Optimal. Leaf size=53 \[ \frac{(c+d x)^3 F^{a+\frac{b}{(c+d x)^3}}}{3 d}-\frac{b F^a \log (F) \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0910202, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2214, 2210} \[ \frac{(c+d x)^3 F^{a+\frac{b}{(c+d x)^3}}}{3 d}-\frac{b F^a \log (F) \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2214
Rule 2210
Rubi steps
\begin{align*} \int F^{a+\frac{b}{(c+d x)^3}} (c+d x)^2 \, dx &=\frac{F^{a+\frac{b}{(c+d x)^3}} (c+d x)^3}{3 d}+(b \log (F)) \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{c+d x} \, dx\\ &=\frac{F^{a+\frac{b}{(c+d x)^3}} (c+d x)^3}{3 d}-\frac{b F^a \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right ) \log (F)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0356271, size = 47, normalized size = 0.89 \[ \frac{F^a \left ((c+d x)^3 F^{\frac{b}{(c+d x)^3}}-b \log (F) \text{Ei}\left (\frac{b \log (F)}{(c+d x)^3}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) ^{2}\, dx \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}{3} \,{\left (F^{a} d^{2} x^{3} + 3 \, F^{a} c d x^{2} + 3 \, F^{a} c^{2} x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int \frac{{\left (F^{a} b d^{3} x^{3} \log \left (F\right ) + 3 \, F^{a} b c d^{2} x^{2} \log \left (F\right ) + 3 \, F^{a} b c^{2} d x \log \left (F\right )\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.67173, size = 300, normalized size = 5.66 \begin{align*} -\frac{F^{a} b{\rm Ei}\left (\frac{b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right ) -{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} F^{\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, d} \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 )}^{2} F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]