Integrand size = 20, antiderivative size = 266 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac {3 e^2 (2 c d-b e) f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac {(2 c d-b e)^3 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{16 c^{7/2} \sqrt {\log (f)}} \]
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Time = 0.21 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2273, 2272, 2266, 2235} \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=-\frac {3 \sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)}+\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e)^3 \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{16 c^{7/2} \sqrt {\log (f)}}+\frac {e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac {e (d+e x) (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)} \]
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Rule 2235
Rule 2266
Rule 2272
Rule 2273
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.44 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.64 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=\frac {f^{a-\frac {b^2}{4 c}} \left ((2 c d-b e) \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)} \left (-6 c e^2+(-2 c d+b e)^2 \log (f)\right )+2 \sqrt {c} e f^{\frac {(b+2 c x)^2}{4 c}} \left (-4 c e^2+\left (b^2 e^2-2 b c e (3 d+e x)+4 c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \log (f)\right )\right )}{16 c^{7/2} \log ^2(f)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(549\) vs. \(2(226)=452\).
Time = 0.30 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.07
method | result | size |
risch | \(-\frac {f^{a} d^{3} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{2 \sqrt {-c \ln \left (f \right )}}+\frac {e^{3} f^{a} x^{2} f^{b x} f^{c \,x^{2}}}{2 \ln \left (f \right ) c}-\frac {e^{3} f^{a} b x \,f^{b x} f^{c \,x^{2}}}{4 c^{2} \ln \left (f \right )}+\frac {e^{3} f^{a} b^{2} f^{b x} f^{c \,x^{2}}}{8 c^{3} \ln \left (f \right )}+\frac {e^{3} f^{a} b^{3} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 c^{3} \sqrt {-c \ln \left (f \right )}}-\frac {3 e^{3} f^{a} b \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 c^{2} \ln \left (f \right ) \sqrt {-c \ln \left (f \right )}}-\frac {e^{3} f^{a} f^{b x} f^{c \,x^{2}}}{2 c^{2} \ln \left (f \right )^{2}}+\frac {3 e^{2} d \,f^{a} x \,f^{b x} f^{c \,x^{2}}}{2 \ln \left (f \right ) c}-\frac {3 e^{2} d \,f^{a} b \,f^{b x} f^{c \,x^{2}}}{4 c^{2} \ln \left (f \right )}-\frac {3 e^{2} d \,f^{a} b^{2} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 c^{2} \sqrt {-c \ln \left (f \right )}}+\frac {3 e^{2} d \,f^{a} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \ln \left (f \right ) c \sqrt {-c \ln \left (f \right )}}+\frac {3 f^{a} d^{2} e \,f^{b x} f^{c \,x^{2}}}{2 \ln \left (f \right ) c}+\frac {3 f^{a} d^{2} e b \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 c \sqrt {-c \ln \left (f \right )}}\) | \(550\) |
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Time = 0.36 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.77 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=-\frac {2 \, {\left (4 \, c^{2} e^{3} - {\left (4 \, c^{3} e^{3} x^{2} + 12 \, c^{3} d^{2} e - 6 \, b c^{2} d e^{2} + b^{2} c e^{3} + 2 \, {\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x\right )} \log \left (f\right )\right )} f^{c x^{2} + b x + a} - \frac {\sqrt {\pi } {\left (12 \, c^{2} d e^{2} - 6 \, b c e^{3} - {\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{16 \, c^{4} \log \left (f\right )^{2}} \]
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\[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=\int f^{a + b x + c x^{2}} \left (d + e x\right )^{3}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (226) = 452\).
Time = 0.44 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.03 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=-\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 \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac {3}{2}}}\right )} d^{2} e f^{a - \frac {b^{2}}{4 \, c}}}{4 \, \sqrt {c \log \left (f\right )}} + \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 \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\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 \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}\right )^{\frac {3}{2}} \left (c \log \left (f\right )\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}}}\right )} d e^{2} f^{a - \frac {b^{2}}{4 \, c}}}{8 \, \sqrt {c \log \left (f\right )}} - \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 \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{4}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\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 \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{4}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}\right )^{\frac {3}{2}} \left (c \log \left (f\right )\right )^{\frac {7}{2}}} - \frac {6 \, b^{2} c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{3}}{\left (c \log \left (f\right )\right )^{\frac {7}{2}}} + \frac {8 \, c^{2} \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac {7}{2}}}\right )} e^{3} f^{a - \frac {b^{2}}{4 \, c}}}{16 \, \sqrt {c \log \left (f\right )}} + \frac {\sqrt {\pi } d^{3} f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )}{2 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
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Time = 0.33 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.90 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=-\frac {\frac {\sqrt {\pi } {\left (8 \, c^{3} d^{3} \log \left (f\right ) - 12 \, b c^{2} d^{2} e \log \left (f\right ) + 6 \, b^{2} c d e^{2} \log \left (f\right ) - b^{3} e^{3} \log \left (f\right ) - 12 \, c^{2} d e^{2} + 6 \, b c e^{3}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{\sqrt {-c \log \left (f\right )} \log \left (f\right )} - \frac {2 \, {\left (c^{2} e^{3} {\left (2 \, x + \frac {b}{c}\right )}^{2} \log \left (f\right ) + 6 \, c^{2} d e^{2} {\left (2 \, x + \frac {b}{c}\right )} \log \left (f\right ) - 3 \, b c e^{3} {\left (2 \, x + \frac {b}{c}\right )} \log \left (f\right ) + 12 \, c^{2} d^{2} e \log \left (f\right ) - 12 \, b c d e^{2} \log \left (f\right ) + 3 \, b^{2} e^{3} \log \left (f\right ) - 4 \, c e^{3}\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right )\right )}}{\log \left (f\right )^{2}}}{16 \, c^{3}} \]
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Time = 0.60 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.94 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=\frac {e^3\,f^a\,f^{c\,x^2}\,f^{b\,x}\,x^2}{2\,c\,\ln \left (f\right )}-\frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \left (f\right )}{2}+c\,x\,\ln \left (f\right )}{\sqrt {c\,\ln \left (f\right )}}\right )\,\left (\frac {\ln \left (f\right )\,b^3\,e^3}{16}-\frac {3\,\ln \left (f\right )\,b^2\,c\,d\,e^2}{8}+\frac {3\,\ln \left (f\right )\,b\,c^2\,d^2\,e}{4}-\frac {3\,b\,c\,e^3}{8}-\frac {\ln \left (f\right )\,c^3\,d^3}{2}+\frac {3\,c^2\,d\,e^2}{4}\right )}{c^3\,\ln \left (f\right )\,\sqrt {c\,\ln \left (f\right )}}-\frac {f^a\,f^{c\,x^2}\,f^{b\,x}\,x\,\left (b\,e^3-6\,c\,d\,e^2\right )}{4\,c^2\,\ln \left (f\right )}-f^a\,f^{c\,x^2}\,f^{b\,x}\,\left (\frac {e^3}{2\,c^2\,{\ln \left (f\right )}^2}-\frac {3\,d^2\,e}{2\,c\,\ln \left (f\right )}-\frac {b^2\,e^3}{8\,c^3\,\ln \left (f\right )}+\frac {3\,b\,d\,e^2}{4\,c^2\,\ln \left (f\right )}\right ) \]
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