Optimal. Leaf size=170 \[ \frac {8 \sqrt {\pi } b^{7/2} F^a \log ^{\frac {7}{2}}(F) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{105 d}-\frac {8 b^3 \log ^3(F) F^{a+b (c+d x)^2}}{105 d (c+d x)}-\frac {4 b^2 \log ^2(F) F^{a+b (c+d x)^2}}{105 d (c+d x)^3}-\frac {F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac {2 b \log (F) F^{a+b (c+d x)^2}}{35 d (c+d x)^5} \]
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Rubi [A] time = 0.29, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2214, 2204} \[ \frac {8 \sqrt {\pi } b^{7/2} F^a \log ^{\frac {7}{2}}(F) \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{105 d}-\frac {8 b^3 \log ^3(F) F^{a+b (c+d x)^2}}{105 d (c+d x)}-\frac {4 b^2 \log ^2(F) F^{a+b (c+d x)^2}}{105 d (c+d x)^3}-\frac {F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac {2 b \log (F) F^{a+b (c+d x)^2}}{35 d (c+d x)^5} \]
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)^8} \, dx &=-\frac {F^{a+b (c+d x)^2}}{7 d (c+d x)^7}+\frac {1}{7} (2 b \log (F)) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{35 d (c+d x)^5}+\frac {1}{35} \left (4 b^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^4} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{35 d (c+d x)^5}-\frac {4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{105 d (c+d x)^3}+\frac {1}{105} \left (8 b^3 \log ^3(F)\right ) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^2} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{35 d (c+d x)^5}-\frac {4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{105 d (c+d x)^3}-\frac {8 b^3 F^{a+b (c+d x)^2} \log ^3(F)}{105 d (c+d x)}+\frac {1}{105} \left (16 b^4 \log ^4(F)\right ) \int F^{a+b (c+d x)^2} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{35 d (c+d x)^5}-\frac {4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{105 d (c+d x)^3}-\frac {8 b^3 F^{a+b (c+d x)^2} \log ^3(F)}{105 d (c+d x)}+\frac {8 b^{7/2} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {7}{2}}(F)}{105 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 112, normalized size = 0.66 \[ \frac {F^a \left (8 \sqrt {\pi } b^{7/2} \log ^{\frac {7}{2}}(F) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )+\frac {F^{b (c+d x)^2} \left (-8 b^3 \log ^3(F) (c+d x)^6-4 b^2 \log ^2(F) (c+d x)^4-6 b \log (F) (c+d x)^2-15\right )}{(c+d x)^7}\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 429, normalized size = 2.52 \[ -\frac {8 \, \sqrt {\pi } {\left (b^{3} d^{7} x^{7} + 7 \, b^{3} c d^{6} x^{6} + 21 \, b^{3} c^{2} d^{5} x^{5} + 35 \, b^{3} c^{3} d^{4} x^{4} + 35 \, b^{3} c^{4} d^{3} x^{3} + 21 \, b^{3} c^{5} d^{2} x^{2} + 7 \, b^{3} c^{6} d x + b^{3} c^{7}\right )} \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) \log \relax (F)^{3} + {\left (8 \, {\left (b^{3} d^{7} x^{6} + 6 \, b^{3} c d^{6} x^{5} + 15 \, b^{3} c^{2} d^{5} x^{4} + 20 \, b^{3} c^{3} d^{4} x^{3} + 15 \, b^{3} c^{4} d^{3} x^{2} + 6 \, b^{3} c^{5} d^{2} x + b^{3} c^{6} d\right )} \log \relax (F)^{3} + 4 \, {\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x + b^{2} c^{4} d\right )} \log \relax (F)^{2} + 6 \, {\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \log \relax (F) + 15 \, d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{105 \, {\left (d^{9} x^{7} + 7 \, c d^{8} x^{6} + 21 \, c^{2} d^{7} x^{5} + 35 \, c^{3} d^{6} x^{4} + 35 \, c^{4} d^{5} x^{3} + 21 \, c^{5} d^{4} x^{2} + 7 \, c^{6} d^{3} x + c^{7} d^{2}\right )}} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 162, normalized size = 0.95 \[ \frac {8 \sqrt {\pi }\, b^{4} F^{a} \erf \left (\sqrt {-b \ln \relax (F )}\, \left (d x +c \right )\right ) \ln \relax (F )^{4}}{105 \sqrt {-b \ln \relax (F )}\, d}-\frac {8 b^{3} F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )^{3}}{105 \left (d x +c \right ) d}-\frac {4 b^{2} F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )^{2}}{105 \left (d x +c \right )^{3} d}-\frac {2 b \,F^{a} F^{\left (d x +c \right )^{2} b} \ln \relax (F )}{35 \left (d x +c \right )^{5} d}-\frac {F^{a} F^{\left (d x +c \right )^{2} b}}{7 \left (d x +c \right )^{7} d} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.13, size = 201, normalized size = 1.18 \[ \frac {8\,F^a\,\sqrt {\pi }\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}^{7/2}}{105\,d\,{\left (c+d\,x\right )}^7}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^2}}{7\,d\,{\left (c+d\,x\right )}^7}-\frac {4\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^2\,{\ln \relax (F)}^2}{105\,d\,{\left (c+d\,x\right )}^3}-\frac {8\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^3\,{\ln \relax (F)}^3}{105\,d\,\left (c+d\,x\right )}-\frac {2\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b\,\ln \relax (F)}{35\,d\,{\left (c+d\,x\right )}^5}-\frac {8\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2}\right )\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^2\right )}^{7/2}}{105\,d\,{\left (c+d\,x\right )}^7} \]
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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