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

Optimal. Leaf size=61 \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^3\right )^{\frac{1}{3} (-m-1)} \text{Gamma}\left (\frac{m+1}{3},-b \log (F) (c+d x)^3\right )}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.061417, antiderivative size = 61, 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} \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^3\right )^{\frac{1}{3} (-m-1)} \text{Gamma}\left (\frac{m+1}{3},-b \log (F) (c+d x)^3\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^m \, dx &=-\frac{F^a (c+d x)^{1+m} \Gamma \left (\frac{1+m}{3},-b (c+d x)^3 \log (F)\right ) \left (-b (c+d x)^3 \log (F)\right )^{\frac{1}{3} (-1-m)}}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0401902, size = 61, normalized size = 1. \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^3\right )^{\frac{1}{3} (-m-1)} \text{Gamma}\left (\frac{m+1}{3},-b \log (F) (c+d x)^3\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b \left ( dx+c \right ) ^{3}} \left ( dx+c \right ) ^{m}\, 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)^m,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} 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)^m,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.90598, size = 188, normalized size = 3.08 \begin{align*} \frac{e^{\left (-\frac{1}{3} \,{\left (m - 2\right )} \log \left (-b \log \left (F\right )\right ) + a \log \left (F\right )\right )} \Gamma \left (\frac{1}{3} \, m + \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 \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*e^(-1/3*(m - 2)*log(-b*log(F)) + a*log(F))*gamma(1/3*m + 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*log(F))

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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**(a+b*(d*x+c)**3)*(d*x+c)**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} 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)^m,x, algorithm="giac")

[Out]

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