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

Optimal. Leaf size=148 \[ \frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^4 (d+e x)^4}-\frac{14 d \left (d^2-e^2 x^2\right )^{3/2}}{15 e^4 (d+e x)^3}+\frac{8 d \sqrt{d^2-e^2 x^2}}{e^4 (d+e x)}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}+\frac{4 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]

[Out]

(8*d*Sqrt[d^2 - e^2*x^2])/(e^4*(d + e*x)) + (d^2*(d^2 - e^2*x^2)^(3/2))/(5*e^4*(
d + e*x)^4) - (14*d*(d^2 - e^2*x^2)^(3/2))/(15*e^4*(d + e*x)^3) - (d^2 - e^2*x^2
)^(3/2)/(e^4*(d + e*x)^2) + (4*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^4

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Rubi [A]  time = 0.423754, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^4 (d+e x)^4}-\frac{14 d \left (d^2-e^2 x^2\right )^{3/2}}{15 e^4 (d+e x)^3}+\frac{8 d \sqrt{d^2-e^2 x^2}}{e^4 (d+e x)}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}+\frac{4 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

(8*d*Sqrt[d^2 - e^2*x^2])/(e^4*(d + e*x)) + (d^2*(d^2 - e^2*x^2)^(3/2))/(5*e^4*(
d + e*x)^4) - (14*d*(d^2 - e^2*x^2)^(3/2))/(15*e^4*(d + e*x)^3) - (d^2 - e^2*x^2
)^(3/2)/(e^4*(d + e*x)^2) + (4*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^4

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Rubi in Sympy [A]  time = 40.9128, size = 124, normalized size = 0.84 \[ \frac{d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 e^{4} \left (d + e x\right )^{4}} + \frac{4 d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{4}} + \frac{6 d \sqrt{d^{2} - e^{2} x^{2}}}{e^{4} \left (d + e x\right )} - \frac{14 d \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{15 e^{4} \left (d + e x\right )^{3}} + \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(d**2 - e**2*x**2)**(3/2)/(5*e**4*(d + e*x)**4) + 4*d*atan(e*x/sqrt(d**2 -
e**2*x**2))/e**4 + 6*d*sqrt(d**2 - e**2*x**2)/(e**4*(d + e*x)) - 14*d*(d**2 - e*
*2*x**2)**(3/2)/(15*e**4*(d + e*x)**3) + sqrt(d**2 - e**2*x**2)/e**4

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Mathematica [A]  time = 0.121448, size = 85, normalized size = 0.57 \[ \frac{60 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (94 d^3+222 d^2 e x+149 d e^2 x^2+15 e^3 x^3\right )}{(d+e x)^3}}{15 e^4} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(94*d^3 + 222*d^2*e*x + 149*d*e^2*x^2 + 15*e^3*x^3))/(d +
e*x)^3 + 60*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^4)

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Maple [A]  time = 0.015, size = 212, normalized size = 1.4 \[ 4\,{\frac{1}{{e}^{4}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+4\,{\frac{d}{{e}^{3}\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ) }+3\,{\frac{1}{{e}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}}-{\frac{14\,d}{15\,{e}^{7}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}}+{\frac{{d}^{2}}{5\,{e}^{8}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+4/e^3*d/(e^2)^(1/2)*arctan((e^2)^(1/2
)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))+3/e^6/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*
(x+d/e))^(3/2)-14/15*d/e^7/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+1/5*d^
2/e^8/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)

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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(sqrt(-e^2*x^2 + d^2)*x^3/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297332, size = 605, normalized size = 4.09 \[ -\frac{15 \, e^{7} x^{7} + 100 \, d e^{6} x^{6} + 643 \, d^{2} e^{5} x^{5} + 730 \, d^{3} e^{4} x^{4} - 560 \, d^{4} e^{3} x^{3} - 1200 \, d^{5} e^{2} x^{2} - 480 \, d^{6} e x + 120 \,{\left (d e^{6} x^{6} - d^{2} e^{5} x^{5} - 13 \, d^{3} e^{4} x^{4} - 15 \, d^{4} e^{3} x^{3} + 8 \, d^{5} e^{2} x^{2} + 20 \, d^{6} e x + 8 \, d^{7} +{\left (d e^{5} x^{5} + 6 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} - 12 \, d^{4} e^{2} x^{2} - 20 \, d^{5} e x - 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (15 \, e^{6} x^{6} + 183 \, d e^{5} x^{5} + 130 \, d^{2} e^{4} x^{4} - 800 \, d^{3} e^{3} x^{3} - 1200 \, d^{4} e^{2} x^{2} - 480 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{10} x^{6} - d e^{9} x^{5} - 13 \, d^{2} e^{8} x^{4} - 15 \, d^{3} e^{7} x^{3} + 8 \, d^{4} e^{6} x^{2} + 20 \, d^{5} e^{5} x + 8 \, d^{6} e^{4} +{\left (e^{9} x^{5} + 6 \, d e^{8} x^{4} + 5 \, d^{2} e^{7} x^{3} - 12 \, d^{3} e^{6} x^{2} - 20 \, d^{4} e^{5} x - 8 \, d^{5} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(15*e^7*x^7 + 100*d*e^6*x^6 + 643*d^2*e^5*x^5 + 730*d^3*e^4*x^4 - 560*d^4*
e^3*x^3 - 1200*d^5*e^2*x^2 - 480*d^6*e*x + 120*(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)) - (15*e^6*x^6 + 183*d*e^5*x^5
+ 130*d^2*e^4*x^4 - 800*d^3*e^3*x^3 - 1200*d^4*e^2*x^2 - 480*d^5*e*x)*sqrt(-e^2
*x^2 + d^2))/(e^10*x^6 - d*e^9*x^5 - 13*d^2*e^8*x^4 - 15*d^3*e^7*x^3 + 8*d^4*e^6
*x^2 + 20*d^5*e^5*x + 8*d^6*e^4 + (e^9*x^5 + 6*d*e^8*x^4 + 5*d^2*e^7*x^3 - 12*d^
3*e^6*x^2 - 20*d^4*e^5*x - 8*d^5*e^4)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.296806, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done