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 = 0.0659408, 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 \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^n\right )}{d n} \]

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)

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Rubi in Sympy [A]  time = 6.21404, size = 31, normalized size = 1. \[ \frac{F^{a} b^{5} \Gamma{\left (-5,- b \left (c + d x\right )^{n} \log{\left (F \right )} \right )} \log{\left (F \right )}^{5}}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

F**a*b**5*Gamma(-5, -b*(c + d*x)**n*log(F))*log(F)**5/(d*n)

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Mathematica [B]  time = 0.118455, size = 131, normalized size = 4.23 \[ \frac{F^a (c+d x)^{-5 n} \left (b^5 \log ^5(F) (c+d x)^{5 n} \text{ExpIntegralEi}\left (b \log (F) (c+d x)^n\right )-F^{b (c+d x)^n} \left (b^4 \log ^4(F) (c+d x)^{4 n}+b^3 \log ^3(F) (c+d x)^{3 n}+2 b^2 \log ^2(F) (c+d x)^{2 n}+6 b \log (F) (c+d x)^n+24\right )\right )}{120 d n} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(b^5*(c + d*x)^(5*n)*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F]^5 - F^(b*(c
+ d*x)^n)*(24 + 6*b*(c + d*x)^n*Log[F] + 2*b^2*(c + d*x)^(2*n)*Log[F]^2 + b^3*(
c + d*x)^(3*n)*Log[F]^3 + b^4*(c + d*x)^(4*n)*Log[F]^4)))/(120*d*n*(c + d*x)^(5*
n))

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Maple [B]  time = 0.043, size = 208, normalized size = 6.7 \[ -{\frac{{F}^{a+b \left ( dx+c \right ) ^{n}}}{5\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{5}}}-{\frac{b\ln \left ( F \right ){F}^{a+b \left ( dx+c \right ) ^{n}}}{20\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{4}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{a+b \left ( dx+c \right ) ^{n}}}{60\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{3}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{F}^{a+b \left ( dx+c \right ) ^{n}}}{120\,dn \left ( \left ( dx+c \right ) ^{n} \right ) ^{2}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{F}^{a+b \left ( dx+c \right ) ^{n}}}{120\,dn \left ( dx+c \right ) ^{n}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{5}{b}^{5}{F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) \right ) }{120\,dn}} \]

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^(a+b*(d*x+c)^n)/((d*x+c)^n)^5-1/20/n/d*b*ln(F)*F^(a+b*(d*x+c)^n)/((d*
x+c)^n)^4-1/60/n/d*b^2*ln(F)^2*F^(a+b*(d*x+c)^n)/((d*x+c)^n)^3-1/120/n/d*b^3*ln(
F)^3*F^(a+b*(d*x+c)^n)/((d*x+c)^n)^2-1/120/n/d*b^4*ln(F)^4*F^(a+b*(d*x+c)^n)/((d
*x+c)^n)-1/120/n/d*b^5*ln(F)^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. \[ \int{\left (d x + c\right )}^{-5 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256495, size = 185, normalized size = 5.97 \[ \frac{{\left (d x + c\right )}^{5 \, n} F^{a} b^{5}{\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right )^{5} -{\left ({\left (d x + c\right )}^{4 \, n} b^{4} \log \left (F\right )^{4} +{\left (d x + c\right )}^{3 \, n} b^{3} \log \left (F\right )^{3} + 2 \,{\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} + 6 \,{\left (d x + c\right )}^{n} b \log \left (F\right ) + 24\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{120 \,{\left (d x + c\right )}^{5 \, n} d n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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