3.369 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1+5 n} \, dx\)

Optimal. Leaf size=94 \[ \frac{F^{a+b (c+d x)^n} \left (12 b^2 \log ^2(F) (c+d x)^{2 n}-4 b^3 \log ^3(F) (c+d x)^{3 n}+b^4 \log ^4(F) (c+d x)^{4 n}-24 b \log (F) (c+d x)^n+24\right )}{b^5 d n \log ^5(F)} \]

[Out]

(F^(a + b*(c + d*x)^n)*(24 - 24*b*(c + d*x)^n*Log[F] + 12*b^2*(c + d*x)^(2*n)*Log[F]^2 - 4*b^3*(c + d*x)^(3*n)
*Log[F]^3 + b^4*(c + d*x)^(4*n)*Log[F]^4))/(b^5*d*n*Log[F]^5)

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Rubi [C]  time = 0.0396707, antiderivative size = 31, normalized size of antiderivative = 0.33, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2218} \[ \frac{F^a \text{Gamma}\left (5,-b \log (F) (c+d x)^n\right )}{b^5 d n \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 5*n),x]

[Out]

(F^a*Gamma[5, -(b*(c + d*x)^n*Log[F])])/(b^5*d*n*Log[F]^5)

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)^n} (c+d x)^{-1+5 n} \, dx &=\frac{F^a \Gamma \left (5,-b (c+d x)^n \log (F)\right )}{b^5 d n \log ^5(F)}\\ \end{align*}

Mathematica [C]  time = 0.0068639, size = 31, normalized size = 0.33 \[ \frac{F^a \text{Gamma}\left (5,-b \log (F) (c+d x)^n\right )}{b^5 d n \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 5*n),x]

[Out]

(F^a*Gamma[5, -(b*(c + d*x)^n*Log[F])])/(b^5*d*n*Log[F]^5)

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Maple [A]  time = 0.022, size = 95, normalized size = 1. \begin{align*}{\frac{ \left ( \left ( \left ( dx+c \right ) ^{n} \right ) ^{4}{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}-4\, \left ( \left ( dx+c \right ) ^{n} \right ) ^{3}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+12\, \left ( \left ( dx+c \right ) ^{n} \right ) ^{2}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-24\,b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) +24 \right ){F}^{a+b \left ( dx+c \right ) ^{n}}}{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}nd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+5*n),x)

[Out]

(((d*x+c)^n)^4*b^4*ln(F)^4-4*((d*x+c)^n)^3*b^3*ln(F)^3+12*((d*x+c)^n)^2*b^2*ln(F)^2-24*b*(d*x+c)^n*ln(F)+24)/b
^5/ln(F)^5/n/d*F^(a+b*(d*x+c)^n)

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Maxima [A]  time = 1.03157, size = 146, normalized size = 1.55 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{4 \, n} F^{a} b^{4} \log \left (F\right )^{4} - 4 \,{\left (d x + c\right )}^{3 \, n} F^{a} b^{3} \log \left (F\right )^{3} + 12 \,{\left (d x + c\right )}^{2 \, n} F^{a} b^{2} \log \left (F\right )^{2} - 24 \,{\left (d x + c\right )}^{n} F^{a} b \log \left (F\right ) + 24 \, F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{5} d n \log \left (F\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+5*n),x, algorithm="maxima")

[Out]

((d*x + c)^(4*n)*F^a*b^4*log(F)^4 - 4*(d*x + c)^(3*n)*F^a*b^3*log(F)^3 + 12*(d*x + c)^(2*n)*F^a*b^2*log(F)^2 -
24*(d*x + c)^n*F^a*b*log(F) + 24*F^a)*F^((d*x + c)^n*b)/(b^5*d*n*log(F)^5)

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Fricas [A]  time = 1.57409, size = 250, normalized size = 2.66 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{4 \, n} b^{4} \log \left (F\right )^{4} - 4 \,{\left (d x + c\right )}^{3 \, n} b^{3} \log \left (F\right )^{3} + 12 \,{\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} - 24 \,{\left (d x + c\right )}^{n} b \log \left (F\right ) + 24\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{5} d n \log \left (F\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+5*n),x, algorithm="fricas")

[Out]

((d*x + c)^(4*n)*b^4*log(F)^4 - 4*(d*x + c)^(3*n)*b^3*log(F)^3 + 12*(d*x + c)^(2*n)*b^2*log(F)^2 - 24*(d*x + c
)^n*b*log(F) + 24)*e^((d*x + c)^n*b*log(F) + a*log(F))/(b^5*d*n*log(F)^5)

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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)**n)*(d*x+c)**(-1+5*n),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 )}^{5 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1+5*n),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5*n - 1)*F^((d*x + c)^n*b + a), x)