3.596 \(\int (c x)^m \left (a+b x^3\right )^p \, dx\)

Optimal. Leaf size=66 \[ \frac{(c x)^{m+1} \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{3},-p;\frac{m+4}{3};-\frac{b x^3}{a}\right )}{c (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0511109, antiderivative size = 66, 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 \[ \frac{(c x)^{m+1} \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{3},-p;\frac{m+4}{3};-\frac{b x^3}{a}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*(a + b*x^3)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.92868, size = 51, normalized size = 0.77 \[ \frac{\left (c x\right )^{m + 1} \left (1 + \frac{b x^{3}}{a}\right )^{- p} \left (a + b x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{c \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0549968, size = 64, normalized size = 0.97 \[ \frac{x (c x)^m \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{3},-p;\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.094, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{m} \left ( b{x}^{3}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(b*x^3+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^p*(c*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^p*(c*x)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^p*(c*x)^m, x)