3.303 \(\int F^{a+\frac{b}{c+d x}} (c+d x)^3 \, dx\)

Optimal. Leaf size=28 \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{c+d x}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0441052, antiderivative size = 28, 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} \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x))*(c + d*x)^3,x]

[Out]

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x))]*Log[F]^4)/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)^3 \, dx &=\frac{b^4 F^a \Gamma \left (-4,-\frac{b \log (F)}{c+d x}\right ) \log ^4(F)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0057956, size = 28, normalized size = 1. \[ \frac{b^4 F^a \log ^4(F) \text{Gamma}\left (-4,-\frac{b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x))*(c + d*x)^3,x]

[Out]

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

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Maple [B]  time = 0.086, size = 368, normalized size = 13.1 \begin{align*}{\frac{{d}^{3}{F}^{a}{x}^{4}}{4}{F}^{{\frac{b}{dx+c}}}}+{d}^{2}{F}^{a}{F}^{{\frac{b}{dx+c}}}c{x}^{3}+{\frac{3\,d{F}^{a}{c}^{2}{x}^{2}}{2}{F}^{{\frac{b}{dx+c}}}}+{F}^{a}{F}^{{\frac{b}{dx+c}}}{c}^{3}x+{\frac{{F}^{a}{c}^{4}}{4\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{\ln \left ( F \right ) b{d}^{2}{F}^{a}{x}^{3}}{12}{F}^{{\frac{b}{dx+c}}}}+{\frac{\ln \left ( F \right ) bd{F}^{a}c{x}^{2}}{4}{F}^{{\frac{b}{dx+c}}}}+{\frac{b\ln \left ( F \right ){F}^{a}{c}^{2}x}{4}{F}^{{\frac{b}{dx+c}}}}+{\frac{b\ln \left ( F \right ){F}^{a}{c}^{3}}{12\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{2}d \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{a}{x}^{2}}{24}{F}^{{\frac{b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{a}cx}{12}{F}^{{\frac{b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{F}^{a}{c}^{2}}{24\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{a}x}{24}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{a}c}{24\,d}{F}^{{\frac{b}{dx+c}}}}+{\frac{{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{a}}{24\,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)^3,x)

[Out]

1/4*d^3*F^a*F^(b/(d*x+c))*x^4+d^2*F^a*F^(b/(d*x+c))*c*x^3+3/2*d*F^a*F^(b/(d*x+c))*c^2*x^2+F^a*F^(b/(d*x+c))*c^
3*x+1/4/d*F^a*F^(b/(d*x+c))*c^4+1/12*d^2*b*ln(F)*F^a*F^(b/(d*x+c))*x^3+1/4*d*b*ln(F)*F^a*F^(b/(d*x+c))*c*x^2+1
/4*b*ln(F)*F^a*F^(b/(d*x+c))*c^2*x+1/12/d*b*ln(F)*F^a*F^(b/(d*x+c))*c^3+1/24*d*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*x
^2+1/12*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c*x+1/24/d*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c^2+1/24*b^3*ln(F)^3*F^a*F^(b/(
d*x+c))*x+1/24/d*b^3*ln(F)^3*F^a*F^(b/(d*x+c))*c+1/24/d*b^4*ln(F)^4*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}{24} \,{\left (6 \, F^{a} d^{3} x^{4} + 2 \,{\left (F^{a} b d^{2} \log \left (F\right ) + 12 \, F^{a} c d^{2}\right )} x^{3} +{\left (F^{a} b^{2} d \log \left (F\right )^{2} + 6 \, F^{a} b c d \log \left (F\right ) + 36 \, F^{a} c^{2} d\right )} x^{2} +{\left (F^{a} b^{3} \log \left (F\right )^{3} + 2 \, F^{a} b^{2} c \log \left (F\right )^{2} + 6 \, F^{a} b c^{2} \log \left (F\right ) + 24 \, F^{a} c^{3}\right )} x\right )} F^{\frac{b}{d x + c}} + \int \frac{{\left (F^{a} b^{4} d x \log \left (F\right )^{4} - F^{a} b^{3} c^{2} \log \left (F\right )^{3} - 2 \, F^{a} b^{2} c^{3} \log \left (F\right )^{2} - 6 \, F^{a} b c^{4} \log \left (F\right )\right )} F^{\frac{b}{d x + c}}}{24 \,{\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)^3,x, algorithm="maxima")

[Out]

1/24*(6*F^a*d^3*x^4 + 2*(F^a*b*d^2*log(F) + 12*F^a*c*d^2)*x^3 + (F^a*b^2*d*log(F)^2 + 6*F^a*b*c*d*log(F) + 36*
F^a*c^2*d)*x^2 + (F^a*b^3*log(F)^3 + 2*F^a*b^2*c*log(F)^2 + 6*F^a*b*c^2*log(F) + 24*F^a*c^3)*x)*F^(b/(d*x + c)
) + integrate(1/24*(F^a*b^4*d*x*log(F)^4 - F^a*b^3*c^2*log(F)^3 - 2*F^a*b^2*c^3*log(F)^2 - 6*F^a*b*c^4*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)^3,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)**3,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 )}^{3} 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)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*F^(a + b/(d*x + c)), x)