Optimal. Leaf size=85 \[ -\frac {b^2 F^a \log ^2(F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{2 d}+\frac {(c+d x)^2 F^{a+\frac {b}{c+d x}}}{2 d}+\frac {b \log (F) (c+d x) F^{a+\frac {b}{c+d x}}}{2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2214, 2206, 2210} \[ -\frac {b^2 F^a \log ^2(F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )}{2 d}+\frac {(c+d x)^2 F^{a+\frac {b}{c+d x}}}{2 d}+\frac {b \log (F) (c+d x) F^{a+\frac {b}{c+d x}}}{2 d} \]
Antiderivative was successfully verified.
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Rule 2206
Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int F^{a+\frac {b}{c+d x}} (c+d x) \, dx &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)^2}{2 d}+\frac {1}{2} (b \log (F)) \int F^{a+\frac {b}{c+d x}} \, dx\\ &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)^2}{2 d}+\frac {b F^{a+\frac {b}{c+d x}} (c+d x) \log (F)}{2 d}+\frac {1}{2} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx\\ &=\frac {F^{a+\frac {b}{c+d x}} (c+d x)^2}{2 d}+\frac {b F^{a+\frac {b}{c+d x}} (c+d x) \log (F)}{2 d}-\frac {b^2 F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log ^2(F)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.68 \[ \frac {F^a \left ((c+d x) F^{\frac {b}{c+d x}} (b \log (F)+c+d x)-b^2 \log ^2(F) \text {Ei}\left (\frac {b \log (F)}{c+d x}\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 77, normalized size = 0.91 \[ -\frac {F^{a} b^{2} {\rm Ei}\left (\frac {b \log \relax (F)}{d x + c}\right ) \log \relax (F)^{2} - {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + {\left (b d x + b c\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{2 \, d} \]
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 \[ \int {\left (d x + c\right )} F^{a + \frac {b}{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 133, normalized size = 1.56 \[ \frac {b^{2} F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{d x +c}\right ) \ln \relax (F )^{2}}{2 d}+\frac {b x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{2}+\frac {d \,x^{2} F^{a} F^{\frac {b}{d x +c}}}{2}+\frac {b c \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{2 d}+c x \,F^{a} F^{\frac {b}{d x +c}}+\frac {c^{2} F^{a} F^{\frac {b}{d x +c}}}{2 d} \]
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 \[ \frac {1}{2} \, {\left (F^{a} d x^{2} + {\left (F^{a} b \log \relax (F) + 2 \, F^{a} c\right )} x\right )} F^{\frac {b}{d x + c}} + \int \frac {{\left (F^{a} b^{2} d x \log \relax (F)^{2} - F^{a} b c^{2} \log \relax (F)\right )} F^{\frac {b}{d x + c}}}{2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.11, size = 82, normalized size = 0.96 \[ \frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^2}{2\,d}+\frac {F^a\,b^2\,{\ln \relax (F)}^2\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{c+d\,x}\right )}{2\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \relax (F)\,\left (c+d\,x\right )}{2\,d} \]
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 \[ \int F^{a + \frac {b}{c + d x}} \left (c + d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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