3.375 \(\int \frac{\sqrt{a+b x^3}}{x^{10}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{5/2}}+\frac{b^2 \sqrt{a+b x^3}}{24 a^2 x^3}-\frac{\sqrt{a+b x^3}}{9 x^9}-\frac{b \sqrt{a+b x^3}}{36 a x^6} \]

[Out]

-Sqrt[a + b*x^3]/(9*x^9) - (b*Sqrt[a + b*x^3])/(36*a*x^6) + (b^2*Sqrt[a + b*x^3]
)/(24*a^2*x^3) - (b^3*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(24*a^(5/2))

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Rubi [A]  time = 0.137731, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{5/2}}+\frac{b^2 \sqrt{a+b x^3}}{24 a^2 x^3}-\frac{\sqrt{a+b x^3}}{9 x^9}-\frac{b \sqrt{a+b x^3}}{36 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^3]/x^10,x]

[Out]

-Sqrt[a + b*x^3]/(9*x^9) - (b*Sqrt[a + b*x^3])/(36*a*x^6) + (b^2*Sqrt[a + b*x^3]
)/(24*a^2*x^3) - (b^3*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(24*a^(5/2))

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Rubi in Sympy [A]  time = 13.5744, size = 82, normalized size = 0.86 \[ - \frac{\sqrt{a + b x^{3}}}{9 x^{9}} - \frac{b \sqrt{a + b x^{3}}}{36 a x^{6}} + \frac{b^{2} \sqrt{a + b x^{3}}}{24 a^{2} x^{3}} - \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{24 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(1/2)/x**10,x)

[Out]

-sqrt(a + b*x**3)/(9*x**9) - b*sqrt(a + b*x**3)/(36*a*x**6) + b**2*sqrt(a + b*x*
*3)/(24*a**2*x**3) - b**3*atanh(sqrt(a + b*x**3)/sqrt(a))/(24*a**(5/2))

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Mathematica [A]  time = 0.276076, size = 79, normalized size = 0.83 \[ \frac{\sqrt{a+b x^3} \left (\frac{a \left (-8 a^2-2 a b x^3+3 b^2 x^6\right )}{x^9}-\frac{3 b^3 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}\right )}{72 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^3]/x^10,x]

[Out]

(Sqrt[a + b*x^3]*((a*(-8*a^2 - 2*a*b*x^3 + 3*b^2*x^6))/x^9 - (3*b^3*ArcTanh[Sqrt
[1 + (b*x^3)/a]])/Sqrt[1 + (b*x^3)/a]))/(72*a^3)

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Maple [A]  time = 0.031, size = 76, normalized size = 0.8 \[ -{\frac{{b}^{3}}{24}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}}-{\frac{1}{9\,{x}^{9}}\sqrt{b{x}^{3}+a}}-{\frac{b}{36\,{x}^{6}a}\sqrt{b{x}^{3}+a}}+{\frac{{b}^{2}}{24\,{x}^{3}{a}^{2}}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/2)/x^10,x)

[Out]

-1/24*b^3*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(5/2)-1/9*(b*x^3+a)^(1/2)/x^9-1/36*
b*(b*x^3+a)^(1/2)/x^6/a+1/24*b^2*(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(sqrt(b*x^3 + a)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22812, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} x^{9} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (3 \, b^{2} x^{6} - 2 \, a b x^{3} - 8 \, a^{2}\right )} \sqrt{b x^{3} + a} \sqrt{a}}{144 \, a^{\frac{5}{2}} x^{9}}, \frac{3 \, b^{3} x^{9} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (3 \, b^{2} x^{6} - 2 \, a b x^{3} - 8 \, a^{2}\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{72 \, \sqrt{-a} a^{2} x^{9}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)/x^10,x, algorithm="fricas")

[Out]

[1/144*(3*b^3*x^9*log(((b*x^3 + 2*a)*sqrt(a) - 2*sqrt(b*x^3 + a)*a)/x^3) + 2*(3*
b^2*x^6 - 2*a*b*x^3 - 8*a^2)*sqrt(b*x^3 + a)*sqrt(a))/(a^(5/2)*x^9), 1/72*(3*b^3
*x^9*arctan(a/(sqrt(b*x^3 + a)*sqrt(-a))) + (3*b^2*x^6 - 2*a*b*x^3 - 8*a^2)*sqrt
(b*x^3 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^9)]

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Sympy [A]  time = 21.4166, size = 129, normalized size = 1.36 \[ - \frac{a}{9 \sqrt{b} x^{\frac{21}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{5 \sqrt{b}}{36 x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b^{\frac{3}{2}}}{72 a x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b^{\frac{5}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{24 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(1/2)/x**10,x)

[Out]

-a/(9*sqrt(b)*x**(21/2)*sqrt(a/(b*x**3) + 1)) - 5*sqrt(b)/(36*x**(15/2)*sqrt(a/(
b*x**3) + 1)) + b**(3/2)/(72*a*x**(9/2)*sqrt(a/(b*x**3) + 1)) + b**(5/2)/(24*a**
2*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b**3*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(24*a*
*(5/2))

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GIAC/XCAS [A]  time = 0.232973, size = 108, normalized size = 1.14 \[ \frac{1}{72} \, b^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 8 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a - 3 \, \sqrt{b x^{3} + a} a^{2}}{a^{2} b^{3} x^{9}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)/x^10,x, algorithm="giac")

[Out]

1/72*b^3*(3*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*x^3 + a)^(5/
2) - 8*(b*x^3 + a)^(3/2)*a - 3*sqrt(b*x^3 + a)*a^2)/(a^2*b^3*x^9))