Optimal. Leaf size=49 \[ \frac {F^a (c+d x)^4 \left (-\frac {b \log (F)}{(c+d x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, 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} \[ \frac {F^a (c+d x)^4 \left (-\frac {b \log (F)}{(c+d x)^3}\right )^{4/3} \text {Gamma}\left (-\frac {4}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^3 \, dx &=\frac {F^a (c+d x)^4 \Gamma \left (-\frac {4}{3},-\frac {b \log (F)}{(c+d x)^3}\right ) \left (-\frac {b \log (F)}{(c+d x)^3}\right )^{4/3}}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ \frac {F^a (c+d x)^4 \left (-\frac {b \log (F)}{(c+d x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 178, normalized size = 3.63 \[ -\frac {3 \, F^{a} b d \left (-\frac {b \log \relax (F)}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {b \log \relax (F)}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \relax (F) - {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4} + 3 \, {\left (b d x + b c\right )} \log \relax (F)\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{3} F^{a +\frac {b}{\left (d x +c \right )^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\left (F^{a} d^{3} x^{4} + 4 \, F^{a} c d^{2} x^{3} + 6 \, F^{a} c^{2} d x^{2} + {\left (4 \, F^{a} c^{3} + 3 \, F^{a} b \log \relax (F)\right )} x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int -\frac {3 \, {\left (F^{a} b c^{4} \log \relax (F) - 3 \, F^{a} b^{2} d x \log \relax (F)^{2}\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{4 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.92, size = 128, normalized size = 2.61 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,{\left (c+d\,x\right )}^4}{4\,d}-\frac {3\,F^a\,\Gamma \left (\frac {2}{3}\right )\,{\left (c+d\,x\right )}^4\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}^{4/3}}{4\,d}+\frac {3\,F^a\,\Gamma \left (\frac {2}{3},-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )\,{\left (c+d\,x\right )}^4\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}^{4/3}}{4\,d}+\frac {3\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,b\,\ln \relax (F)\,\left (c+d\,x\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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