3.2966 \(\int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx\)

Optimal. Leaf size=101 \[ \frac{b^2 c^3 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right )}{18 a^{3/2}}-\frac{b c^3 \sqrt{a+b \left (c x^3\right )^{3/2}}}{18 a \left (c x^3\right )^{3/2}}-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{9 x^9} \]

[Out]

-Sqrt[a + b*(c*x^3)^(3/2)]/(9*x^9) - (b*c^3*Sqrt[a + b*(c*x^3)^(3/2)])/(18*a*(c*
x^3)^(3/2)) + (b^2*c^3*ArcTanh[Sqrt[a + b*(c*x^3)^(3/2)]/Sqrt[a]])/(18*a^(3/2))

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Rubi [A]  time = 0.142428, antiderivative size = 104, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{b^2 c^3 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right )}{18 a^{3/2}}-\frac{b c^6 x^9 \sqrt{a+b \left (c x^3\right )^{3/2}}}{18 a \left (c x^3\right )^{9/2}}-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{9 x^9} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c*x^3)^(3/2)]/x^10,x]

[Out]

-Sqrt[a + b*(c*x^3)^(3/2)]/(9*x^9) - (b*c^6*x^9*Sqrt[a + b*(c*x^3)^(3/2)])/(18*a
*(c*x^3)^(9/2)) + (b^2*c^3*ArcTanh[Sqrt[a + b*(c*x^3)^(3/2)]/Sqrt[a]])/(18*a^(3/
2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}}{x^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*(c*x**3)**(3/2))/x**10, x)

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Mathematica [A]  time = 0.0436025, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*(c*x^3)^(3/2)]/x^10,x]

[Out]

Integrate[Sqrt[a + b*(c*x^3)^(3/2)]/x^10, x]

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10}}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((a+b*(c*x^3)^(3/2))^(1/2)/x^10,x)

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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((c*x^3)^(3/2)*b + a)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)/x^10,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}}{x^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*(c*x**3)**(3/2))/x**10, x)

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GIAC/XCAS [A]  time = 0.740945, size = 158, normalized size = 1.56 \[ -\frac{1}{18} \, b^{2} c^{\frac{13}{2}}{\left (\frac{\arctan \left (\frac{\sqrt{\sqrt{c x} b c^{4} x^{4} + a c^{3}}}{\sqrt{-a c} c}\right )}{\sqrt{-a c} a c^{4}} + \frac{\sqrt{\sqrt{c x} b c^{4} x^{4} + a c^{3}} a c^{3} +{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{3}{2}}}{a b^{2} c^{12} x^{9}}\right )}{\left | c \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)/x^10,x, algorithm="giac")

[Out]

-1/18*b^2*c^(13/2)*(arctan(sqrt(sqrt(c*x)*b*c^4*x^4 + a*c^3)/(sqrt(-a*c)*c))/(sq
rt(-a*c)*a*c^4) + (sqrt(sqrt(c*x)*b*c^4*x^4 + a*c^3)*a*c^3 + (sqrt(c*x)*b*c^4*x^
4 + a*c^3)^(3/2))/(a*b^2*c^12*x^9))*abs(c)