Optimal. Leaf size=115 \[ -\frac {3 \sqrt {\pi } F^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d \log ^2(F) (c+d x)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^3} \]
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Rubi [A] time = 0.15, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2212, 2211, 2204} \[ -\frac {3 \sqrt {\pi } F^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d \log ^2(F) (c+d x)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^3} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2211
Rule 2212
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)}-\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^4} \, dx}{2 b \log (F)}\\ &=\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)}+\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^2} \, dx}{4 b^2 \log ^2(F)}\\ &=\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)}-\frac {3 \operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\frac {1}{c+d x}\right )}{4 b^2 d \log ^2(F)}\\ &=-\frac {3 F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 95, normalized size = 0.83 \[ \frac {F^a \left (-3 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )-\frac {2 \sqrt {b} \sqrt {\log (F)} F^{\frac {b}{(c+d x)^2}} \left (2 b \log (F)-3 (c+d x)^2\right )}{(c+d x)^3}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 199, normalized size = 1.73 \[ \frac {3 \, \sqrt {\pi } {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )} F^{a} \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right ) - 2 \, {\left (2 \, b^{2} \log \relax (F)^{2} - 3 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{8 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d\right )} \log \relax (F)^{3}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 109, normalized size = 0.95 \[ -\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2 \left (d x +c \right )^{3} b d \ln \relax (F )}+\frac {3 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{4 \left (d x +c \right ) b^{2} d \ln \relax (F )^{2}}-\frac {3 \sqrt {\pi }\, F^{a} \erf \left (\frac {\sqrt {-b \ln \relax (F )}}{d x +c}\right )}{8 \sqrt {-b \ln \relax (F )}\, b^{2} d \ln \relax (F )^{2}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 105, normalized size = 0.91 \[ -\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}}{2\,b\,d\,\ln \relax (F)\,{\left (c+d\,x\right )}^3}-\frac {F^a\,\left (3\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (F)}{\sqrt {b\,\ln \relax (F)}\,\left (c+d\,x\right )}\right )-\frac {6\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,\sqrt {b\,\ln \relax (F)}}{c+d\,x}\right )}{8\,b^2\,d\,{\ln \relax (F)}^2\,\sqrt {b\,\ln \relax (F)}} \]
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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