3.368 \(\int F^{a+b (c+d x)^n} (c+d x)^{-1+6 n} \, dx\)

Optimal. Leaf size=114 \[ -\frac{F^{a+b (c+d x)^n} \left (60 b^2 \log ^2(F) (c+d x)^{2 n}-20 b^3 \log ^3(F) (c+d x)^{3 n}+5 b^4 \log ^4(F) (c+d x)^{4 n}-b^5 \log ^5(F) (c+d x)^{5 n}-120 b \log (F) (c+d x)^n+120\right )}{b^6 d n \log ^6(F)} \]

[Out]

-((F^(a + b*(c + d*x)^n)*(120 - 120*b*(c + d*x)^n*Log[F] + 60*b^2*(c + d*x)^(2*n)*Log[F]^2 - 20*b^3*(c + d*x)^
(3*n)*Log[F]^3 + 5*b^4*(c + d*x)^(4*n)*Log[F]^4 - b^5*(c + d*x)^(5*n)*Log[F]^5))/(b^6*d*n*Log[F]^6))

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Rubi [C]  time = 0.0375697, antiderivative size = 32, normalized size of antiderivative = 0.28, 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} \[ -\frac{F^a \text{Gamma}\left (6,-b \log (F) (c+d x)^n\right )}{b^6 d n \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 6*n),x]

[Out]

-((F^a*Gamma[6, -(b*(c + d*x)^n*Log[F])])/(b^6*d*n*Log[F]^6))

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+6 n} \, dx &=-\frac{F^a \Gamma \left (6,-b (c+d x)^n \log (F)\right )}{b^6 d n \log ^6(F)}\\ \end{align*}

Mathematica [C]  time = 0.0066138, size = 32, normalized size = 0.28 \[ -\frac{F^a \text{Gamma}\left (6,-b \log (F) (c+d x)^n\right )}{b^6 d n \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 + 6*n),x]

[Out]

-((F^a*Gamma[6, -(b*(c + d*x)^n*Log[F])])/(b^6*d*n*Log[F]^6))

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Maple [A]  time = 0.027, size = 113, normalized size = 1. \begin{align*}{\frac{ \left ( \left ( \left ( dx+c \right ) ^{n} \right ) ^{5}{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}-5\, \left ( \left ( dx+c \right ) ^{n} \right ) ^{4}{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}+20\, \left ( \left ( dx+c \right ) ^{n} \right ) ^{3}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}-60\, \left ( \left ( dx+c \right ) ^{n} \right ) ^{2}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+120\,b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) -120 \right ){F}^{a+b \left ( dx+c \right ) ^{n}}}{{b}^{6} \left ( \ln \left ( F \right ) \right ) ^{6}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+6*n),x)

[Out]

(((d*x+c)^n)^5*b^5*ln(F)^5-5*((d*x+c)^n)^4*b^4*ln(F)^4+20*((d*x+c)^n)^3*b^3*ln(F)^3-60*((d*x+c)^n)^2*b^2*ln(F)
^2+120*b*(d*x+c)^n*ln(F)-120)/b^6/ln(F)^6/n/d*F^(a+b*(d*x+c)^n)

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Maxima [A]  time = 1.02355, size = 174, normalized size = 1.53 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{5 \, n} F^{a} b^{5} \log \left (F\right )^{5} - 5 \,{\left (d x + c\right )}^{4 \, n} F^{a} b^{4} \log \left (F\right )^{4} + 20 \,{\left (d x + c\right )}^{3 \, n} F^{a} b^{3} \log \left (F\right )^{3} - 60 \,{\left (d x + c\right )}^{2 \, n} F^{a} b^{2} \log \left (F\right )^{2} + 120 \,{\left (d x + c\right )}^{n} F^{a} b \log \left (F\right ) - 120 \, F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{6} d n \log \left (F\right )^{6}} \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+6*n),x, algorithm="maxima")

[Out]

((d*x + c)^(5*n)*F^a*b^5*log(F)^5 - 5*(d*x + c)^(4*n)*F^a*b^4*log(F)^4 + 20*(d*x + c)^(3*n)*F^a*b^3*log(F)^3 -
60*(d*x + c)^(2*n)*F^a*b^2*log(F)^2 + 120*(d*x + c)^n*F^a*b*log(F) - 120*F^a)*F^((d*x + c)^n*b)/(b^6*d*n*log(
F)^6)

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Fricas [A]  time = 1.62008, size = 298, normalized size = 2.61 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{5 \, n} b^{5} \log \left (F\right )^{5} - 5 \,{\left (d x + c\right )}^{4 \, n} b^{4} \log \left (F\right )^{4} + 20 \,{\left (d x + c\right )}^{3 \, n} b^{3} \log \left (F\right )^{3} - 60 \,{\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} + 120 \,{\left (d x + c\right )}^{n} b \log \left (F\right ) - 120\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{6} d n \log \left (F\right )^{6}} \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+6*n),x, algorithm="fricas")

[Out]

((d*x + c)^(5*n)*b^5*log(F)^5 - 5*(d*x + c)^(4*n)*b^4*log(F)^4 + 20*(d*x + c)^(3*n)*b^3*log(F)^3 - 60*(d*x + c
)^(2*n)*b^2*log(F)^2 + 120*(d*x + c)^n*b*log(F) - 120)*e^((d*x + c)^n*b*log(F) + a*log(F))/(b^6*d*n*log(F)^6)

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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+6*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 )}^{6 \, 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+6*n),x, algorithm="giac")

[Out]

integrate((d*x + c)^(6*n - 1)*F^((d*x + c)^n*b + a), x)