3.49 \(\int F^{c (a+b x)} (d+e x)^{4/3} \, dx\)

Optimal. Leaf size=71 \[ -\frac{e \sqrt [3]{d+e x} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{7}{3},-\frac{b c \log (F) (d+e x)}{e}\right )}{b^2 c^2 \log ^2(F) \sqrt [3]{-\frac{b c \log (F) (d+e x)}{e}}} \]

[Out]

-((e*F^(c*(a - (b*d)/e))*(d + e*x)^(1/3)*Gamma[7/3, -((b*c*(d + e*x)*Log[F])/e)])/(b^2*c^2*Log[F]^2*(-((b*c*(d
+ e*x)*Log[F])/e))^(1/3)))

________________________________________________________________________________________

Rubi [A]  time = 0.0313849, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2181} \[ -\frac{e \sqrt [3]{d+e x} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{7}{3},-\frac{b c \log (F) (d+e x)}{e}\right )}{b^2 c^2 \log ^2(F) \sqrt [3]{-\frac{b c \log (F) (d+e x)}{e}}} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*(d + e*x)^(4/3),x]

[Out]

-((e*F^(c*(a - (b*d)/e))*(d + e*x)^(1/3)*Gamma[7/3, -((b*c*(d + e*x)*Log[F])/e)])/(b^2*c^2*Log[F]^2*(-((b*c*(d
+ e*x)*Log[F])/e))^(1/3)))

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int F^{c (a+b x)} (d+e x)^{4/3} \, dx &=-\frac{e F^{c \left (a-\frac{b d}{e}\right )} \sqrt [3]{d+e x} \Gamma \left (\frac{7}{3},-\frac{b c (d+e x) \log (F)}{e}\right )}{b^2 c^2 \log ^2(F) \sqrt [3]{-\frac{b c (d+e x) \log (F)}{e}}}\\ \end{align*}

Mathematica [A]  time = 0.0802949, size = 63, normalized size = 0.89 \[ -\frac{(d+e x)^{7/3} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{7}{3},-\frac{b c \log (F) (d+e x)}{e}\right )}{e \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x))*(d + e*x)^(4/3),x]

[Out]

-((F^(c*(a - (b*d)/e))*(d + e*x)^(7/3)*Gamma[7/3, -((b*c*(d + e*x)*Log[F])/e)])/(e*(-((b*c*(d + e*x)*Log[F])/e
))^(7/3)))

________________________________________________________________________________________

Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))*(e*x+d)^(4/3),x)

[Out]

int(F^(c*(b*x+a))*(e*x+d)^(4/3),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{\frac{4}{3}} F^{{\left (b x + a\right )} c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x+d)^(4/3),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(4/3)*F^((b*x + a)*c), x)

________________________________________________________________________________________

Fricas [A]  time = 1.57135, size = 281, normalized size = 3.96 \begin{align*} \frac{\frac{4 \, \left (-\frac{b c \log \left (F\right )}{e}\right )^{\frac{2}{3}} e^{2} \Gamma \left (\frac{1}{3}, -\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right )}{F^{\frac{b c d - a c e}{e}}} - 3 \,{\left (4 \, b c e \log \left (F\right ) - 3 \,{\left (b^{2} c^{2} e x + b^{2} c^{2} d\right )} \log \left (F\right )^{2}\right )}{\left (e x + d\right )}^{\frac{1}{3}} F^{b c x + a c}}{9 \, b^{3} c^{3} \log \left (F\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x+d)^(4/3),x, algorithm="fricas")

[Out]

1/9*(4*(-b*c*log(F)/e)^(2/3)*e^2*gamma(1/3, -(b*c*e*x + b*c*d)*log(F)/e)/F^((b*c*d - a*c*e)/e) - 3*(4*b*c*e*lo
g(F) - 3*(b^2*c^2*e*x + b^2*c^2*d)*log(F)^2)*(e*x + d)^(1/3)*F^(b*c*x + a*c))/(b^3*c^3*log(F)^3)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*(e*x+d)**(4/3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{\frac{4}{3}} F^{{\left (b x + a\right )} c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x+d)^(4/3),x, algorithm="giac")

[Out]

integrate((e*x + d)^(4/3)*F^((b*x + a)*c), x)