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

Optimal. Leaf size=49 \[ -\frac{F^a (c+d x)^4 \text{Gamma}\left (\frac{4}{3},-b \log (F) (c+d x)^3\right )}{3 d \left (-b \log (F) (c+d x)^3\right )^{4/3}} \]

[Out]

-(F^a*(c + d*x)^4*Gamma[4/3, -(b*(c + d*x)^3*Log[F])])/(3*d*(-(b*(c + d*x)^3*Log[F]))^(4/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0638079, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac{F^a (c+d x)^4 \text{Gamma}\left (\frac{4}{3},-b \log (F) (c+d x)^3\right )}{3 d \left (-b \log (F) (c+d x)^3\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^3)*(c + d*x)^3,x]

[Out]

-(F^a*(c + d*x)^4*Gamma[4/3, -(b*(c + d*x)^3*Log[F])])/(3*d*(-(b*(c + d*x)^3*Log[F]))^(4/3))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int F^{a+b (c+d x)^3} (c+d x)^3 \, dx &=-\frac{F^a (c+d x)^4 \Gamma \left (\frac{4}{3},-b (c+d x)^3 \log (F)\right )}{3 d \left (-b (c+d x)^3 \log (F)\right )^{4/3}}\\ \end{align*}

Mathematica [A]  time = 0.0253807, size = 49, normalized size = 1. \[ -\frac{F^a (c+d x)^4 \text{Gamma}\left (\frac{4}{3},-b \log (F) (c+d x)^3\right )}{3 d \left (-b \log (F) (c+d x)^3\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^3)*(c + d*x)^3,x]

[Out]

-(F^a*(c + d*x)^4*Gamma[4/3, -(b*(c + d*x)^3*Log[F])])/(3*d*(-(b*(c + d*x)^3*Log[F]))^(4/3))

________________________________________________________________________________________

Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b \left ( dx+c \right ) ^{3}} \left ( dx+c \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.62648, size = 282, normalized size = 5.76 \begin{align*} \frac{3 \,{\left (b d^{3} x + b c d^{2}\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} \log \left (F\right ) - \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 )}{9 \, b^{2} d^{3} \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*(b*d^3*x + b*c*d^2)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*log(F) - (-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^2*d^3*log(F)^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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