\(\int (c x)^m (a+b x^3)^{2/3} \, dx\) [596]

Optimal result

Integrand size = 17, antiderivative size = 61 \[ \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx=\frac {(c x)^{1+m} \left (a+b c^3 x^3\right )^{5/3} \operatorname {Hypergeometric2F1}\left (1,\frac {6+m}{3},\frac {4+m}{3},-\frac {b c^3 x^3}{a}\right )}{a c (1+m)} \]

[Out]

(c*x)^(1+m)*(b*c^3*x^3+a)^(5/3)*hypergeom([1, 2+1/3*m],[4/3+1/3*m],-b*c^3*x^3/a)/a/c/(1+m)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {372, 371} \[ \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx=\frac {\left (a+b x^3\right )^{2/3} (c x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {m+1}{3},\frac {m+4}{3},-\frac {b x^3}{a}\right )}{c (m+1) \left (\frac {b x^3}{a}+1\right )^{2/3}} \]

[In]

Int[(c*x)^m*(a + b*x^3)^(2/3),x]

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x^3\right )^{2/3} \int (c x)^m \left (1+\frac {b x^3}{a}\right )^{2/3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \\ & = \frac {(c x)^{1+m} \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{c (1+m) \left (1+\frac {b x^3}{a}\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08 \[ \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx=\frac {x (c x)^m \left (a+b x^3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1+m}{3},1+\frac {1+m}{3},-\frac {b x^3}{a}\right )}{(1+m) \left (1+\frac {b x^3}{a}\right )^{2/3}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (c x \right )^{m} \left (b \,x^{3}+a \right )^{\frac {2}{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (c x\right )^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.95 \[ \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx=\frac {a^{\frac {2}{3}} c^{m} x^{m + 1} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {m}{3} + \frac {1}{3} \\ \frac {m}{3} + \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} \]

[In]

integrate((c*x)**m*(b*x**3+a)**(2/3),x)

[Out]

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

Maxima [F]

\[ \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (c x\right )^{m} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (c x\right )^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (c x)^m \left (a+b x^3\right )^{2/3} \, dx=\int {\left (c\,x\right )}^m\,{\left (b\,x^3+a\right )}^{2/3} \,d x \]

[In]

int((c*x)^m*(a + b*x^3)^(2/3),x)

[Out]

int((c*x)^m*(a + b*x^3)^(2/3), x)