3.42 \(\int \left (a+b x^3\right )^m \, dx\)

Optimal. Leaf size=44 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0267679, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.92162, size = 34, normalized size = 0.77 \[ x \left (1 + \frac{b x^{3}}{a}\right )^{- m} \left (a + b x^{3}\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**m,x)

[Out]

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

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Mathematica [C]  time = 0.287412, size = 196, normalized size = 4.45 \[ \frac{2^{-m} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-m} \left (\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{-m} \left (a+b x^3\right )^m F_1\left (m+1;-m,-m;m+2;-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{\sqrt [3]{b} (m+1)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^3)^m,x]

[Out]

(((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*(a + b*x^3)^m*AppellF1[1 + m, -m, -m, 2 + m, (
(-I)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(Sqrt[3]*a^(1/3)), (I + Sqrt[3] - ((2*I)*
b^(1/3)*x)/a^(1/3))/(3*I + Sqrt[3])])/(2^m*b^(1/3)*(1 + m)*((a^(1/3) + (-1)^(2/3
)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3)))^m*((I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I +
Sqrt[3]))^m)

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^m,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^m, x)

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Sympy [A]  time = 54.2667, size = 34, normalized size = 0.77 \[ \frac{a^{m} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, - m \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**m,x)

[Out]

a**m*x*gamma(1/3)*hyper((1/3, -m), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/
3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^m, x)