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

Optimal. Leaf size=31 \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^n\right )}{d n} \]

[Out]

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

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Rubi [A]  time = 0.0392378, antiderivative size = 31, normalized size of antiderivative = 1., 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} \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0072041, size = 31, normalized size = 1. \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.095, size = 213, normalized size = 6.9 \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{5\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{5}}}-{\frac{b\ln \left ( F \right ){F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{20\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{4}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{60\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{3}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{120\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{2}}}-{\frac{{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{b \left ( dx+c \right ) ^{n}}{F}^{a}}{120\,dn \left ( dx+c \right ) ^{n}}}-{\frac{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}{F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) \right ) }{120\,dn}} \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]

-1/5/n/d*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^5-1/20/n/d*ln(F)*b*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^4-1/60/n/d*ln(F)^2
*b^2*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^3-1/120/n/d*ln(F)^3*b^3*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)^2-1/120/n/d*ln(F)
^4*b^4*F^(b*(d*x+c)^n)*F^a/((d*x+c)^n)-1/120/n/d*ln(F)^5*b^5*F^a*Ei(1,-b*(d*x+c)^n*ln(F))

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Maxima [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="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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]

Exception raised: TypeError

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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)