3.374 \(\int \frac{\sqrt{a+b x^3}}{x^7} \, dx\)

Optimal. Leaf size=71 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}-\frac{b \sqrt{a+b x^3}}{12 a x^3}-\frac{\sqrt{a+b x^3}}{6 x^6} \]

[Out]

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

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Rubi [A]  time = 0.104662, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}-\frac{b \sqrt{a+b x^3}}{12 a x^3}-\frac{\sqrt{a+b x^3}}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^3]/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.88967, size = 60, normalized size = 0.85 \[ - \frac{\sqrt{a + b x^{3}}}{6 x^{6}} - \frac{b \sqrt{a + b x^{3}}}{12 a x^{3}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{12 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**3)/(6*x**6) - b*sqrt(a + b*x**3)/(12*a*x**3) + b**2*atanh(sqrt(a
+ b*x**3)/sqrt(a))/(12*a**(3/2))

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Mathematica [A]  time = 0.195269, size = 67, normalized size = 0.94 \[ \frac{\sqrt{a+b x^3} \left (\frac{b^2 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a \left (2 a+b x^3\right )}{x^6}\right )}{12 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^3]/x^7,x]

[Out]

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

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Maple [A]  time = 0.03, size = 56, normalized size = 0.8 \[{\frac{{b}^{2}}{12}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{1}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{b}{12\,a{x}^{3}}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/2)/x^7,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224143, size = 1, normalized size = 0.01 \[ \left [\frac{b^{2} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) - 2 \,{\left (b x^{3} + 2 \, a\right )} \sqrt{b x^{3} + a} \sqrt{a}}{24 \, a^{\frac{3}{2}} x^{6}}, -\frac{b^{2} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (b x^{3} + 2 \, a\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{12 \, \sqrt{-a} a x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.7865, size = 100, normalized size = 1.41 \[ - \frac{a}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b}}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{b^{\frac{3}{2}}}{12 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{12 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.246616, size = 84, normalized size = 1.18 \[ -\frac{1}{12} \, b^{2}{\left (\frac{\arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}} + \sqrt{b x^{3} + a} a}{a b^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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