\(\int f^{c (a+b x)^3} \, dx\) [204]

Optimal result

Integrand size = 11, antiderivative size = 44 \[ \int f^{c (a+b x)^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b \sqrt [3]{-c (a+b x)^3 \log (f)}} \]

[Out]

-1/3*(b*x+a)*GAMMA(1/3,-c*(b*x+a)^3*ln(f))/b/(-c*(b*x+a)^3*ln(f))^(1/3)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2239} \[ \int f^{c (a+b x)^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b \sqrt [3]{-c \log (f) (a+b x)^3}} \]

[In]

Int[f^(c*(a + b*x)^3),x]

[Out]

-1/3*((a + b*x)*Gamma[1/3, -(c*(a + b*x)^3*Log[f])])/(b*(-(c*(a + b*x)^3*Log[f]))^(1/3))

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b \sqrt [3]{-c (a+b x)^3 \log (f)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int f^{c (a+b x)^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b \sqrt [3]{-c (a+b x)^3 \log (f)}} \]

[In]

Integrate[f^(c*(a + b*x)^3),x]

[Out]

-1/3*((a + b*x)*Gamma[1/3, -(c*(a + b*x)^3*Log[f])])/(b*(-(c*(a + b*x)^3*Log[f]))^(1/3))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.36 \[ \int f^{c (a+b x)^3} \, dx=\frac {\left (-b^{3} c \log \left (f\right )\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right )}{3 \, b^{3} c \log \left (f\right )} \]

[In]

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

[Out]

1/3*(-b^3*c*log(f))^(2/3)*gamma(1/3, -(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)*log(f))/(b^3*c*log(f))

Sympy [F]

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

[In]

integrate(f**(c*(b*x+a)**3),x)

[Out]

Integral(f**(c*(a + b*x)**3), x)

Maxima [F]

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

[In]

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

[Out]

integrate(f^((b*x + a)^3*c), x)

Giac [F]

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

[In]

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

[Out]

integrate(f^((b*x + a)^3*c), x)

Mupad [F(-1)]

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

[In]

int(f^(c*(a + b*x)^3),x)

[Out]

int(f^(c*(a + b*x)^3), x)