\(\int F^{a+b (c+d x)^3} \, dx\) [295]

Optimal result

Integrand size = 13, antiderivative size = 47 \[ \int F^{a+b (c+d x)^3} \, dx=-\frac {F^a (c+d x) \Gamma \left (\frac {1}{3},-b (c+d x)^3 \log (F)\right )}{3 d \sqrt [3]{-b (c+d x)^3 \log (F)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2239} \[ \int F^{a+b (c+d x)^3} \, dx=-\frac {F^a (c+d x) \Gamma \left (\frac {1}{3},-b (c+d x)^3 \log (F)\right )}{3 d \sqrt [3]{-b \log (F) (c+d x)^3}} \]

[In]

Int[F^(a + b*(c + d*x)^3),x]

[Out]

-1/3*(F^a*(c + d*x)*Gamma[1/3, -(b*(c + d*x)^3*Log[F])])/(d*(-(b*(c + d*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 {F^a (c+d x) \Gamma \left (\frac {1}{3},-b (c+d x)^3 \log (F)\right )}{3 d \sqrt [3]{-b (c+d x)^3 \log (F)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int F^{a+b (c+d x)^3} \, dx=-\frac {F^a (c+d x) \Gamma \left (\frac {1}{3},-b (c+d x)^3 \log (F)\right )}{3 d \sqrt [3]{-b (c+d x)^3 \log (F)}} \]

[In]

Integrate[F^(a + b*(c + d*x)^3),x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.34 \[ \int F^{a+b (c+d x)^3} \, dx=\frac {\left (-b d^{3} \log \left (F\right )\right )^{\frac {2}{3}} F^{a} \Gamma \left (\frac {1}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )}{3 \, b d^{3} \log \left (F\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate(F**(a+b*(d*x+c)**3),x)

[Out]

Integral(F**(a + b*(c + d*x)**3), x)

Maxima [F]

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

[In]

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

[Out]

integrate(F^((d*x + c)^3*b + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(F^((d*x + c)^3*b + a), x)

Mupad [F(-1)]

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

[In]

int(F^(a + b*(c + d*x)^3),x)

[Out]

int(F^(a + b*(c + d*x)^3), x)