Optimal. Leaf size=111 \[ \frac {3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}-\frac {3 F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^3}{2 b d \log (F)} \]
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Rubi [A]
time = 0.11, antiderivative size = 111, 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 = {2243, 2235}
\begin {gather*} \frac {3 \sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}-\frac {3 (c+d x) F^{a+b (c+d x)^2}}{4 b^2 d \log ^2(F)}+\frac {(c+d x)^3 F^{a+b (c+d x)^2}}{2 b d \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2243
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} (c+d x)^4 \, dx &=\frac {F^{a+b (c+d x)^2} (c+d x)^3}{2 b d \log (F)}-\frac {3 \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{2 b \log (F)}\\ &=-\frac {3 F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^3}{2 b d \log (F)}+\frac {3 \int F^{a+b (c+d x)^2} \, dx}{4 b^2 \log ^2(F)}\\ &=\frac {3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}-\frac {3 F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^3}{2 b d \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 90, normalized size = 0.81 \begin {gather*} \frac {F^a \left (3 \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )+2 \sqrt {b} F^{b (c+d x)^2} (c+d x) \sqrt {\log (F)} \left (-3+2 b (c+d x)^2 \log (F)\right )\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs.
\(2(95)=190\).
time = 0.12, size = 300, normalized size = 2.70
method | result | size |
risch | \(\frac {d^{2} x^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}+\frac {3 d c \,x^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}+\frac {3 c^{2} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 \ln \left (F \right ) b}+\frac {c^{3} F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{2 d \ln \left (F \right ) b}-\frac {3 c \,F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{4 d \ln \left (F \right )^{2} b^{2}}-\frac {3 x \,F^{b \,d^{2} x^{2}} F^{2 b c d x} F^{b \,c^{2}} F^{a}}{4 \ln \left (F \right )^{2} b^{2}}-\frac {3 \sqrt {\pi }\, F^{a} \erf \left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{8 d \ln \left (F \right )^{2} b^{2} \sqrt {-b \ln \left (F \right )}}\) | \(300\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1037 vs.
\(2 (95) = 190\).
time = 0.63, size = 1037, normalized size = 9.34 \begin {gather*} -\frac {2 \, {\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 \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {3}{2}} d^{2} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b \log \left (F\right )}{\left (b \log \left (F\right )\right )^{\frac {3}{2}} d}\right )} F^{a} c^{3}}{\sqrt {b \log \left (F\right )}} + \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 \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{3} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {2 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{2} c \log \left (F\right )^{2}}{\left (b \log \left (F\right )\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 \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{5} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right )^{\frac {3}{2}}}\right )} F^{a} c^{2} d}{\sqrt {b \log \left (F\right )}} - \frac {2 \, {\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 \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{4}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{4} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{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 \left (F\right )^{3}}{\left (b \log \left (F\right )\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 \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{4}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{6} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{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 \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{3}}\right )} F^{a} c d^{2}}{\sqrt {b \log \left (F\right )}} + \frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{4} c^{4} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{5}}{\left (b \log \left (F\right )\right )^{\frac {9}{2}} d^{5} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {4 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{4} c^{3} \log \left (F\right )^{4}}{\left (b \log \left (F\right )\right )^{\frac {9}{2}} d^{4}} - \frac {6 \, {\left (b d^{2} x + b c d\right )}^{3} b^{2} c^{2} \Gamma \left (\frac {3}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{5}}{\left (b \log \left (F\right )\right )^{\frac {9}{2}} d^{7} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right )^{\frac {3}{2}}} + \frac {4 \, b^{3} c \Gamma \left (2, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {9}{2}} d^{4}} - \frac {{\left (b d^{2} x + b c d\right )}^{5} \Gamma \left (\frac {5}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{5}}{\left (b \log \left (F\right )\right )^{\frac {9}{2}} d^{9} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right )^{\frac {5}{2}}}\right )} F^{a} d^{3}}{2 \, \sqrt {b \log \left (F\right )}} + \frac {\sqrt {\pi } F^{b c^{2} + a} c^{4} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} d x - \frac {b c \log \left (F\right )}{\sqrt {-b \log \left (F\right )}}\right )}{2 \, \sqrt {-b \log \left (F\right )} F^{b c^{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 141, normalized size = 1.27 \begin {gather*} -\frac {3 \, \sqrt {\pi } \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 ) - 2 \, {\left (2 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \left (F\right )^{2} - 3 \, {\left (b d^{2} x + b c d\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{8 \, b^{3} d^{2} \log \left (F\right )^{3}} \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 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 [A]
time = 3.15, size = 111, normalized size = 1.00 \begin {gather*} \frac {{\left (2 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{3} \log \left (F\right ) - 3 \, x - \frac {3 \, c}{d}\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{4 \, b^{2} \log \left (F\right )^{2}} - \frac {3 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{8 \, \sqrt {-b \log \left (F\right )} b^{2} d \log \left (F\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.60, size = 243, normalized size = 2.19 \begin {gather*} \frac {3\,F^a\,\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 )}{8\,b^2\,{\ln \left (F\right )}^2\,\sqrt {b\,d^2\,\ln \left (F\right )}}-F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (\frac {3\,c}{4\,b^2\,d\,{\ln \left (F\right )}^2}-\frac {c^3}{2\,b\,d\,\ln \left (F\right )}\right )+\frac {3\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (2\,b\,c^2\,\ln \left (F\right )-1\right )}{4\,b^2\,{\ln \left (F\right )}^2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,d^2\,x^3}{2\,b\,\ln \left (F\right )}+\frac {3\,F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,c\,d\,x^2}{2\,b\,\ln \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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