Optimal. Leaf size=62 \[ \frac {(c+d x)^2 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)} \]
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Rubi [A] time = 0.10, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {(c+d x)^2 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx &=\frac {F^{a+b (c+d x)^2} (c+d x)^2}{2 b d \log (F)}-\frac {\int F^{a+b (c+d x)^2} (c+d x) \, dx}{b \log (F)}\\ &=-\frac {F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^2}{2 b d \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.65 \[ \frac {F^{a+b (c+d x)^2} \left (b \log (F) (c+d x)^2-1\right )}{2 b^2 d \log ^2(F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 60, normalized size = 0.97 \[ \frac {{\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \relax (F) - 1\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{2} d \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 1186, normalized size = 19.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 1.02 \[ \frac {\left (b \,d^{2} x^{2} \ln \relax (F )+2 b c d x \ln \relax (F )+b \,c^{2} \ln \relax (F )-1\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 b^{2} d \ln \relax (F )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 4.15, size = 683, normalized size = 11.02 \[ -\frac {3 \, {\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b c {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}\right ) - 1\right )} \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {3}{2}} d^{2} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b \log \relax (F)}{\left (b \log \relax (F)\right )^{\frac {3}{2}} d}\right )} F^{a} c^{2}}{2 \, \sqrt {b \log \relax (F)}} + \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{2} c^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}\right ) - 1\right )} \log \relax (F)^{3}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{3} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {2 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{2} c \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{2}} - \frac {{\left (b d^{2} x + b c d\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right ) \log \relax (F)^{3}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{5} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right )^{\frac {3}{2}}}\right )} F^{a} c d}{2 \, \sqrt {b \log \relax (F)}} - \frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{3} c^{3} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}\right ) - 1\right )} \log \relax (F)^{4}}{\left (b \log \relax (F)\right )^{\frac {7}{2}} d^{4} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {3 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{3} c^{2} \log \relax (F)^{3}}{\left (b \log \relax (F)\right )^{\frac {7}{2}} d^{3}} - \frac {3 \, {\left (b d^{2} x + b c d\right )}^{3} b c \Gamma \left (\frac {3}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right ) \log \relax (F)^{4}}{\left (b \log \relax (F)\right )^{\frac {7}{2}} d^{6} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right )^{\frac {3}{2}}} + \frac {b^{2} \Gamma \left (2, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right ) \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {7}{2}} d^{3}}\right )} F^{a} d^{2}}{2 \, \sqrt {b \log \relax (F)}} + \frac {\sqrt {\pi } F^{b c^{2} + a} c^{3} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} d x - \frac {b c \log \relax (F)}{\sqrt {-b \log \relax (F)}}\right )}{2 \, \sqrt {-b \log \relax (F)} F^{b c^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 67, normalized size = 1.08 \[ \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (b\,\ln \relax (F)\,c^2+2\,b\,\ln \relax (F)\,c\,d\,x+b\,\ln \relax (F)\,d^2\,x^2-1\right )}{2\,b^2\,d\,{\ln \relax (F)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 100, normalized size = 1.61 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )^{2}} \left (b c^{2} \log {\relax (F )} + 2 b c d x \log {\relax (F )} + b d^{2} x^{2} \log {\relax (F )} - 1\right )}{2 b^{2} d \log {\relax (F )}^{2}} & \text {for}\: 2 b^{2} d \log {\relax (F )}^{2} \neq 0 \\c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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