3.302 \(\int F^{a+\frac{b}{c+d x}} (c+d x)^4 \, dx\)

Optimal. Leaf size=29 \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{c+d x}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0446677, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x))*(c + d*x)^4,x]

[Out]

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

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+\frac{b}{c+d x}} (c+d x)^4 \, dx &=-\frac{b^5 F^a \Gamma \left (-5,-\frac{b \log (F)}{c+d x}\right ) \log ^5(F)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0060069, size = 29, normalized size = 1. \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x))*(c + d*x)^4,x]

[Out]

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

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Maple [B]  time = 0.122, size = 534, normalized size = 18.4 \begin{align*}{\frac{{d}^{4}{F}^{a}{x}^{5}}{5}{F}^{{\frac{b}{dx+c}}}}+{d}^{3}{F}^{a}{F}^{{\frac{b}{dx+c}}}c{x}^{4}+2\,{d}^{2}{F}^{a}{F}^{{\frac{b}{dx+c}}}{c}^{2}{x}^{3}+2\,d{F}^{a}{F}^{{\frac{b}{dx+c}}}{c}^{3}{x}^{2}+{F}^{a}{F}^{{\frac{b}{dx+c}}}{c}^{4}x+{\frac{{F}^{a}{c}^{5}}{5\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{\ln \left ( F \right ) b{d}^{3}{F}^{a}{x}^{4}}{20}{F}^{{\frac{b}{dx+c}}}}+{\frac{\ln \left ( F \right ) b{d}^{2}{F}^{a}c{x}^{3}}{5}{F}^{{\frac{b}{dx+c}}}}+{\frac{3\,\ln \left ( F \right ) bd{F}^{a}{c}^{2}{x}^{2}}{10}{F}^{{\frac{b}{dx+c}}}}+{\frac{b\ln \left ( F \right ){F}^{a}{c}^{3}x}{5}{F}^{{\frac{b}{dx+c}}}}+{\frac{b\ln \left ( F \right ){F}^{a}{c}^{4}}{20\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{2}{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{a}{x}^{3}}{60}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{2}d \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{a}c{x}^{2}}{20}{F}^{{\frac{b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{a}{c}^{2}x}{20}{F}^{{\frac{b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{a}{c}^{3}}{60\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{3}d \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{a}{x}^{2}}{120}{F}^{{\frac{b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{F}^{a}cx}{60}{F}^{{\frac{b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{F}^{a}{c}^{2}}{120\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{a}x}{120}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{a}c}{120\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}{F}^{a}}{120\,d}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c))*(d*x+c)^4,x)

[Out]

1/5*d^4*F^a*F^(b/(d*x+c))*x^5+d^3*F^a*F^(b/(d*x+c))*c*x^4+2*d^2*F^a*F^(b/(d*x+c))*c^2*x^3+2*d*F^a*F^(b/(d*x+c)
)*c^3*x^2+F^a*F^(b/(d*x+c))*c^4*x+1/5/d*F^a*F^(b/(d*x+c))*c^5+1/20*d^3*b*ln(F)*F^a*F^(b/(d*x+c))*x^4+1/5*d^2*b
*ln(F)*F^a*F^(b/(d*x+c))*c*x^3+3/10*d*b*ln(F)*F^a*F^(b/(d*x+c))*c^2*x^2+1/5*b*ln(F)*F^a*F^(b/(d*x+c))*c^3*x+1/
20/d*b*ln(F)*F^a*F^(b/(d*x+c))*c^4+1/60*d^2*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*x^3+1/20*d*b^2*ln(F)^2*F^a*F^(b/(d*x
+c))*c*x^2+1/20*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c^2*x+1/60/d*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c^3+1/120*d*b^3*ln(F)
^3*F^a*F^(b/(d*x+c))*x^2+1/60*b^3*ln(F)^3*F^a*F^(b/(d*x+c))*c*x+1/120/d*b^3*ln(F)^3*F^a*F^(b/(d*x+c))*c^2+1/12
0*b^4*ln(F)^4*F^a*F^(b/(d*x+c))*x+1/120/d*b^4*ln(F)^4*F^a*F^(b/(d*x+c))*c+1/120/d*b^5*ln(F)^5*F^a*Ei(1,-b*ln(F
)/(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{120} \,{\left (24 \, F^{a} d^{4} x^{5} + 6 \,{\left (F^{a} b d^{3} \log \left (F\right ) + 20 \, F^{a} c d^{3}\right )} x^{4} + 2 \,{\left (F^{a} b^{2} d^{2} \log \left (F\right )^{2} + 12 \, F^{a} b c d^{2} \log \left (F\right ) + 120 \, F^{a} c^{2} d^{2}\right )} x^{3} +{\left (F^{a} b^{3} d \log \left (F\right )^{3} + 6 \, F^{a} b^{2} c d \log \left (F\right )^{2} + 36 \, F^{a} b c^{2} d \log \left (F\right ) + 240 \, F^{a} c^{3} d\right )} x^{2} +{\left (F^{a} b^{4} \log \left (F\right )^{4} + 2 \, F^{a} b^{3} c \log \left (F\right )^{3} + 6 \, F^{a} b^{2} c^{2} \log \left (F\right )^{2} + 24 \, F^{a} b c^{3} \log \left (F\right ) + 120 \, F^{a} c^{4}\right )} x\right )} F^{\frac{b}{d x + c}} + \int \frac{{\left (F^{a} b^{5} d x \log \left (F\right )^{5} - F^{a} b^{4} c^{2} \log \left (F\right )^{4} - 2 \, F^{a} b^{3} c^{3} \log \left (F\right )^{3} - 6 \, F^{a} b^{2} c^{4} \log \left (F\right )^{2} - 24 \, F^{a} b c^{5} \log \left (F\right )\right )} F^{\frac{b}{d x + c}}}{120 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c))*(d*x+c)^4,x, algorithm="maxima")

[Out]

1/120*(24*F^a*d^4*x^5 + 6*(F^a*b*d^3*log(F) + 20*F^a*c*d^3)*x^4 + 2*(F^a*b^2*d^2*log(F)^2 + 12*F^a*b*c*d^2*log
(F) + 120*F^a*c^2*d^2)*x^3 + (F^a*b^3*d*log(F)^3 + 6*F^a*b^2*c*d*log(F)^2 + 36*F^a*b*c^2*d*log(F) + 240*F^a*c^
3*d)*x^2 + (F^a*b^4*log(F)^4 + 2*F^a*b^3*c*log(F)^3 + 6*F^a*b^2*c^2*log(F)^2 + 24*F^a*b*c^3*log(F) + 120*F^a*c
^4)*x)*F^(b/(d*x + c)) + integrate(1/120*(F^a*b^5*d*x*log(F)^5 - F^a*b^4*c^2*log(F)^4 - 2*F^a*b^3*c^3*log(F)^3
- 6*F^a*b^2*c^4*log(F)^2 - 24*F^a*b*c^5*log(F))*F^(b/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), 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))*(d*x+c)^4,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))*(d*x+c)**4,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 )}^{4} F^{a + \frac{b}{d x + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c))*(d*x+c)^4,x, algorithm="giac")

[Out]

integrate((d*x + c)^4*F^(a + b/(d*x + c)), x)