Optimal. Leaf size=54 \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b (c+d x)^n \log (F)\right )}{d n (c+d x)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 54, 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} \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{2/n} \text {Gamma}\left (-\frac {2}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^n}}{(c+d x)^3} \, dx &=-\frac {F^a \Gamma \left (-\frac {2}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{2/n}}{d n (c+d x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 1.00 \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-b (c+d x)^n \log (F)\right )}{d n (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{{\left (d x + c\right )}^{n} b + a}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, x\right ) \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \left (d x +c \right )^{n}+a}}{\left (d x +c \right )^{3}}\, dx \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.72, size = 67, normalized size = 1.24 \[ -\frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n}{2}}\,{\mathrm {M}}_{\frac {1}{n}+\frac {1}{2},-\frac {1}{n}}\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n\right )\,{\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n\right )}^{\frac {1}{n}-\frac {1}{2}}}{2\,d\,{\left (c+d\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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