3.593 \(\int (c x)^m \left (a+b x^3\right )^{4/3} \, dx\)

Optimal. Leaf size=69 \[ \frac{a \sqrt [3]{a+b x^3} (c x)^{m+1} \, _2F_1\left (-\frac{4}{3},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{c (m+1) \sqrt [3]{\frac{b x^3}{a}+1}} \]

[Out]

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

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Rubi [A]  time = 0.0673379, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a \sqrt [3]{a+b x^3} (c x)^{m+1} \, _2F_1\left (-\frac{4}{3},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{c (m+1) \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*(a + b*x^3)^(4/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.79481, size = 58, normalized size = 0.84 \[ \frac{a \left (c x\right )^{m + 1} \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{c \sqrt [3]{1 + \frac{b x^{3}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**m*(b*x**3+a)**(4/3),x)

[Out]

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

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Mathematica [A]  time = 0.12274, size = 110, normalized size = 1.59 \[ \frac{x \sqrt [3]{a+b x^3} (c x)^m \left (b (m+1) x^3 \, _2F_1\left (-\frac{1}{3},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right )+a (m+4) \, _2F_1\left (-\frac{1}{3},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )\right )}{(m+1) (m+4) \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^m*(a + b*x^3)^(4/3),x]

[Out]

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

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{m} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(b*x^3+a)^(4/3),x)

[Out]

int((c*x)^m*(b*x^3+a)^(4/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{4}{3}} \left (c x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(4/3)*(c*x)^m,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(4/3)*(c*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{4}{3}} \left (c x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(4/3)*(c*x)^m,x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(4/3)*(c*x)^m, x)

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Sympy [A]  time = 162.541, size = 58, normalized size = 0.84 \[ \frac{a^{\frac{4}{3}} c^{m} x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**m*(b*x**3+a)**(4/3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{4}{3}} \left (c x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(4/3)*(c*x)^m,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(4/3)*(c*x)^m, x)