Optimal. Leaf size=266 \[ -\frac {3 \sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e)^3 \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{16 c^{7/2} \sqrt {\log (f)}}+\frac {e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac {e (d+e x) (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)} \]
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Rubi [A] time = 0.32, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2241, 2240, 2234, 2204} \[ -\frac {3 \sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e)^3 \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{16 c^{7/2} \sqrt {\log (f)}}+\frac {e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac {e (d+e x) (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2240
Rule 2241
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} (d+e x)^3 \, dx &=\frac {e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}-\frac {(-2 c d+b e) \int f^{a+b x+c x^2} (d+e x)^2 \, dx}{2 c}-\frac {e^2 \int f^{a+b x+c x^2} (d+e x) \, dx}{c \log (f)}\\ &=-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac {(2 c d-b e)^2 \int f^{a+b x+c x^2} (d+e x) \, dx}{4 c^2}-\frac {\left (e^2 (2 c d-b e)\right ) \int f^{a+b x+c x^2} \, dx}{4 c^2 \log (f)}-\frac {\left (e^2 (2 c d-b e)\right ) \int f^{a+b x+c x^2} \, dx}{2 c^2 \log (f)}\\ &=-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac {e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac {(2 c d-b e)^3 \int f^{a+b x+c x^2} \, dx}{8 c^3}-\frac {\left (e^2 (2 c d-b e) f^{a-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{4 c^2 \log (f)}-\frac {\left (e^2 (2 c d-b e) f^{a-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{2 c^2 \log (f)}\\ &=-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac {3 e^2 (2 c d-b e) f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac {\left ((2 c d-b e)^3 f^{a-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{8 c^3}\\ &=-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac {3 e^2 (2 c d-b e) f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac {(2 c d-b e)^3 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{16 c^{7/2} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 169, normalized size = 0.64 \[ \frac {f^{a-\frac {b^2}{4 c}} \left (2 \sqrt {c} e f^{\frac {(b+2 c x)^2}{4 c}} \left (\log (f) \left (b^2 e^2-2 b c e (3 d+e x)+4 c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right )-4 c e^2\right )+\sqrt {\pi } \sqrt {\log (f)} (2 c d-b e) \left (\log (f) (b e-2 c d)^2-6 c e^2\right ) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )\right )}{16 c^{7/2} \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 204, normalized size = 0.77 \[ -\frac {2 \, {\left (4 \, c^{2} e^{3} - {\left (4 \, c^{3} e^{3} x^{2} + 12 \, c^{3} d^{2} e - 6 \, b c^{2} d e^{2} + b^{2} c e^{3} + 2 \, {\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x\right )} \log \relax (f)\right )} f^{c x^{2} + b x + a} - \frac {\sqrt {\pi } {\left (12 \, c^{2} d e^{2} - 6 \, b c e^{3} - {\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \relax (f)\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{16 \, c^{4} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 401, normalized size = 1.51 \[ -\frac {\sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f)}{4 \, c}\right )}}{2 \, \sqrt {-c \log \relax (f)}} + \frac {3 \, {\left (\frac {\sqrt {\pi } b d^{2} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 4 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)}} + \frac {2 \, d^{2} e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 1\right )}}{\log \relax (f)}\right )}}{4 \, c} - \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (b^{2} d \log \relax (f) - 2 \, c d\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 8 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)} \log \relax (f)} - \frac {2 \, {\left (c d {\left (2 \, x + \frac {b}{c}\right )} - 2 \, b d\right )} e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 2\right )}}{\log \relax (f)}\right )}}{8 \, c^{2}} + \frac {\frac {\sqrt {\pi } {\left (b^{3} \log \relax (f) - 6 \, b c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 12 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)} \log \relax (f)} + \frac {2 \, {\left (c^{2} {\left (2 \, x + \frac {b}{c}\right )}^{2} \log \relax (f) - 3 \, b c {\left (2 \, x + \frac {b}{c}\right )} \log \relax (f) + 3 \, b^{2} \log \relax (f) - 4 \, c\right )} e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 3\right )}}{\log \relax (f)^{2}}}{16 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 550, normalized size = 2.07 \[ \frac {\sqrt {\pi }\, b^{3} e^{3} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{16 \sqrt {-c \ln \relax (f )}\, c^{3}}-\frac {3 \sqrt {\pi }\, b^{2} d \,e^{2} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{8 \sqrt {-c \ln \relax (f )}\, c^{2}}+\frac {3 \sqrt {\pi }\, b \,d^{2} e \,f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}\, c}+\frac {e^{3} x^{2} f^{a} f^{b x} f^{c \,x^{2}}}{2 c \ln \relax (f )}-\frac {\sqrt {\pi }\, d^{3} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{2 \sqrt {-c \ln \relax (f )}}-\frac {b \,e^{3} x \,f^{a} f^{b x} f^{c \,x^{2}}}{4 c^{2} \ln \relax (f )}+\frac {3 d \,e^{2} x \,f^{a} f^{b x} f^{c \,x^{2}}}{2 c \ln \relax (f )}+\frac {b^{2} e^{3} f^{a} f^{b x} f^{c \,x^{2}}}{8 c^{3} \ln \relax (f )}-\frac {3 b d \,e^{2} f^{a} f^{b x} f^{c \,x^{2}}}{4 c^{2} \ln \relax (f )}-\frac {3 \sqrt {\pi }\, b \,e^{3} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{8 \sqrt {-c \ln \relax (f )}\, c^{2} \ln \relax (f )}+\frac {3 d^{2} e \,f^{a} f^{b x} f^{c \,x^{2}}}{2 c \ln \relax (f )}+\frac {3 \sqrt {\pi }\, d \,e^{2} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}\, c \ln \relax (f )}-\frac {e^{3} f^{a} f^{b x} f^{c \,x^{2}}}{2 c^{2} \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.71, size = 539, normalized size = 2.03 \[ -\frac {3 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)}{\left (c \log \relax (f)\right )^{\frac {3}{2}}}\right )} d^{2} e f^{a - \frac {b^{2}}{4 \, c}}}{4 \, \sqrt {c \log \relax (f)}} + \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}\right )^{\frac {3}{2}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {5}{2}}}\right )} d e^{2} f^{a - \frac {b^{2}}{4 \, c}}}{8 \, \sqrt {c \log \relax (f)}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{4}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {7}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{4}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}\right )^{\frac {3}{2}} \left (c \log \relax (f)\right )^{\frac {7}{2}}} - \frac {6 \, b^{2} c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)^{3}}{\left (c \log \relax (f)\right )^{\frac {7}{2}}} + \frac {8 \, c^{2} \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {7}{2}}}\right )} e^{3} f^{a - \frac {b^{2}}{4 \, c}}}{16 \, \sqrt {c \log \relax (f)}} + \frac {\sqrt {\pi } d^{3} f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{2 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.89, size = 251, normalized size = 0.94 \[ \frac {e^3\,f^a\,f^{c\,x^2}\,f^{b\,x}\,x^2}{2\,c\,\ln \relax (f)}-\frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \relax (f)}{2}+c\,x\,\ln \relax (f)}{\sqrt {c\,\ln \relax (f)}}\right )\,\left (\frac {\ln \relax (f)\,b^3\,e^3}{16}-\frac {3\,\ln \relax (f)\,b^2\,c\,d\,e^2}{8}+\frac {3\,\ln \relax (f)\,b\,c^2\,d^2\,e}{4}-\frac {3\,b\,c\,e^3}{8}-\frac {\ln \relax (f)\,c^3\,d^3}{2}+\frac {3\,c^2\,d\,e^2}{4}\right )}{c^3\,\ln \relax (f)\,\sqrt {c\,\ln \relax (f)}}-\frac {f^a\,f^{c\,x^2}\,f^{b\,x}\,x\,\left (b\,e^3-6\,c\,d\,e^2\right )}{4\,c^2\,\ln \relax (f)}-f^a\,f^{c\,x^2}\,f^{b\,x}\,\left (\frac {e^3}{2\,c^2\,{\ln \relax (f)}^2}-\frac {3\,d^2\,e}{2\,c\,\ln \relax (f)}-\frac {b^2\,e^3}{8\,c^3\,\ln \relax (f)}+\frac {3\,b\,d\,e^2}{4\,c^2\,\ln \relax (f)}\right ) \]
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 \[ \int f^{a + b x + c x^{2}} \left (d + e x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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