Optimal. Leaf size=189 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e)^2 \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{8 c^{5/2} \sqrt {\log (f)}}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)} \]
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Rubi [A] time = 0.11, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2241, 2240, 2234, 2204} \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e)^2 \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{8 c^{5/2} \sqrt {\log (f)}}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)} \]
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)^2 \, dx &=\frac {e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}-\frac {(-2 c d+b e) \int f^{a+b x+c x^2} (d+e x) \, dx}{2 c}-\frac {e^2 \int f^{a+b x+c x^2} \, dx}{2 c \log (f)}\\ &=\frac {e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}+\frac {(2 c d-b e)^2 \int f^{a+b x+c x^2} \, dx}{4 c^2}-\frac {\left (e^2 f^{a-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{2 c \log (f)}\\ &=-\frac {e^2 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}+\frac {\left ((2 c d-b e)^2 f^{a-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{4 c^2}\\ &=-\frac {e^2 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}+\frac {(2 c d-b e)^2 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} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 123, normalized size = 0.65 \[ \frac {f^{a-\frac {b^2}{4 c}} \left (\sqrt {\pi } \left (\log (f) (b e-2 c d)^2-2 c e^2\right ) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )+2 \sqrt {c} e \sqrt {\log (f)} f^{\frac {(b+2 c x)^2}{4 c}} (-b e+4 c d+2 c e x)\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 130, normalized size = 0.69 \[ \frac {2 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} f^{c x^{2} + b x + a} \log \relax (f) + \frac {\sqrt {\pi } {\left (2 \, c e^{2} - {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \log \relax (f)\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, c^{3} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 252, normalized size = 1.33 \[ -\frac {\sqrt {\pi } d^{2} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f)}{4 \, c}\right )}}{2 \, \sqrt {-c \log \relax (f)}} + \frac {\frac {\sqrt {\pi } b d \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 4 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)}} + \frac {2 \, d e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 1\right )}}{\log \relax (f)}}{2 \, c} - \frac {\frac {\sqrt {\pi } {\left (b^{2} \log \relax (f) - 2 \, c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 8 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)} \log \relax (f)} - \frac {2 \, {\left (c {\left (2 \, x + \frac {b}{c}\right )} - 2 \, b\right )} e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 2\right )}}{\log \relax (f)}}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 307, normalized size = 1.62 \[ -\frac {\sqrt {\pi }\, b^{2} e^{2} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{8 \sqrt {-c \ln \relax (f )}\, c^{2}}+\frac {\sqrt {\pi }\, b d e \,f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{2 \sqrt {-c \ln \relax (f )}\, c}-\frac {\sqrt {\pi }\, d^{2} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{2 \sqrt {-c \ln \relax (f )}}+\frac {e^{2} x \,f^{a} f^{b x} f^{c \,x^{2}}}{2 c \ln \relax (f )}-\frac {b \,e^{2} f^{a} f^{b x} f^{c \,x^{2}}}{4 c^{2} \ln \relax (f )}+\frac {d e \,f^{a} f^{b x} f^{c \,x^{2}}}{c \ln \relax (f )}+\frac {\sqrt {\pi }\, e^{2} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}\, c \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.07, size = 332, normalized size = 1.76 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)}{\left (c \log \relax (f)\right )^{\frac {3}{2}}}\right )} d e f^{a - \frac {b^{2}}{4 \, c}}}{2 \, \sqrt {c \log \relax (f)}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}\right )^{\frac {3}{2}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {5}{2}}}\right )} e^{2} f^{a - \frac {b^{2}}{4 \, c}}}{8 \, \sqrt {c \log \relax (f)}} + \frac {\sqrt {\pi } d^{2} f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{2 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.85, size = 153, normalized size = 0.81 \[ f^a\,f^{c\,x^2}\,f^{b\,x}\,\left (\frac {d\,e}{c\,\ln \relax (f)}-\frac {b\,e^2}{4\,c^2\,\ln \relax (f)}\right )+\frac {e^2\,f^a\,f^{c\,x^2}\,f^{b\,x}\,x}{2\,c\,\ln \relax (f)}-\frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \relax (f)}{2}+c\,x\,\ln \relax (f)}{\sqrt {c\,\ln \relax (f)}}\right )\,\left (-\frac {\ln \relax (f)\,b^2\,e^2}{8}+\frac {\ln \relax (f)\,b\,c\,d\,e}{2}-\frac {\ln \relax (f)\,c^2\,d^2}{2}+\frac {c\,e^2}{4}\right )}{c^2\,\ln \relax (f)\,\sqrt {c\,\ln \relax (f)}} \]
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 \[ \int f^{a + b x + c x^{2}} \left (d + e x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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