3.187 \(\int \frac{1}{x^3 (d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{4 e^2 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (90 d-107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}+\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{13 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^6}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}+\frac{e^2 (25 d-31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(4*e^2*(d - e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e^2*(25*d - 31*e*x))/(15*d^4*
(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d - 107*e*x))/(15*d^6*Sqrt[d^2 - e^2*x^2]) - S
qrt[d^2 - e^2*x^2]/(2*d^5*x^2) + (3*e*Sqrt[d^2 - e^2*x^2])/(d^6*x) - (13*e^2*Arc
Tanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^6)

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Rubi [A]  time = 0.648789, antiderivative size = 183, 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 \[ \frac{4 e^2 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (90 d-107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}+\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{13 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^6}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}+\frac{e^2 (25 d-31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*e^2*(d - e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e^2*(25*d - 31*e*x))/(15*d^4*
(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d - 107*e*x))/(15*d^6*Sqrt[d^2 - e^2*x^2]) - S
qrt[d^2 - e^2*x^2]/(2*d^5*x^2) + (3*e*Sqrt[d^2 - e^2*x^2])/(d^6*x) - (13*e^2*Arc
Tanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^6)

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Rubi in Sympy [A]  time = 57.7122, size = 158, normalized size = 0.86 \[ \frac{e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{4} \left (d + e x\right )^{3}} + \frac{17 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{5} \left (d + e x\right )^{2}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{2 d^{5} x^{2}} - \frac{13 e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{6}} + \frac{107 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{6} \left (d + e x\right )} + \frac{3 e \sqrt{d^{2} - e^{2} x^{2}}}{d^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**2*sqrt(d**2 - e**2*x**2)/(5*d**4*(d + e*x)**3) + 17*e**2*sqrt(d**2 - e**2*x**
2)/(15*d**5*(d + e*x)**2) - sqrt(d**2 - e**2*x**2)/(2*d**5*x**2) - 13*e**2*atanh
(sqrt(d**2 - e**2*x**2)/d)/(2*d**6) + 107*e**2*sqrt(d**2 - e**2*x**2)/(15*d**6*(
d + e*x)) + 3*e*sqrt(d**2 - e**2*x**2)/(d**6*x)

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Mathematica [A]  time = 0.186904, size = 107, normalized size = 0.58 \[ \frac{-195 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (-15 d^4+45 d^3 e x+479 d^2 e^2 x^2+717 d e^3 x^3+304 e^4 x^4\right )}{x^2 (d+e x)^3}+195 e^2 \log (x)}{30 d^6} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(-15*d^4 + 45*d^3*e*x + 479*d^2*e^2*x^2 + 717*d*e^3*x^3 +
304*e^4*x^4))/(x^2*(d + e*x)^3) + 195*e^2*Log[x] - 195*e^2*Log[d + Sqrt[d^2 - e^
2*x^2]])/(30*d^6)

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Maple [A]  time = 0.019, size = 222, normalized size = 1.2 \[ -{\frac{1}{2\,{d}^{5}{x}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{13\,{e}^{2}}{2\,{d}^{5}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{107\,e}{15\,{d}^{6}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}}+3\,{\frac{e\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{{d}^{6}x}}+{\frac{17}{15\,{d}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{1}{5\,{d}^{4}e}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-e^2*x^2+d^2)^(1/2)/d^5/x^2-13/2/d^5*e^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/
2)*(-e^2*x^2+d^2)^(1/2))/x)+107/15/d^6*e/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^
(1/2)+3*e*(-e^2*x^2+d^2)^(1/2)/d^6/x+17/15/d^5/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(
x+d/e))^(1/2)+1/5/d^4/e/(x+d/e)^3*(-(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. \[ \int \frac{1}{\sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )}^{3} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3*x^3),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3*x^3), x)

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Fricas [A]  time = 0.300411, size = 767, normalized size = 4.19 \[ \frac{558 \, e^{9} x^{9} + 975 \, d e^{8} x^{8} - 4776 \, d^{2} e^{7} x^{7} - 9880 \, d^{3} e^{6} x^{6} + 2540 \, d^{4} e^{5} x^{5} + 13215 \, d^{5} e^{4} x^{4} + 3540 \, d^{6} e^{3} x^{3} - 3540 \, d^{7} e^{2} x^{2} - 840 \, d^{8} e x + 240 \, d^{9} + 195 \,{\left (e^{9} x^{9} + 7 \, d e^{8} x^{8} + 3 \, d^{2} e^{7} x^{7} - 31 \, d^{3} e^{6} x^{6} - 40 \, d^{4} e^{5} x^{5} + 12 \, d^{5} e^{4} x^{4} + 40 \, d^{6} e^{3} x^{3} + 16 \, d^{7} e^{2} x^{2} -{\left (e^{8} x^{8} - 2 \, d e^{7} x^{7} - 19 \, d^{2} e^{6} x^{6} - 20 \, d^{3} e^{5} x^{5} + 20 \, d^{4} e^{4} x^{4} + 40 \, d^{5} e^{3} x^{3} + 16 \, d^{6} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (50 \, e^{8} x^{8} + 2441 \, d e^{7} x^{7} + 4525 \, d^{2} e^{6} x^{6} - 3995 \, d^{3} e^{5} x^{5} - 11535 \, d^{4} e^{4} x^{4} - 3120 \, d^{5} e^{3} x^{3} + 3420 \, d^{6} e^{2} x^{2} + 840 \, d^{7} e x - 240 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (d^{6} e^{7} x^{9} + 7 \, d^{7} e^{6} x^{8} + 3 \, d^{8} e^{5} x^{7} - 31 \, d^{9} e^{4} x^{6} - 40 \, d^{10} e^{3} x^{5} + 12 \, d^{11} e^{2} x^{4} + 40 \, d^{12} e x^{3} + 16 \, d^{13} x^{2} -{\left (d^{6} e^{6} x^{8} - 2 \, d^{7} e^{5} x^{7} - 19 \, d^{8} e^{4} x^{6} - 20 \, d^{9} e^{3} x^{5} + 20 \, d^{10} e^{2} x^{4} + 40 \, d^{11} e x^{3} + 16 \, d^{12} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3*x^3),x, algorithm="fricas")

[Out]

1/30*(558*e^9*x^9 + 975*d*e^8*x^8 - 4776*d^2*e^7*x^7 - 9880*d^3*e^6*x^6 + 2540*d
^4*e^5*x^5 + 13215*d^5*e^4*x^4 + 3540*d^6*e^3*x^3 - 3540*d^7*e^2*x^2 - 840*d^8*e
*x + 240*d^9 + 195*(e^9*x^9 + 7*d*e^8*x^8 + 3*d^2*e^7*x^7 - 31*d^3*e^6*x^6 - 40*
d^4*e^5*x^5 + 12*d^5*e^4*x^4 + 40*d^6*e^3*x^3 + 16*d^7*e^2*x^2 - (e^8*x^8 - 2*d*
e^7*x^7 - 19*d^2*e^6*x^6 - 20*d^3*e^5*x^5 + 20*d^4*e^4*x^4 + 40*d^5*e^3*x^3 + 16
*d^6*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (50*e^8
*x^8 + 2441*d*e^7*x^7 + 4525*d^2*e^6*x^6 - 3995*d^3*e^5*x^5 - 11535*d^4*e^4*x^4
- 3120*d^5*e^3*x^3 + 3420*d^6*e^2*x^2 + 840*d^7*e*x - 240*d^8)*sqrt(-e^2*x^2 + d
^2))/(d^6*e^7*x^9 + 7*d^7*e^6*x^8 + 3*d^8*e^5*x^7 - 31*d^9*e^4*x^6 - 40*d^10*e^3
*x^5 + 12*d^11*e^2*x^4 + 40*d^12*e*x^3 + 16*d^13*x^2 - (d^6*e^6*x^8 - 2*d^7*e^5*
x^7 - 19*d^8*e^4*x^6 - 20*d^9*e^3*x^5 + 20*d^10*e^2*x^4 + 40*d^11*e*x^3 + 16*d^1
2*x^2)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**3), x)

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GIAC/XCAS [A]  time = 0.319938, size = 1, normalized size = 0.01 \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3*x^3),x, algorithm="giac")

[Out]

+Infinity