e^2*x^2 + 5*e^3*x^3))/(d + e*
x)^3 + 15*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(5*e^5)

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Maple [A]  time = 0.018, size = 187, normalized size = 1.3 \[ -{\frac{1}{{e}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-3\,{\frac{d}{{e}^{4}\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) }-{\frac{{d}^{3}}{5\,{e}^{8}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-3}}+{\frac{6\,{d}^{2}}{5\,{e}^{7}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-2}}-{\frac{24\,d}{5\,{e}^{6}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-(-e^2*x^2+d^2)^(1/2)/e^5-3*d/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^8/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+6/5*d^2/e^7/
(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-24/5/e^6*d/(x+d/e)*(-(x+d/e)^2*e^
2+2*d*e*(x+d/e))^(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/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286621, size = 605, normalized size = 4.14 \[ \frac{5 \, e^{7} x^{7} + 30 \, d e^{6} x^{6} + 158 \, d^{2} e^{5} x^{5} + 175 \, d^{3} e^{4} x^{4} - 140 \, d^{4} e^{3} x^{3} - 300 \, d^{5} e^{2} x^{2} - 120 \, d^{6} e x + 30 \,{\left (d e^{6} x^{6} - d^{2} e^{5} x^{5} - 13 \, d^{3} e^{4} x^{4} - 15 \, d^{4} e^{3} x^{3} + 8 \, d^{5} e^{2} x^{2} + 20 \, d^{6} e x + 8 \, d^{7} +{\left (d e^{5} x^{5} + 6 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} - 12 \, d^{4} e^{2} x^{2} - 20 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (5 \, e^{6} x^{6} + 43 \, d e^{5} x^{5} + 25 \, d^{2} e^{4} x^{4} - 200 \, d^{3} e^{3} x^{3} - 300 \, d^{4} e^{2} x^{2} - 120 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (e^{11} x^{6} - d e^{10} x^{5} - 13 \, d^{2} e^{9} x^{4} - 15 \, d^{3} e^{8} x^{3} + 8 \, d^{4} e^{7} x^{2} + 20 \, d^{5} e^{6} x + 8 \, d^{6} e^{5} +{\left (e^{10} x^{5} + 6 \, d e^{9} x^{4} + 5 \, d^{2} e^{8} x^{3} - 12 \, d^{3} e^{7} x^{2} - 20 \, d^{4} e^{6} x - 8 \, d^{5} e^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(5*e^7*x^7 + 30*d*e^6*x^6 + 158*d^2*e^5*x^5 + 175*d^3*e^4*x^4 - 140*d^4*e^3*
x^3 - 300*d^5*e^2*x^2 - 120*d^6*e*x + 30*(d*e^6*x^6 - d^2*e^5*x^5 - 13*d^3*e^4*x
^4 - 15*d^4*e^3*x^3 + 8*d^5*e^2*x^2 + 20*d^6*e*x + 8*d^7 + (d*e^5*x^5 + 6*d^2*e^
4*x^4 + 5*d^3*e^3*x^3 - 12*d^4*e^2*x^2 - 20*d^5*e*x - 8*d^6)*sqrt(-e^2*x^2 + d^2
))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (5*e^6*x^6 + 43*d*e^5*x^5 + 25*d^
2*e^4*x^4 - 200*d^3*e^3*x^3 - 300*d^4*e^2*x^2 - 120*d^5*e*x)*sqrt(-e^2*x^2 + d^2
))/(e^11*x^6 - d*e^10*x^5 - 13*d^2*e^9*x^4 - 15*d^3*e^8*x^3 + 8*d^4*e^7*x^2 + 20
*d^5*e^6*x + 8*d^6*e^5 + (e^10*x^5 + 6*d*e^9*x^4 + 5*d^2*e^8*x^3 - 12*d^3*e^7*x^
2 - 20*d^4*e^6*x - 8*d^5*e^5)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError