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]
________________________________________________________________________________________
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]
[Out]
Rule 2218
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]