3.2969 \(\int \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{9 a x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{13 \sqrt{a+b \left (c x^3\right )^{3/2}}}+\frac{4}{13} x \sqrt{a+b \left (c x^3\right )^{3/2}} \]

[Out]

(4*x*Sqrt[a + b*(c*x^3)^(3/2)])/13 + (9*a*x*Sqrt[1 + (b*(c*x^3)^(3/2))/a]*Hyperg
eometric2F1[2/9, 1/2, 11/9, -((b*(c*x^3)^(3/2))/a)])/(13*Sqrt[a + b*(c*x^3)^(3/2
)])

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Rubi [A]  time = 0.123822, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{9 a x \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac{2}{9},\frac{1}{2};\frac{11}{9};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{13 \sqrt{a+b \left (c x^3\right )^{3/2}}}+\frac{4}{13} x \sqrt{a+b \left (c x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*x*Sqrt[a + b*(c*x^3)^(3/2)])/13 + (9*a*x*Sqrt[1 + (b*(c*x^3)^(3/2))/a]*Hyperg
eometric2F1[2/9, 1/2, 11/9, -((b*(c*x^3)^(3/2))/a)])/(13*Sqrt[a + b*(c*x^3)^(3/2
)])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0158161, size = 0, normalized size = 0. \[ \int \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int \sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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