Optimal. Leaf size=121 \[ \frac {b^3 F^a \log ^3(F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{12 d}-\frac {b^2 \log ^2(F) F^{a+b (c+d x)^2}}{12 d (c+d x)^2}-\frac {F^{a+b (c+d x)^2}}{6 d (c+d x)^6}-\frac {b \log (F) F^{a+b (c+d x)^2}}{12 d (c+d x)^4} \]
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Rubi [A] time = 0.26, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2214, 2210} \[ \frac {b^3 F^a \log ^3(F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{12 d}-\frac {b^2 \log ^2(F) F^{a+b (c+d x)^2}}{12 d (c+d x)^2}-\frac {F^{a+b (c+d x)^2}}{6 d (c+d x)^6}-\frac {b \log (F) F^{a+b (c+d x)^2}}{12 d (c+d x)^4} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^7} \, dx &=-\frac {F^{a+b (c+d x)^2}}{6 d (c+d x)^6}+\frac {1}{3} (b \log (F)) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^5} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{6 d (c+d x)^6}-\frac {b F^{a+b (c+d x)^2} \log (F)}{12 d (c+d x)^4}+\frac {1}{6} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^3} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{6 d (c+d x)^6}-\frac {b F^{a+b (c+d x)^2} \log (F)}{12 d (c+d x)^4}-\frac {b^2 F^{a+b (c+d x)^2} \log ^2(F)}{12 d (c+d x)^2}+\frac {1}{6} \left (b^3 \log ^3(F)\right ) \int \frac {F^{a+b (c+d x)^2}}{c+d x} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{6 d (c+d x)^6}-\frac {b F^{a+b (c+d x)^2} \log (F)}{12 d (c+d x)^4}-\frac {b^2 F^{a+b (c+d x)^2} \log ^2(F)}{12 d (c+d x)^2}+\frac {b^3 F^a \text {Ei}\left (b (c+d x)^2 \log (F)\right ) \log ^3(F)}{12 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 79, normalized size = 0.65 \[ \frac {F^a \left (b^3 \log ^3(F) \text {Ei}\left (b (c+d x)^2 \log (F)\right )-\frac {F^{b (c+d x)^2} \left (b^2 \log ^2(F) (c+d x)^4+b \log (F) (c+d x)^2+2\right )}{(c+d x)^6}\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 292, normalized size = 2.41 \[ \frac {{\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} F^{a} {\rm Ei}\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right ) \log \relax (F)^{3} - {\left ({\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \relax (F)^{2} + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F) + 2\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{12 \, {\left (d^{7} x^{6} + 6 \, c d^{6} x^{5} + 15 \, c^{2} d^{5} x^{4} + 20 \, c^{3} d^{4} x^{3} + 15 \, c^{4} d^{3} x^{2} + 6 \, c^{5} d^{2} x + c^{6} d\right )}} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 119, normalized size = 0.98 \[ -\frac {b^{3} F^{a} \Ei \left (1, -\left (d x +c \right )^{2} b \ln \relax (F )\right ) \ln \relax (F )^{3}}{12 d}-\frac {b^{2} F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )^{2}}{12 \left (d x +c \right )^{2} d}-\frac {b \,F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )}{12 \left (d x +c \right )^{4} d}-\frac {F^{a} F^{\left (d x +c \right )^{2} b}}{6 \left (d x +c \right )^{6} d} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.82, size = 104, normalized size = 0.86 \[ -\frac {F^a\,b^3\,{\ln \relax (F)}^3\,\mathrm {expint}\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}{12\,d}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^3\,{\ln \relax (F)}^3\,\left (\frac {1}{6\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2}+\frac {1}{6\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^4}+\frac {1}{3\,b^3\,{\ln \relax (F)}^3\,{\left (c+d\,x\right )}^6}\right )}{2\,d} \]
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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