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

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

[Out]

-F^a/((d*x+c)^n)^5*Ei(6,-b*(d*x+c)^n*ln(F))/d/n

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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 = 0.01, size = 31, normalized size = 1.00 \[ \frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-b (c+d x)^n \log (F)\right )}{d n} \]

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)

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 137, normalized size = 4.42 \[ \frac {{\left (d x + c\right )}^{5 \, n} F^{a} b^{5} {\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \relax (F)\right ) \log \relax (F)^{5} - {\left ({\left (d x + c\right )}^{4 \, n} b^{4} \log \relax (F)^{4} + {\left (d x + c\right )}^{3 \, n} b^{3} \log \relax (F)^{3} + 2 \, {\left (d x + c\right )}^{2 \, n} b^{2} \log \relax (F)^{2} + 6 \, {\left (d x + c\right )}^{n} b \log \relax (F) + 24\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \relax (F) + a \log \relax (F)\right )}}{120 \, {\left (d x + c\right )}^{5 \, n} d n} \]

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]

1/120*((d*x + c)^(5*n)*F^a*b^5*Ei((d*x + c)^n*b*log(F))*log(F)^5 - ((d*x + c)^(4*n)*b^4*log(F)^4 + (d*x + c)^(
3*n)*b^3*log(F)^3 + 2*(d*x + c)^(2*n)*b^2*log(F)^2 + 6*(d*x + c)^n*b*log(F) + 24)*e^((d*x + c)^n*b*log(F) + a*
log(F)))/((d*x + c)^(5*n)*d*n)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{-5 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

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)

________________________________________________________________________________________

maple [B]  time = 0.12, size = 213, normalized size = 6.87 \[ -\frac {b^{5} F^{a} \Ei \left (1, -b \left (d x +c \right )^{n} \ln \relax (F )\right ) \ln \relax (F )^{5}}{120 d n}-\frac {b^{4} F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-n} \ln \relax (F )^{4}}{120 d n}-\frac {b^{3} F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-2 n} \ln \relax (F )^{3}}{120 d n}-\frac {b^{2} F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-3 n} \ln \relax (F )^{2}}{60 d n}-\frac {b \,F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-4 n} \ln \relax (F )}{20 d n}-\frac {F^{a} F^{b \left (d x +c \right )^{n}} \left (d x +c \right )^{-5 n}}{5 d n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(b*(d*x+c)^n+a)*(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))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{-5 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {F^{a+b\,{\left (c+d\,x\right )}^n}}{{\left (c+d\,x\right )}^{5\,n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________