3.2968 \(\int x^9 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=170 \[ \frac{792 a^3 x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{19747 b^2 c^3 \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{792 a^2 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac{36 a x \left (c x^3\right )^{3/2} \sqrt{a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}} \]

[Out]

(-792*a^2*x*Sqrt[a + b*(c*x^3)^(3/2)])/(19747*b^2*c^3) + (4*x^10*Sqrt[a + b*(c*x
^3)^(3/2)])/49 + (36*a*x*(c*x^3)^(3/2)*Sqrt[a + b*(c*x^3)^(3/2)])/(1519*b*c^3) +
(792*a^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)])/(19747*b^2*c^3*Sqrt[a + b*(c*x^3)^(3/2)])

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Rubi [A]  time = 0.315746, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{792 a^3 x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{19747 b^2 c^3 \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{792 a^2 x \sqrt{a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac{36 a x \left (c x^3\right )^{3/2} \sqrt{a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\frac{4}{49} x^{10} \sqrt{a+b \left (c x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-792*a^2*x*Sqrt[a + b*(c*x^3)^(3/2)])/(19747*b^2*c^3) + (4*x^10*Sqrt[a + b*(c*x
^3)^(3/2)])/49 + (36*a*x*(c*x^3)^(3/2)*Sqrt[a + b*(c*x^3)^(3/2)])/(1519*b*c^3) +
(792*a^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)])/(19747*b^2*c^3*Sqrt[a + b*(c*x^3)^(3/2)])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0540448, size = 0, normalized size = 0. \[ \int x^9 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{x}^{9}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)