Optimal. Leaf size=53 \[ \frac {(c+d x)^3 F^{a+\frac {b}{(c+d x)^3}}}{3 d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]
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Rubi [A] time = 0.09, antiderivative size = 53, normalized size of antiderivative = 1.00, 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.
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Rule 2210
Rule 2214
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*}
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Mathematica [A] time = 0.04, 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.
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fricas [B] time = 0.42, size = 141, normalized size = 2.66 \[ -\frac {F^{a} b {\rm Ei}\left (\frac {b \log \relax (F)}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \relax (F) - {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, d} \]
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 \[ \int {\left (d x + c\right )}^{2} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}\,{d x} \]
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 \[ \int \left (d x +c \right )^{2} F^{a +\frac {b}{\left (d x +c \right )^{3}}}\, dx \]
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 \[ \frac {1}{3} \, {\left (F^{a} d^{2} x^{3} + 3 \, F^{a} c d x^{2} + 3 \, F^{a} c^{2} x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int \frac {{\left (F^{a} b d^{3} x^{3} \log \relax (F) + 3 \, F^{a} b c d^{2} x^{2} \log \relax (F) + 3 \, F^{a} b c^{2} d x \log \relax (F)\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.72, size = 51, normalized size = 0.96 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,{\left (c+d\,x\right )}^3}{3\,d}+\frac {F^a\,b\,\ln \relax (F)\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}{3\,d} \]
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 \[ \int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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