3.589 \(\int x^m \left (a+b x^3\right )^{3/2} \, dx\)

Optimal. Leaf size=64 \[ \frac{a x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt{\frac{b x^3}{a}+1}} \]

[Out]

(a*x^(1 + m)*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, (1 + m)/3, (4 + m)/3, -((b*
x^3)/a)])/((1 + m)*Sqrt[1 + (b*x^3)/a])

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Rubi [A]  time = 0.0592202, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^3)^(3/2),x]

[Out]

(a*x^(1 + m)*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, (1 + m)/3, (4 + m)/3, -((b*
x^3)/a)])/((1 + m)*Sqrt[1 + (b*x^3)/a])

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Rubi in Sympy [A]  time = 7.01364, size = 54, normalized size = 0.84 \[ \frac{a x^{m + 1} \sqrt{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{\sqrt{1 + \frac{b x^{3}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**(m + 1)*sqrt(a + b*x**3)*hyper((-3/2, m/3 + 1/3), (m/3 + 4/3,), -b*x**3/a)/
(sqrt(1 + b*x**3/a)*(m + 1))

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Mathematica [A]  time = 0.122575, size = 109, normalized size = 1.7 \[ \frac{x^{m+1} \sqrt{a+b x^3} \left (b (m+1) x^3 \, _2F_1\left (-\frac{1}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right )+a (m+4) \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )\right )}{(m+1) (m+4) \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)*Sqrt[a + b*x^3]*(a*(4 + m)*Hypergeometric2F1[-1/2, (1 + m)/3, (4 + m)
/3, -((b*x^3)/a)] + b*(1 + m)*x^3*Hypergeometric2F1[-1/2, (4 + m)/3, (7 + m)/3,
-((b*x^3)/a)]))/((1 + m)*(4 + m)*Sqrt[1 + (b*x^3)/a])

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^3+a)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(3/2)*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^(3/2)*x^m, x)

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Sympy [A]  time = 21.2882, size = 54, normalized size = 0.84 \[ \frac{a^{\frac{3}{2}} x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*x*x**m*gamma(m/3 + 1/3)*hyper((-3/2, m/3 + 1/3), (m/3 + 4/3,), b*x**3*e
xp_polar(I*pi)/a)/(3*gamma(m/3 + 4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(3/2)*x^m, x)