Optimal. Leaf size=102 \[ -\frac {F^{a+b (c+d x)^2}}{3 d (c+d x)^3}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{3 d (c+d x)}+\frac {2 b^{3/2} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {3}{2}}(F)}{3 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2245, 2235}
\begin {gather*} \frac {2 \sqrt {\pi } b^{3/2} F^a \log ^{\frac {3}{2}}(F) \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{3 d}-\frac {F^{a+b (c+d x)^2}}{3 d (c+d x)^3}-\frac {2 b \log (F) F^{a+b (c+d x)^2}}{3 d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2245
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^4} \, dx &=-\frac {F^{a+b (c+d x)^2}}{3 d (c+d x)^3}+\frac {1}{3} (2 b \log (F)) \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^2} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{3 d (c+d x)^3}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{3 d (c+d x)}+\frac {1}{3} \left (4 b^2 \log ^2(F)\right ) \int F^{a+b (c+d x)^2} \, dx\\ &=-\frac {F^{a+b (c+d x)^2}}{3 d (c+d x)^3}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{3 d (c+d x)}+\frac {2 b^{3/2} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {3}{2}}(F)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 81, normalized size = 0.79 \begin {gather*} \frac {F^a \left (2 b^{3/2} \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {3}{2}}(F)-\frac {F^{b (c+d x)^2} \left (1+2 b (c+d x)^2 \log (F)\right )}{(c+d x)^3}\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 96, normalized size = 0.94
method | result | size |
risch | \(-\frac {F^{b \left (d x +c \right )^{2}} F^{a}}{3 d \left (d x +c \right )^{3}}-\frac {2 b \ln \left (F \right ) F^{b \left (d x +c \right )^{2}} F^{a}}{3 d \left (d x +c \right )}+\frac {2 b^{2} \ln \left (F \right )^{2} \sqrt {\pi }\, F^{a} \erf \left (\sqrt {-b \ln \left (F \right )}\, \left (d x +c \right )\right )}{3 d \sqrt {-b \ln \left (F \right )}}\) | \(96\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 163, normalized size = 1.60 \begin {gather*} -\frac {2 \, \sqrt {\pi } {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) \log \left (F\right ) + {\left (2 \, {\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \log \left (F\right ) + d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{3 \, {\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (c + d x\right )^{4}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.03, size = 201, normalized size = 1.97 \begin {gather*} \frac {2\,F^a\,b^2\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (F\right )\,d^2+b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )\,{\ln \left (F\right )}^2}{3\,\sqrt {b\,d^2\,\ln \left (F\right )}}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (\frac {1}{3\,d}+\frac {2\,b\,c^2\,\ln \left (F\right )}{3\,d}\right )+\frac {4\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b\,c\,x\,\ln \left (F\right )}{3}+\frac {2\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,b\,d\,x^2\,\ln \left (F\right )}{3}}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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