Optimal. Leaf size=136 \[ \frac {4 \sqrt {\pi } b^{5/2} F^a \log ^{\frac {5}{2}}(F) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{15 d}-\frac {4 b^2 \log ^2(F) F^{a+b (c+d x)^2}}{15 d (c+d x)}-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}-\frac {2 b \log (F) F^{a+b (c+d x)^2}}{15 d (c+d x)^3} \]
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Rubi [A] time = 0.22, antiderivative size = 136, 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, 2204} \[ \frac {4 \sqrt {\pi } b^{5/2} F^a \log ^{\frac {5}{2}}(F) \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{15 d}-\frac {4 b^2 \log ^2(F) F^{a+b (c+d x)^2}}{15 d (c+d x)}-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}-\frac {2 b \log (F) F^{a+b (c+d x)^2}}{15 d (c+d x)^3} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2214
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx &=-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}+\frac {1}{5} (2 b \log (F)) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^4} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{15 d (c+d x)^3}+\frac {1}{15} \left (4 b^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^2} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{15 d (c+d x)^3}-\frac {4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{15 d (c+d x)}+\frac {1}{15} \left (8 b^3 \log ^3(F)\right ) \int F^{a+b (c+d x)^2} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{15 d (c+d x)^3}-\frac {4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{15 d (c+d x)}+\frac {4 b^{5/2} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {5}{2}}(F)}{15 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 97, normalized size = 0.71 \[ \frac {F^a \left (4 \sqrt {\pi } b^{5/2} \log ^{\frac {5}{2}}(F) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )-\frac {F^{b (c+d x)^2} \left (4 b^2 \log ^2(F) (c+d x)^4+2 b \log (F) (c+d x)^2+3\right )}{(c+d x)^5}\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 288, normalized size = 2.12 \[ -\frac {4 \, \sqrt {\pi } {\left (b^{2} d^{5} x^{5} + 5 \, b^{2} c d^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} x^{3} + 10 \, b^{2} c^{3} d^{2} x^{2} + 5 \, b^{2} c^{4} d x + b^{2} c^{5}\right )} \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) \log \relax (F)^{2} + {\left (4 \, {\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x + b^{2} c^{4} d\right )} \log \relax (F)^{2} + 2 \, {\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \log \relax (F) + 3 \, d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{15 \, {\left (d^{7} x^{5} + 5 \, c d^{6} x^{4} + 10 \, c^{2} d^{5} x^{3} + 10 \, c^{3} d^{4} x^{2} + 5 \, c^{4} d^{3} x + c^{5} d^{2}\right )}} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 129, normalized size = 0.95 \[ \frac {4 \sqrt {\pi }\, b^{3} F^{a} \erf \left (\sqrt {-b \ln \relax (F )}\, \left (d x +c \right )\right ) \ln \relax (F )^{3}}{15 \sqrt {-b \ln \relax (F )}\, d}-\frac {4 b^{2} F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )^{2}}{15 \left (d x +c \right ) d}-\frac {2 b \,F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )}{15 \left (d x +c \right )^{3} d}-\frac {F^{a} F^{\left (d x +c \right )^{2} b}}{5 \left (d x +c \right )^{5} d} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.87, size = 168, normalized size = 1.24 \[ \frac {4\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2}\right )\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}^{5/2}}{15\,d\,{\left (c+d\,x\right )}^5}-\frac {4\,F^a\,\sqrt {\pi }\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}^{5/2}}{15\,d\,{\left (c+d\,x\right )}^5}-\frac {4\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^2\,{\ln \relax (F)}^2}{15\,d\,\left (c+d\,x\right )}-\frac {2\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b\,\ln \relax (F)}{15\,d\,{\left (c+d\,x\right )}^3}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^2}}{5\,d\,{\left (c+d\,x\right )}^5} \]
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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