3.410 \(\int \frac{1}{x^7 \sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{b \sqrt{a+b x^3}}{4 a^2 x^3}-\frac{\sqrt{a+b x^3}}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.105396, antiderivative size = 74, 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 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{b \sqrt{a+b x^3}}{4 a^2 x^3}-\frac{\sqrt{a+b x^3}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[a + b*x^3]),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.1412, size = 63, normalized size = 0.85 \[ - \frac{\sqrt{a + b x^{3}}}{6 a x^{6}} + \frac{b \sqrt{a + b x^{3}}}{4 a^{2} x^{3}} - \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.201573, size = 68, normalized size = 0.92 \[ \frac{\sqrt{a+b x^3} \left (\frac{a \left (3 b x^3-2 a\right )}{x^6}-\frac{3 b^2 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}\right )}{12 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.028, size = 59, normalized size = 0.8 \[ -{\frac{{b}^{2}}{4}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}}-{\frac{1}{6\,{x}^{6}a}\sqrt{b{x}^{3}+a}}+{\frac{b}{4\,{x}^{3}{a}^{2}}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^3+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246884, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (3 \, b x^{3} - 2 \, a\right )} \sqrt{b x^{3} + a} \sqrt{a}}{24 \, a^{\frac{5}{2}} x^{6}}, \frac{3 \, b^{2} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (3 \, b x^{3} - 2 \, a\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{12 \, \sqrt{-a} a^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 14.5315, size = 104, normalized size = 1.41 \[ - \frac{1}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{\sqrt{b}}{12 a x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b^{\frac{3}{2}}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.2193, size = 89, normalized size = 1.2 \[ \frac{1}{12} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} - 5 \, \sqrt{b x^{3} + a} a}{a^{2} b^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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