Optimal. Leaf size=53 \[ \frac {(c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{2 d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2214, 2210} \[ \frac {(c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{2 d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int F^{a+\frac {b}{(c+d x)^2}} (c+d x) \, dx &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2}{2 d}+(b \log (F)) \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{c+d x} \, dx\\ &=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2}{2 d}-\frac {b F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right ) \log (F)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 0.89 \[ \frac {F^a \left ((c+d x)^2 F^{\frac {b}{(c+d x)^2}}-b \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 96, normalized size = 1.81 \[ -\frac {F^{a} b {\rm Ei}\left (\frac {b \log \relax (F)}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \relax (F) - {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, d} \]
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 \[ \int {\left (d x + c\right )} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 86, normalized size = 1.62 \[ \frac {d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2}+c x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b \,F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}{2 d}+\frac {c^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2 d} \]
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 \[ \frac {1}{2} \, {\left (F^{a} d x^{2} + 2 \, F^{a} c x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {{\left (F^{a} b d^{2} x^{2} \log \relax (F) + 2 \, F^{a} b c d x \log \relax (F)\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.45, size = 51, normalized size = 0.96 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^2}{2\,d}+\frac {F^a\,b\,\ln \relax (F)\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}{2\,d} \]
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 \[ \int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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