Optimal. Leaf size=258 \[ -\frac {3 \sqrt {\pi } f^2 F^a (d e-c f) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac {3}{2}}(F)}-\frac {f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^3 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^4 \sqrt {\log (F)}}+\frac {3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)} \]
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Rubi [A] time = 0.44, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2226, 2204, 2209, 2212} \[ -\frac {3 \sqrt {\pi } f^2 F^a (d e-c f) \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac {3}{2}}(F)}-\frac {f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^3 \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^4 \sqrt {\log (F)}}+\frac {3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2209
Rule 2212
Rule 2226
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} (e+f x)^3 \, dx &=\int \left (\frac {(d e-c f)^3 F^{a+b (c+d x)^2}}{d^3}+\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)}{d^3}+\frac {3 f^2 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^2}{d^3}+\frac {f^3 F^{a+b (c+d x)^2} (c+d x)^3}{d^3}\right ) \, dx\\ &=\frac {f^3 \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{d^3}+\frac {\left (3 f^2 (d e-c f)\right ) \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{d^3}+\frac {\left (3 f (d e-c f)^2\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{d^3}+\frac {(d e-c f)^3 \int F^{a+b (c+d x)^2} \, dx}{d^3}\\ &=\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {3 f^2 (d e-c f) F^{a+b (c+d x)^2} (c+d x)}{2 b d^4 \log (F)}+\frac {f^3 F^{a+b (c+d x)^2} (c+d x)^2}{2 b d^4 \log (F)}+\frac {(d e-c f)^3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d^4 \sqrt {\log (F)}}-\frac {f^3 \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b d^3 \log (F)}-\frac {\left (3 f^2 (d e-c f)\right ) \int F^{a+b (c+d x)^2} \, dx}{2 b d^3 \log (F)}\\ &=-\frac {f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}-\frac {3 f^2 (d e-c f) F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{4 b^{3/2} d^4 \log ^{\frac {3}{2}}(F)}+\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {3 f^2 (d e-c f) F^{a+b (c+d x)^2} (c+d x)}{2 b d^4 \log (F)}+\frac {f^3 F^{a+b (c+d x)^2} (c+d x)^2}{2 b d^4 \log (F)}+\frac {(d e-c f)^3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d^4 \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 148, normalized size = 0.57 \[ \frac {F^a \left (2 f F^{b (c+d x)^2} \left (b \log (F) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-f^2\right )+\sqrt {\pi } \sqrt {b} \sqrt {\log (F)} (d e-c f) \left (2 b \log (F) (d e-c f)^2-3 f^2\right ) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )\right )}{4 b^2 d^4 \log ^2(F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 208, normalized size = 0.81 \[ \frac {\sqrt {\pi } {\left (3 \, d e f^{2} - 3 \, c f^{3} - 2 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \relax (F)\right )} \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (d f^{3} - {\left (b d^{3} f^{3} x^{2} + 3 \, b d^{3} e^{2} f - 3 \, b c d^{2} e f^{2} + b c^{2} d f^{3} + {\left (3 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x\right )} \log \relax (F)\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{4 \, b^{2} d^{5} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 426, normalized size = 1.65 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 3\right )}}{2 \, \sqrt {-b \log \relax (F)} d} + \frac {3 \, {\left (\frac {\sqrt {\pi } c f \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 2\right )}}{\sqrt {-b \log \relax (F)} d} + \frac {f e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F) + 2\right )}}{b d \log \relax (F)}\right )}}{2 \, d} - \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (2 \, b c^{2} f^{2} \log \relax (F) - f^{2}\right )} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 1\right )}}{\sqrt {-b \log \relax (F)} b d \log \relax (F)} - \frac {2 \, {\left (d f^{2} {\left (x + \frac {c}{d}\right )} - 2 \, c f^{2}\right )} e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F) + 1\right )}}{b d \log \relax (F)}\right )}}{4 \, d^{2}} + \frac {\frac {\sqrt {\pi } {\left (2 \, b c^{3} f^{3} \log \relax (F) - 3 \, c f^{3}\right )} F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {-b \log \relax (F)} b d \log \relax (F)} + \frac {2 \, {\left (b d^{2} f^{3} {\left (x + \frac {c}{d}\right )}^{2} \log \relax (F) - 3 \, b c d f^{3} {\left (x + \frac {c}{d}\right )} \log \relax (F) + 3 \, b c^{2} f^{3} \log \relax (F) - f^{3}\right )} e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F)\right )}}{b^{2} d \log \relax (F)^{2}}}{4 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 617, normalized size = 2.39 \[ \frac {f^{3} x^{2} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{2} \ln \relax (F )}-\frac {c \,f^{3} x \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{3} \ln \relax (F )}+\frac {3 e \,f^{2} x \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{2} \ln \relax (F )}+\frac {\sqrt {\pi }\, c^{3} f^{3} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d^{4}}-\frac {3 \sqrt {\pi }\, c^{2} e \,f^{2} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d^{3}}+\frac {3 \sqrt {\pi }\, c \,e^{2} f \,F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d^{2}}-\frac {\sqrt {\pi }\, e^{3} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d}+\frac {c^{2} f^{3} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{4} \ln \relax (F )}-\frac {3 c e \,f^{2} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{3} \ln \relax (F )}+\frac {3 e^{2} f \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{2} \ln \relax (F )}-\frac {3 \sqrt {\pi }\, c \,f^{3} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{4 \sqrt {-b \ln \relax (F )}\, b \,d^{4} \ln \relax (F )}+\frac {3 \sqrt {\pi }\, e \,f^{2} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{4 \sqrt {-b \ln \relax (F )}\, b \,d^{3} \ln \relax (F )}-\frac {f^{3} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b^{2} d^{4} \ln \relax (F )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.28, size = 695, normalized size = 2.69 \[ -\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} e^{2} f}{2 \, \sqrt {b \log \relax (F)} d} + \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} e f^{2}}{2 \, \sqrt {b \log \relax (F)} d} - \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} f^{3}}{2 \, \sqrt {b \log \relax (F)} d} + \frac {\sqrt {\pi } F^{b c^{2} + a} e^{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 = 3.76, size = 313, normalized size = 1.21 \[ \frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (F)\,d^2+b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )\,\left (-2\,b\,\ln \relax (F)\,c^3\,f^3+6\,b\,\ln \relax (F)\,c^2\,d\,e\,f^2-6\,b\,\ln \relax (F)\,c\,d^2\,e^2\,f+3\,c\,f^3+2\,b\,\ln \relax (F)\,d^3\,e^3-3\,d\,e\,f^2\right )}{4\,b\,d^3\,\ln \relax (F)\,\sqrt {b\,d^2\,\ln \relax (F)}}-F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (\frac {f^3}{2\,b^2\,d^4\,{\ln \relax (F)}^2}-\frac {3\,e^2\,f}{2\,b\,d^2\,\ln \relax (F)}-\frac {c^2\,f^3}{2\,b\,d^4\,\ln \relax (F)}+\frac {3\,c\,e\,f^2}{2\,b\,d^3\,\ln \relax (F)}\right )-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (c\,f^3-3\,d\,e\,f^2\right )}{2\,b\,d^3\,\ln \relax (F)}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f^3\,x^2}{2\,b\,d^2\,\ln \relax (F)} \]
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 \[ \int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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