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

Optimal. Leaf size=119 \[ \frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}+\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.33132, antiderivative size = 119, 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 \[ \frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}+\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.3862, size = 97, normalized size = 0.82 \[ \frac{d - e x}{5 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{5 d - 4 e x}{15 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{15 d - 8 e x}{15 d^{6} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d - e*x)/(5*d**2*(d**2 - e**2*x**2)**(5/2)) + (5*d - 4*e*x)/(15*d**4*(d**2 - e*
*2*x**2)**(3/2)) + (15*d - 8*e*x)/(15*d**6*sqrt(d**2 - e**2*x**2)) - atanh(sqrt(
d**2 - e**2*x**2)/d)/d**6

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Mathematica [A]  time = 0.110204, size = 106, normalized size = 0.89 \[ \frac{-15 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (23 d^4+8 d^3 e x-27 d^2 e^2 x^2-7 d e^3 x^3+8 e^4 x^4\right )}{(d-e x)^2 (d+e x)^3}+15 \log (x)}{15 d^6} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(23*d^4 + 8*d^3*e*x - 27*d^2*e^2*x^2 - 7*d*e^3*x^3 + 8*e^4
*x^4))/((d - e*x)^2*(d + e*x)^3) + 15*Log[x] - 15*Log[d + Sqrt[d^2 - e^2*x^2]])/
(15*d^6)

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Maple [A]  time = 0.019, size = 196, normalized size = 1.7 \[{\frac{1}{3\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{d}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{5}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{1}{5\,e{d}^{2}} \left ( x+{\frac{d}{e}} \right ) ^{-1} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,ex}{15\,{d}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,ex}{15\,{d}^{6}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/d^3/(-e^2*x^2+d^2)^(3/2)+1/d^5/(-e^2*x^2+d^2)^(1/2)-1/d^5/(d^2)^(1/2)*ln((2*
d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+1/5/d^2/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d
*e*(x+d/e))^(3/2)-4/15/d^4*e/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x-8/15/d^6*e/(
-(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. \[ \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )} x}\,{d x} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.307014, size = 717, normalized size = 6.03 \[ \frac{8 \, e^{8} x^{8} + 85 \, d e^{7} x^{7} + d^{2} e^{6} x^{6} - 304 \, d^{3} e^{5} x^{5} - 65 \, d^{4} e^{4} x^{4} + 340 \, d^{5} e^{3} x^{3} + 60 \, d^{6} e^{2} x^{2} - 120 \, d^{7} e x + 15 \,{\left (4 \, d e^{7} x^{7} + 4 \, d^{2} e^{6} x^{6} - 16 \, d^{3} e^{5} x^{5} - 16 \, d^{4} e^{4} x^{4} + 20 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} - 8 \, d^{7} e x - 8 \, d^{8} -{\left (e^{7} x^{7} + d e^{6} x^{6} - 9 \, d^{2} e^{5} x^{5} - 9 \, d^{3} e^{4} x^{4} + 16 \, d^{4} e^{3} x^{3} + 16 \, d^{5} e^{2} x^{2} - 8 \, d^{6} e x - 8 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (23 \, e^{7} x^{7} - 9 \, d e^{6} x^{6} - 179 \, d^{2} e^{5} x^{5} - 35 \, d^{3} e^{4} x^{4} + 280 \, d^{4} e^{3} x^{3} + 60 \, d^{5} e^{2} x^{2} - 120 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{7} e^{7} x^{7} + 4 \, d^{8} e^{6} x^{6} - 16 \, d^{9} e^{5} x^{5} - 16 \, d^{10} e^{4} x^{4} + 20 \, d^{11} e^{3} x^{3} + 20 \, d^{12} e^{2} x^{2} - 8 \, d^{13} e x - 8 \, d^{14} -{\left (d^{6} e^{7} x^{7} + d^{7} e^{6} x^{6} - 9 \, d^{8} e^{5} x^{5} - 9 \, d^{9} e^{4} x^{4} + 16 \, d^{10} e^{3} x^{3} + 16 \, d^{11} e^{2} x^{2} - 8 \, d^{12} e x - 8 \, d^{13}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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),x, algorithm="fricas")

[Out]

1/15*(8*e^8*x^8 + 85*d*e^7*x^7 + d^2*e^6*x^6 - 304*d^3*e^5*x^5 - 65*d^4*e^4*x^4
+ 340*d^5*e^3*x^3 + 60*d^6*e^2*x^2 - 120*d^7*e*x + 15*(4*d*e^7*x^7 + 4*d^2*e^6*x
^6 - 16*d^3*e^5*x^5 - 16*d^4*e^4*x^4 + 20*d^5*e^3*x^3 + 20*d^6*e^2*x^2 - 8*d^7*e
*x - 8*d^8 - (e^7*x^7 + d*e^6*x^6 - 9*d^2*e^5*x^5 - 9*d^3*e^4*x^4 + 16*d^4*e^3*x
^3 + 16*d^5*e^2*x^2 - 8*d^6*e*x - 8*d^7)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e
^2*x^2 + d^2))/x) - (23*e^7*x^7 - 9*d*e^6*x^6 - 179*d^2*e^5*x^5 - 35*d^3*e^4*x^4
+ 280*d^4*e^3*x^3 + 60*d^5*e^2*x^2 - 120*d^6*e*x)*sqrt(-e^2*x^2 + d^2))/(4*d^7*
e^7*x^7 + 4*d^8*e^6*x^6 - 16*d^9*e^5*x^5 - 16*d^10*e^4*x^4 + 20*d^11*e^3*x^3 + 2
0*d^12*e^2*x^2 - 8*d^13*e*x - 8*d^14 - (d^6*e^7*x^7 + d^7*e^6*x^6 - 9*d^8*e^5*x^
5 - 9*d^9*e^4*x^4 + 16*d^10*e^3*x^3 + 16*d^11*e^2*x^2 - 8*d^12*e*x - 8*d^13)*sqr
t(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(-(-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),x, algorithm="giac")

[Out]

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