Optimal. Leaf size=49 \[ \frac {F^a \Gamma \left (\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}}} \]
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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} \[ \frac {F^a \text {Gamma}\left (\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}}} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^2} \, dx &=\frac {F^a \Gamma \left (\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ \frac {F^a \Gamma \left (\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 59, normalized size = 1.20 \[ -\frac {F^{a} d \left (-\frac {b \log \relax (F)}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {b \log \relax (F)}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{3 \, b \log \relax (F)} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{a +\frac {b}{\left (d x +c \right )^{3}}}}{\left (d x +c \right )^{2}}\, dx \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 58, normalized size = 1.18 \[ \frac {F^a\,\left (3\,\Gamma \left (\frac {2}{3}\right )\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )-2\,\pi \,\sqrt {3}\right )}{9\,d\,\Gamma \left (\frac {2}{3}\right )\,\left (c+d\,x\right )\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}^{1/3}} \]
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 \[ \int \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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