3.597 \(\int (c x)^m \sqrt [3]{a+b x^3} \, dx\)

Optimal. Leaf size=61 \[ \frac{(c x)^{m+1} \left (a+b c^3 x^3\right )^{4/3} \, _2F_1\left (1,\frac{m+5}{3};\frac{m+4}{3};-\frac{b c^3 x^3}{a}\right )}{a c (m+1)} \]

[Out]

((c*x)^(1 + m)*(a + b*c^3*x^3)^(4/3)*Hypergeometric2F1[1, (5 + m)/3, (4 + m)/3, -((b*c^3*x^3)/a)])/(a*c*(1 + m
))

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Rubi [A]  time = 0.0191093, antiderivative size = 68, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {365, 364} \[ \frac{\sqrt [3]{a+b x^3} (c x)^{m+1} \, _2F_1\left (-\frac{1}{3},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{c (m+1) \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(a + b*x^3)^(1/3),x]

[Out]

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

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (c x)^m \sqrt [3]{a+b x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int (c x)^m \sqrt [3]{1+\frac{b x^3}{a}} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{(c x)^{1+m} \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{c (1+m) \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}

Mathematica [A]  time = 0.0138202, size = 66, normalized size = 1.08 \[ \frac{x \sqrt [3]{a+b x^3} (c x)^m \, _2F_1\left (-\frac{1}{3},\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{(m+1) \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{m}\sqrt [3]{b{x}^{3}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (c x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(1/3)*(c*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (c x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^3+a)^(1/3),x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(1/3)*(c*x)^m, x)

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Sympy [C]  time = 1.8689, size = 58, normalized size = 0.95 \begin{align*} \frac{\sqrt [3]{a} c^{m} x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(b*x**3+a)**(1/3),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (c x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(b*x^3+a)^(1/3),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)*(c*x)^m, x)