Optimal. Leaf size=47 \[ \frac {F^a (c+d x) \Gamma \left (-\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right ) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}}}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2239}
\begin {gather*} \frac {F^a (c+d x) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2239
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^3}} \, dx &=\frac {F^a (c+d x) \Gamma \left (-\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right ) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}}}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 47, normalized size = 1.00 \begin {gather*} \frac {F^a (c+d x) \Gamma \left (-\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right ) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}}}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int F^{a +\frac {b}{\left (d x +c \right )^{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (41) = 82\).
time = 0.09, size = 129, normalized size = 2.74 \begin {gather*} -\frac {F^{a} d \left (-\frac {b \log \left (F\right )}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (d x + c\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}}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{a + \frac {b}{\left (c + d x\right )^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.93, size = 71, normalized size = 1.51 \begin {gather*} \frac {F^a\,\left (c+d\,x\right )\,\left (F^{\frac {b}{{\left (c+d\,x\right )}^3}}-\Gamma \left (\frac {2}{3},-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )\,{\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}^{1/3}+\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}^{1/3}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________