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

Optimal. Leaf size=68 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{b \sqrt{a+b x^3}}{4 x^3}-\frac{\left (a+b x^3\right )^{3/2}}{6 x^6} \]

[Out]

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

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Rubi [A]  time = 0.0964115, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{b \sqrt{a+b x^3}}{4 x^3}-\frac{\left (a+b x^3\right )^{3/2}}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(3/2)/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.0394, size = 60, normalized size = 0.88 \[ - \frac{b \sqrt{a + b x^{3}}}{4 x^{3}} - \frac{\left (a + b x^{3}\right )^{\frac{3}{2}}}{6 x^{6}} - \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.196606, size = 70, normalized size = 1.03 \[ \frac{1}{4} \sqrt{a+b x^3} \left (-\frac{b^2 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{a \sqrt{\frac{b x^3}{a}+1}}-\frac{2 a+5 b x^3}{3 x^6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^3]*(-(2*a + 5*b*x^3)/(3*x^6) - (b^2*ArcTanh[Sqrt[1 + (b*x^3)/a]])/
(a*Sqrt[1 + (b*x^3)/a])))/4

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Maple [A]  time = 0.03, size = 54, normalized size = 0.8 \[ -{\frac{a}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{5\,b}{12\,{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{{b}^{2}}{4}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(3/2)/x^7,x)

[Out]

-1/6*a*(b*x^3+a)^(1/2)/x^6-5/12*b*(b*x^3+a)^(1/2)/x^3-1/4*b^2*arctanh((b*x^3+a)^
(1/2)/a^(1/2))/a^(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((b*x^3 + a)^(3/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235056, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) - 2 \,{\left (5 \, b x^{3} + 2 \, a\right )} \sqrt{b x^{3} + a} \sqrt{a}}{24 \, \sqrt{a} x^{6}}, \frac{3 \, b^{2} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) -{\left (5 \, b x^{3} + 2 \, a\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{12 \, \sqrt{-a} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.0387, size = 78, normalized size = 1.15 \[ - \frac{a \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{6 x^{\frac{9}{2}}} - \frac{5 b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}}{12 x^{\frac{3}{2}}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219138, size = 82, normalized size = 1.21 \[ \frac{1}{12} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{3} + a} a}{b^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*b^2*(3*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) - (5*(b*x^3 + a)^(3/2) - 3
*sqrt(b*x^3 + a)*a)/(b^2*x^6))