3.89 \(\int \frac{(d+e x)^3}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=114 \[ \frac{4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d+22 e x}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]

[Out]

(4*(d + e*x))/(5*(d^2 - e^2*x^2)^(5/2)) + (5*d + 11*e*x)/(15*d^2*(d^2 - e^2*x^2)
^(3/2)) + (15*d + 22*e*x)/(15*d^4*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2*
x^2]/d]/d^4

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Rubi [A]  time = 0.346936, antiderivative size = 114, 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 \[ \frac{4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d+22 e x}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(x*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(4*(d + e*x))/(5*(d^2 - e^2*x^2)^(5/2)) + (5*d + 11*e*x)/(15*d^2*(d^2 - e^2*x^2)
^(3/2)) + (15*d + 22*e*x)/(15*d^4*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2*
x^2]/d]/d^4

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Rubi in Sympy [A]  time = 46.6765, size = 97, normalized size = 0.85 \[ \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d^{2} \left (d - e x\right )^{3}} + \frac{7 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{3} \left (d - e x\right )^{2}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{4}} + \frac{22 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{4} \left (d - e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/x/(-e**2*x**2+d**2)**(7/2),x)

[Out]

sqrt(d**2 - e**2*x**2)/(5*d**2*(d - e*x)**3) + 7*sqrt(d**2 - e**2*x**2)/(15*d**3
*(d - e*x)**2) - atanh(sqrt(d**2 - e**2*x**2)/d)/d**4 + 22*sqrt(d**2 - e**2*x**2
)/(15*d**4*(d - e*x))

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Mathematica [A]  time = 0.145697, size = 77, normalized size = 0.68 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (32 d^2-51 d e x+22 e^2 x^2\right )}{(d-e x)^3}-15 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+15 \log (x)}{15 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(x*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(32*d^2 - 51*d*e*x + 22*e^2*x^2))/(d - e*x)^3 + 15*Log[x]
- 15*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^4)

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Maple [A]  time = 0.017, size = 158, normalized size = 1.4 \[{\frac{4\,d}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{1}{3\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{d}^{3}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{4\,ex}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{11\,ex}{15\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{22\,ex}{15\,{d}^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/x/(-e^2*x^2+d^2)^(7/2),x)

[Out]

4/5*d/(-e^2*x^2+d^2)^(5/2)+1/3/d/(-e^2*x^2+d^2)^(3/2)+1/d^3/(-e^2*x^2+d^2)^(1/2)
-1/d^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+4/5*e*x/(-e^
2*x^2+d^2)^(5/2)+11/15*e/d^2*x/(-e^2*x^2+d^2)^(3/2)+22/15*e/d^4*x/(-e^2*x^2+d^2)
^(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((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289158, size = 447, normalized size = 3.92 \[ \frac{54 \, e^{5} x^{5} - 145 \, d e^{4} x^{4} - 5 \, d^{2} e^{3} x^{3} + 270 \, d^{3} e^{2} x^{2} - 180 \, d^{4} e x + 15 \,{\left (e^{5} x^{5} - 5 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} + 5 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x + 4 \, d^{5} +{\left (e^{4} x^{4} - 7 \, d^{2} e^{2} x^{2} + 10 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 5 \,{\left (2 \, e^{4} x^{4} + 19 \, d e^{3} x^{3} - 54 \, d^{2} e^{2} x^{2} + 36 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{4} e^{5} x^{5} - 5 \, d^{5} e^{4} x^{4} + 5 \, d^{6} e^{3} x^{3} + 5 \, d^{7} e^{2} x^{2} - 10 \, d^{8} e x + 4 \, d^{9} +{\left (d^{4} e^{4} x^{4} - 7 \, d^{6} e^{2} x^{2} + 10 \, d^{7} e x - 4 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="fricas")

[Out]

1/15*(54*e^5*x^5 - 145*d*e^4*x^4 - 5*d^2*e^3*x^3 + 270*d^3*e^2*x^2 - 180*d^4*e*x
+ 15*(e^5*x^5 - 5*d*e^4*x^4 + 5*d^2*e^3*x^3 + 5*d^3*e^2*x^2 - 10*d^4*e*x + 4*d^
5 + (e^4*x^4 - 7*d^2*e^2*x^2 + 10*d^3*e*x - 4*d^4)*sqrt(-e^2*x^2 + d^2))*log(-(d
- sqrt(-e^2*x^2 + d^2))/x) + 5*(2*e^4*x^4 + 19*d*e^3*x^3 - 54*d^2*e^2*x^2 + 36*
d^3*e*x)*sqrt(-e^2*x^2 + d^2))/(d^4*e^5*x^5 - 5*d^5*e^4*x^4 + 5*d^6*e^3*x^3 + 5*
d^7*e^2*x^2 - 10*d^8*e*x + 4*d^9 + (d^4*e^4*x^4 - 7*d^6*e^2*x^2 + 10*d^7*e*x - 4
*d^8)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/x/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**3/(x*(-(-d + e*x)*(d + e*x))**(7/2)), x)

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GIAC/XCAS [A]  time = 0.293, size = 158, normalized size = 1.39 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{22 \, x e^{5}}{d^{4}} + \frac{15 \, e^{4}}{d^{3}}\right )} - \frac{55 \, e^{3}}{d^{2}}\right )} x - \frac{35 \, e^{2}}{d}\right )} x + 45 \, e\right )} x + 32 \, d\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="giac")

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*((((x*(22*x*e^5/d^4 + 15*e^4/d^3) - 55*e^3/d^2)*x - 3
5*e^2/d)*x + 45*e)*x + 32*d)/(x^2*e^2 - d^2)^3 - ln(1/2*abs(-2*d*e - 2*sqrt(-x^2
*e^2 + d^2)*e)*e^(-2)/abs(x))/d^4