3.145 \(\int \frac{1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=154 \[ \frac{1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 d^7 x}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^7}+\frac{8 d-5 e x}{5 d^6 x \sqrt{d^2-e^2 x^2}}+\frac{6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(6*d - 5*e*x)/(15*d^4*x*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x*(d + e*x)*(d^2 - e^2
*x^2)^(3/2)) + (8*d - 5*e*x)/(5*d^6*x*Sqrt[d^2 - e^2*x^2]) - (16*Sqrt[d^2 - e^2*
x^2])/(5*d^7*x) + (e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^7

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Rubi [A]  time = 0.430702, antiderivative size = 154, 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 \[ \frac{1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 d^7 x}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^7}+\frac{8 d-5 e x}{5 d^6 x \sqrt{d^2-e^2 x^2}}+\frac{6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(6*d - 5*e*x)/(15*d^4*x*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x*(d + e*x)*(d^2 - e^2
*x^2)^(3/2)) + (8*d - 5*e*x)/(5*d^6*x*Sqrt[d^2 - e^2*x^2]) - (16*Sqrt[d^2 - e^2*
x^2])/(5*d^7*x) + (e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^7

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Rubi in Sympy [A]  time = 50.9282, size = 126, normalized size = 0.82 \[ \frac{d - e x}{5 d^{2} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{6 d - 5 e x}{15 d^{4} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{24 d - 15 e x}{15 d^{6} x \sqrt{d^{2} - e^{2} x^{2}}} + \frac{e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{7}} - \frac{16 \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{7} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

(d - e*x)/(5*d**2*x*(d**2 - e**2*x**2)**(5/2)) + (6*d - 5*e*x)/(15*d**4*x*(d**2
- e**2*x**2)**(3/2)) + (24*d - 15*e*x)/(15*d**6*x*sqrt(d**2 - e**2*x**2)) + e*at
anh(sqrt(d**2 - e**2*x**2)/d)/d**7 - 16*sqrt(d**2 - e**2*x**2)/(5*d**7*x)

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Mathematica [A]  time = 0.148569, size = 122, normalized size = 0.79 \[ -\frac{-15 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (15 d^5+38 d^4 e x-52 d^3 e^2 x^2-87 d^2 e^3 x^3+33 d e^4 x^4+48 e^5 x^5\right )}{x (d-e x)^2 (d+e x)^3}+15 e \log (x)}{15 d^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

-((Sqrt[d^2 - e^2*x^2]*(15*d^5 + 38*d^4*e*x - 52*d^3*e^2*x^2 - 87*d^2*e^3*x^3 +
33*d*e^4*x^4 + 48*e^5*x^5))/(x*(d - e*x)^2*(d + e*x)^3) + 15*e*Log[x] - 15*e*Log
[d + Sqrt[d^2 - e^2*x^2]])/(15*d^7)

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Maple [A]  time = 0.02, size = 268, normalized size = 1.7 \[ -{\frac{1}{{d}^{3}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,{e}^{2}x}{3\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{e}^{2}x}{3\,{d}^{7}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{5\,{d}^{3}} \left ( x+{\frac{d}{e}} \right ) ^{-1} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,{e}^{2}x}{15\,{d}^{5}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{e}^{2}x}{15\,{d}^{7}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{e}{3\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{e}{{d}^{6}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{e}{{d}^{6}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)*x^2), x)

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Fricas [A]  time = 0.307297, size = 871, normalized size = 5.66 \[ -\frac{23 \, e^{10} x^{10} - 217 \, d e^{9} x^{9} - 487 \, d^{2} e^{8} x^{8} + 1073 \, d^{3} e^{7} x^{7} + 1863 \, d^{4} e^{6} x^{6} - 1755 \, d^{5} e^{5} x^{5} - 2655 \, d^{6} e^{4} x^{4} + 1140 \, d^{7} e^{3} x^{3} + 1500 \, d^{8} e^{2} x^{2} - 240 \, d^{9} e x - 240 \, d^{10} + 15 \,{\left (e^{10} x^{10} + d e^{9} x^{9} - 14 \, d^{2} e^{8} x^{8} - 14 \, d^{3} e^{7} x^{7} + 41 \, d^{4} e^{6} x^{6} + 41 \, d^{5} e^{5} x^{5} - 44 \, d^{6} e^{4} x^{4} - 44 \, d^{7} e^{3} x^{3} + 16 \, d^{8} e^{2} x^{2} + 16 \, d^{9} e x +{\left (5 \, d e^{8} x^{8} + 5 \, d^{2} e^{7} x^{7} - 25 \, d^{3} e^{6} x^{6} - 25 \, d^{4} e^{5} x^{5} + 36 \, d^{5} e^{4} x^{4} + 36 \, d^{6} e^{3} x^{3} - 16 \, d^{7} e^{2} x^{2} - 16 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (48 \, e^{9} x^{9} + 148 \, d e^{8} x^{8} - 548 \, d^{2} e^{7} x^{7} - 1023 \, d^{3} e^{6} x^{6} + 1275 \, d^{4} e^{5} x^{5} + 1995 \, d^{5} e^{4} x^{4} - 1020 \, d^{6} e^{3} x^{3} - 1380 \, d^{7} e^{2} x^{2} + 240 \, d^{8} e x + 240 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{7} e^{9} x^{10} + d^{8} e^{8} x^{9} - 14 \, d^{9} e^{7} x^{8} - 14 \, d^{10} e^{6} x^{7} + 41 \, d^{11} e^{5} x^{6} + 41 \, d^{12} e^{4} x^{5} - 44 \, d^{13} e^{3} x^{4} - 44 \, d^{14} e^{2} x^{3} + 16 \, d^{15} e x^{2} + 16 \, d^{16} x +{\left (5 \, d^{8} e^{7} x^{8} + 5 \, d^{9} e^{6} x^{7} - 25 \, d^{10} e^{5} x^{6} - 25 \, d^{11} e^{4} x^{5} + 36 \, d^{12} e^{3} x^{4} + 36 \, d^{13} e^{2} x^{3} - 16 \, d^{14} e x^{2} - 16 \, d^{15} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)*x^2),x, algorithm="fricas")

[Out]

-1/15*(23*e^10*x^10 - 217*d*e^9*x^9 - 487*d^2*e^8*x^8 + 1073*d^3*e^7*x^7 + 1863*
d^4*e^6*x^6 - 1755*d^5*e^5*x^5 - 2655*d^6*e^4*x^4 + 1140*d^7*e^3*x^3 + 1500*d^8*
e^2*x^2 - 240*d^9*e*x - 240*d^10 + 15*(e^10*x^10 + d*e^9*x^9 - 14*d^2*e^8*x^8 -
14*d^3*e^7*x^7 + 41*d^4*e^6*x^6 + 41*d^5*e^5*x^5 - 44*d^6*e^4*x^4 - 44*d^7*e^3*x
^3 + 16*d^8*e^2*x^2 + 16*d^9*e*x + (5*d*e^8*x^8 + 5*d^2*e^7*x^7 - 25*d^3*e^6*x^6
- 25*d^4*e^5*x^5 + 36*d^5*e^4*x^4 + 36*d^6*e^3*x^3 - 16*d^7*e^2*x^2 - 16*d^8*e*
x)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (48*e^9*x^9 + 148*
d*e^8*x^8 - 548*d^2*e^7*x^7 - 1023*d^3*e^6*x^6 + 1275*d^4*e^5*x^5 + 1995*d^5*e^4
*x^4 - 1020*d^6*e^3*x^3 - 1380*d^7*e^2*x^2 + 240*d^8*e*x + 240*d^9)*sqrt(-e^2*x^
2 + d^2))/(d^7*e^9*x^10 + d^8*e^8*x^9 - 14*d^9*e^7*x^8 - 14*d^10*e^6*x^7 + 41*d^
11*e^5*x^6 + 41*d^12*e^4*x^5 - 44*d^13*e^3*x^4 - 44*d^14*e^2*x^3 + 16*d^15*e*x^2
+ 16*d^16*x + (5*d^8*e^7*x^8 + 5*d^9*e^6*x^7 - 25*d^10*e^5*x^6 - 25*d^11*e^4*x^
5 + 36*d^12*e^3*x^4 + 36*d^13*e^2*x^3 - 16*d^14*e*x^2 - 16*d^15*x)*sqrt(-e^2*x^2
+ d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/(x**2*(-(-d + e*x)*(d + e*x))**(5/2)*(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(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)*x^2),x, algorithm="giac")

[Out]

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