3.427 \(\int \frac{1}{x^7 \left (a+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{5 b \sqrt{a+b x^3}}{4 a^3 x^3}-\frac{5 \sqrt{a+b x^3}}{6 a^2 x^6}+\frac{2}{3 a x^6 \sqrt{a+b x^3}} \]

[Out]

2/(3*a*x^6*Sqrt[a + b*x^3]) - (5*Sqrt[a + b*x^3])/(6*a^2*x^6) + (5*b*Sqrt[a + b*
x^3])/(4*a^3*x^3) - (5*b^2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*a^(7/2))

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Rubi [A]  time = 0.138171, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{5 b \sqrt{a+b x^3}}{4 a^3 x^3}-\frac{5 \sqrt{a+b x^3}}{6 a^2 x^6}+\frac{2}{3 a x^6 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(a + b*x^3)^(3/2)),x]

[Out]

2/(3*a*x^6*Sqrt[a + b*x^3]) - (5*Sqrt[a + b*x^3])/(6*a^2*x^6) + (5*b*Sqrt[a + b*
x^3])/(4*a^3*x^3) - (5*b^2*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*a^(7/2))

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Rubi in Sympy [A]  time = 14.0509, size = 88, normalized size = 0.93 \[ \frac{2}{3 a x^{6} \sqrt{a + b x^{3}}} - \frac{5 \sqrt{a + b x^{3}}}{6 a^{2} x^{6}} + \frac{5 b \sqrt{a + b x^{3}}}{4 a^{3} x^{3}} - \frac{5 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(b*x**3+a)**(3/2),x)

[Out]

2/(3*a*x**6*sqrt(a + b*x**3)) - 5*sqrt(a + b*x**3)/(6*a**2*x**6) + 5*b*sqrt(a +
b*x**3)/(4*a**3*x**3) - 5*b**2*atanh(sqrt(a + b*x**3)/sqrt(a))/(4*a**(7/2))

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Mathematica [A]  time = 0.331747, size = 73, normalized size = 0.77 \[ \frac{-\frac{2 a^2}{x^6}-15 b^2 \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+\frac{5 a b}{x^3}+15 b^2}{12 a^3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(a + b*x^3)^(3/2)),x]

[Out]

(15*b^2 - (2*a^2)/x^6 + (5*a*b)/x^3 - 15*b^2*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1
+ (b*x^3)/a]])/(12*a^3*Sqrt[a + b*x^3])

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Maple [A]  time = 0.038, size = 80, normalized size = 0.8 \[ -{\frac{1}{6\,{a}^{2}{x}^{6}}\sqrt{b{x}^{3}+a}}+{\frac{7\,b}{12\,{a}^{3}{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{2\,{b}^{2}}{3\,{a}^{3}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{5\,{b}^{2}}{4}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^3+a)^(3/2),x)

[Out]

-1/6*(b*x^3+a)^(1/2)/a^2/x^6+7/12*b*(b*x^3+a)^(1/2)/a^3/x^3+2/3*b^2/a^3/((x^3+a/
b)*b)^(1/2)-5/4*b^2*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(7/2)

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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(1/((b*x^3 + a)^(3/2)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242336, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b x^{3} + a} b^{2} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (15 \, b^{2} x^{6} + 5 \, a b x^{3} - 2 \, a^{2}\right )} \sqrt{a}}{24 \, \sqrt{b x^{3} + a} a^{\frac{7}{2}} x^{6}}, \frac{15 \, \sqrt{b x^{3} + a} b^{2} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (15 \, b^{2} x^{6} + 5 \, a b x^{3} - 2 \, a^{2}\right )} \sqrt{-a}}{12 \, \sqrt{b x^{3} + a} \sqrt{-a} a^{3} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(3/2)*x^7),x, algorithm="fricas")

[Out]

[1/24*(15*sqrt(b*x^3 + a)*b^2*x^6*log(((b*x^3 + 2*a)*sqrt(a) - 2*sqrt(b*x^3 + a)
*a)/x^3) + 2*(15*b^2*x^6 + 5*a*b*x^3 - 2*a^2)*sqrt(a))/(sqrt(b*x^3 + a)*a^(7/2)*
x^6), 1/12*(15*sqrt(b*x^3 + a)*b^2*x^6*arctan(a/(sqrt(b*x^3 + a)*sqrt(-a))) + (1
5*b^2*x^6 + 5*a*b*x^3 - 2*a^2)*sqrt(-a))/(sqrt(b*x^3 + a)*sqrt(-a)*a^3*x^6)]

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Sympy [A]  time = 21.4905, size = 112, normalized size = 1.18 \[ - \frac{1}{6 a \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{5 \sqrt{b}}{12 a^{2} x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{5 b^{\frac{3}{2}}}{4 a^{3} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{5 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(b*x**3+a)**(3/2),x)

[Out]

-1/(6*a*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) + 5*sqrt(b)/(12*a**2*x**(9/2)*sq
rt(a/(b*x**3) + 1)) + 5*b**(3/2)/(4*a**3*x**(3/2)*sqrt(a/(b*x**3) + 1)) - 5*b**2
*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*a**(7/2))

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GIAC/XCAS [A]  time = 0.212856, size = 108, normalized size = 1.14 \[ \frac{1}{12} \, b^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{8}{\sqrt{b x^{3} + a} a^{3}} + \frac{7 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} - 9 \, \sqrt{b x^{3} + a} a}{a^{3} b^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(3/2)*x^7),x, algorithm="giac")

[Out]

1/12*b^2*(15*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a^3) + 8/(sqrt(b*x^3 + a
)*a^3) + (7*(b*x^3 + a)^(3/2) - 9*sqrt(b*x^3 + a)*a)/(a^3*b^2*x^6))