\(\int (c x)^m (a+b x^3)^p \, dx\) [598]

Optimal result

Integrand size = 15, antiderivative size = 66 \[ \int (c x)^m \left (a+b x^3\right )^p \, dx=\frac {(c x)^{1+m} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{3},-p,\frac {4+m}{3},-\frac {b x^3}{a}\right )}{c (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {372, 371} \[ \int (c x)^m \left (a+b x^3\right )^p \, dx=\frac {(c x)^{m+1} \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{3},-p,\frac {m+4}{3},-\frac {b x^3}{a}\right )}{c (m+1)} \]

[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)

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int (c x)^m \left (a+b x^3\right )^p \, dx=\frac {x (c x)^m \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{3},-p,1+\frac {1+m}{3},-\frac {b x^3}{a}\right )}{1+m} \]

[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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*x^3 + a)^p*(c*x)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 116.74 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int (c x)^m \left (a+b x^3\right )^p \, dx=\frac {a^{p} c^{m} x^{m + 1} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{3} + \frac {1}{3} \\ \frac {m}{3} + \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((b*x^3 + a)^p*(c*x)^m, x)

Giac [F]

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

[In]

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

[Out]

integrate((b*x^3 + a)^p*(c*x)^m, x)

Mupad [F(-1)]

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

[In]

int((c*x)^m*(a + b*x^3)^p,x)

[Out]

int((c*x)^m*(a + b*x^3)^p, x)