3.2970 \(\int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx\)

Optimal. Leaf size=139 \[ -\frac{45 b^2 c^3 x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{9 b c^3 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{3/2}}-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{8 x^8} \]

[Out]

-Sqrt[a + b*(c*x^3)^(3/2)]/(8*x^8) - (9*b*c^3*x*Sqrt[a + b*(c*x^3)^(3/2)])/(112*
a*(c*x^3)^(3/2)) - (45*b^2*c^3*x*Sqrt[1 + (b*(c*x^3)^(3/2))/a]*Hypergeometric2F1
[2/9, 1/2, 11/9, -((b*(c*x^3)^(3/2))/a)])/(448*a*Sqrt[a + b*(c*x^3)^(3/2)])

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Rubi [A]  time = 0.216143, antiderivative size = 141, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{45 b^2 c^3 x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{9 b c^5 x^7 \sqrt{a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c*x^3)^(3/2)]/x^9,x]

[Out]

-Sqrt[a + b*(c*x^3)^(3/2)]/(8*x^8) - (9*b*c^5*x^7*Sqrt[a + b*(c*x^3)^(3/2)])/(11
2*a*(c*x^3)^(7/2)) - (45*b^2*c^3*x*Sqrt[1 + (b*(c*x^3)^(3/2))/a]*Hypergeometric2
F1[2/9, 1/2, 11/9, -((b*(c*x^3)^(3/2))/a)])/(448*a*Sqrt[a + b*(c*x^3)^(3/2)])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**3)**(3/2))**(1/2)/x**9,x)

[Out]

Integral(sqrt(a + b*(c*x**3)**(3/2))/x**9, x)

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Mathematica [A]  time = 0.043311, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*(c*x^3)^(3/2)]/x^9,x]

[Out]

Integrate[Sqrt[a + b*(c*x^3)^(3/2)]/x^9, x]

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9}}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^3)^(3/2))^(1/2)/x^9,x)

[Out]

int((a+b*(c*x^3)^(3/2))^(1/2)/x^9,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (c x^{3}\right )^{\frac{3}{2}} b + a}}{x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)/x^9,x, algorithm="maxima")

[Out]

integrate(sqrt((c*x^3)^(3/2)*b + a)/x^9, x)

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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^9,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**3)**(3/2))**(1/2)/x**9,x)

[Out]

Integral(sqrt(a + b*(c*x**3)**(3/2))/x**9, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (c x^{3}\right )^{\frac{3}{2}} b + a}}{x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)/x^9,x, algorithm="giac")

[Out]

integrate(sqrt((c*x^3)^(3/2)*b + a)/x^9, x)