Optimal. Leaf size=81 \[ \frac {\sqrt {\pi } F^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)} \]
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Rubi [A] time = 0.10, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2212, 2211, 2204} \[ \frac {\sqrt {\pi } F^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)} \]
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)^4} \, dx &=-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x) \log (F)}-\frac {\int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^2} \, dx}{2 b \log (F)}\\ &=-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x) \log (F)}+\frac {\operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\frac {1}{c+d x}\right )}{2 b d \log (F)}\\ &=\frac {F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x) \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 81, normalized size = 1.00 \[ \frac {\sqrt {\pi } F^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 117, normalized size = 1.44 \[ -\frac {\sqrt {\pi } {\left (d^{2} x + c d\right )} F^{a} \sqrt {-\frac {b \log \relax (F)}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \relax (F)}{d^{2}}}}{d x + c}\right ) + 2 \, F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} b \log \relax (F)}{4 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \log \relax (F)^{2}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 76, normalized size = 0.94 \[ -\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2 \left (d x +c \right ) b d \ln \relax (F )}+\frac {\sqrt {\pi }\, F^{a} \erf \left (\frac {\sqrt {-b \ln \relax (F )}}{d x +c}\right )}{4 \sqrt {-b \ln \relax (F )}\, b d \ln \relax (F )} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.98, size = 76, normalized size = 0.94 \[ \frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (F)}{\sqrt {b\,\ln \relax (F)}\,\left (c+d\,x\right )}\right )}{4\,b\,d\,\ln \relax (F)\,\sqrt {b\,\ln \relax (F)}}-\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}}{2\,b\,d\,\ln \relax (F)\,\left (c+d\,x\right )} \]
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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