Integrand size = 20, antiderivative size = 189 \[ \int f^{a+b x+c x^2} (d+e x)^2 \, dx=-\frac {e^2 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)}{2 c \log (f)}+\frac {(2 c d-b e)^2 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{8 c^{5/2} \sqrt {\log (f)}} \]
1/4*e*(-b*e+2*c*d)*f^(c*x^2+b*x+a)/c^2/ln(f)+1/2*e*f^(c*x^2+b*x+a)*(e*x+d) /c/ln(f)-1/4*e^2*f^(a-1/4*b^2/c)*erfi(1/2*(2*c*x+b)*ln(f)^(1/2)/c^(1/2))*P i^(1/2)/c^(3/2)/ln(f)^(3/2)+1/8*(-b*e+2*c*d)^2*f^(a-1/4*b^2/c)*erfi(1/2*(2 *c*x+b)*ln(f)^(1/2)/c^(1/2))*Pi^(1/2)/c^(5/2)/ln(f)^(1/2)
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.65 \[ \int f^{a+b x+c x^2} (d+e x)^2 \, dx=\frac {f^{a-\frac {b^2}{4 c}} \left (2 \sqrt {c} e f^{\frac {(b+2 c x)^2}{4 c}} (4 c d-b e+2 c e x) \sqrt {\log (f)}+\sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \left (-2 c e^2+(-2 c d+b e)^2 \log (f)\right )\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)} \]
(f^(a - b^2/(4*c))*(2*Sqrt[c]*e*f^((b + 2*c*x)^2/(4*c))*(4*c*d - b*e + 2*c *e*x)*Sqrt[Log[f]] + Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])] *(-2*c*e^2 + (-2*c*d + b*e)^2*Log[f])))/(8*c^(5/2)*Log[f]^(3/2))
Time = 0.50 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2671, 2664, 2633, 2670, 2664, 2633}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x)^2 f^{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 2671 |
| \(\displaystyle \frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)dx}{2 c}-\frac {e^2 \int f^{c x^2+b x+a}dx}{2 c \log (f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle -\frac {e^2 f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}}dx}{2 c \log (f)}+\frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)dx}{2 c}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)dx}{2 c}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2670 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \int f^{c x^2+b x+a}dx}{2 c}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {f^{a-\frac {b^2}{4 c}} (2 c d-b e) \int f^{\frac {(b+2 c x)^2}{4 c}}dx}{2 c}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\) |
((2*c*d - b*e)*((e*f^(a + b*x + c*x^2))/(2*c*Log[f]) + ((2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^ (3/2)*Sqrt[Log[f]])))/(2*c) - (e^2*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2 *c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Log[f]^(3/2)) + (e*f^(a + b*x + c*x^2)*(d + e*x))/(2*c*Log[f])
3.5.45.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ (4*c)) Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol ] :> Simp[e*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] - Simp[(b*e - 2*c*d)/(2* c) Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ [b*e - 2*c*d, 0]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] + (-Simp[(b*e - 2*c*d)/(2*c) Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2), x] , x] - Simp[(m - 1)*(e^2/(2*c*Log[F])) Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
Time = 0.36 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(-\frac {f^{a} d^{2} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{2 \sqrt {-c \ln \left (f \right )}}+\frac {e^{2} f^{a} x \,f^{b x} f^{c \,x^{2}}}{2 \ln \left (f \right ) c}-\frac {e^{2} f^{a} b \,f^{b x} f^{c \,x^{2}}}{4 c^{2} \ln \left (f \right )}-\frac {e^{2} f^{a} b^{2} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 c^{2} \sqrt {-c \ln \left (f \right )}}+\frac {e^{2} f^{a} \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \ln \left (f \right ) c \sqrt {-c \ln \left (f \right )}}+\frac {f^{a} d e \,f^{b x} f^{c \,x^{2}}}{\ln \left (f \right ) c}+\frac {f^{a} d e b \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{2 c \sqrt {-c \ln \left (f \right )}}\) | \(307\) |
-1/2*f^a*d^2*Pi^(1/2)*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2 )*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))+1/2*e^2*f^a/ln(f)/c*x*f^(b*x)*f^(c*x^2)- 1/4*e^2*f^a*b/c^2/ln(f)*f^(b*x)*f^(c*x^2)-1/8*e^2*f^a*b^2/c^2*Pi^(1/2)*f^( -1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f) )^(1/2))+1/4*e^2*f^a/ln(f)/c*Pi^(1/2)*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf( -(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))+f^a*d*e/ln(f)/c*f^(b*x)* f^(c*x^2)+1/2*f^a*d*e*b/c*Pi^(1/2)*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(- c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))
Time = 0.35 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.69 \[ \int f^{a+b x+c x^2} (d+e x)^2 \, dx=\frac {2 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} f^{c x^{2} + b x + a} \log \left (f\right ) + \frac {\sqrt {\pi } {\left (2 \, c e^{2} - {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, c^{3} \log \left (f\right )^{2}} \]
1/8*(2*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*f^(c*x^2 + b*x + a)*log(f) + sq rt(pi)*(2*c*e^2 - (4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*log(f))*sqrt(-c*log(f) )*erf(1/2*(2*c*x + b)*sqrt(-c*log(f))/c)/f^(1/4*(b^2 - 4*a*c)/c))/(c^3*log (f)^2)
\[ \int f^{a+b x+c x^2} (d+e x)^2 \, dx=\int f^{a + b x + c x^{2}} \left (d + e x\right )^{2}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (153) = 306\).
Time = 0.35 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.76 \[ \int f^{a+b x+c x^2} (d+e x)^2 \, dx=-\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac {3}{2}}}\right )} d e f^{a - \frac {b^{2}}{4 \, c}}}{2 \, \sqrt {c \log \left (f\right )}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}\right )^{\frac {3}{2}} \left (c \log \left (f\right )\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}}}\right )} e^{2} f^{a - \frac {b^{2}}{4 \, c}}}{8 \, \sqrt {c \log \left (f\right )}} + \frac {\sqrt {\pi } d^{2} f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )}{2 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
-1/2*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)* log(f)^2/(sqrt(-(2*c*x + b)^2*log(f)/c)*(c*log(f))^(3/2)) - 2*c*f^(1/4*(2* c*x + b)^2/c)*log(f)/(c*log(f))^(3/2))*d*e*f^(a - 1/4*b^2/c)/sqrt(c*log(f) ) + 1/8*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^3/(sqrt(-(2*c*x + b)^2*log(f)/c)*(c*log(f))^(5/2)) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^3/((-(2*c*x + b)^2*l og(f)/c)^(3/2)*(c*log(f))^(5/2)) - 4*b*c*f^(1/4*(2*c*x + b)^2/c)*log(f)^2/ (c*log(f))^(5/2))*e^2*f^(a - 1/4*b^2/c)/sqrt(c*log(f)) + 1/2*sqrt(pi)*d^2* f^a*erf(sqrt(-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log(f)))/(sqrt(-c*log(f)) *f^(1/4*b^2/c))
Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.76 \[ \int f^{a+b x+c x^2} (d+e x)^2 \, dx=-\frac {\frac {\sqrt {\pi } {\left (4 \, c^{2} d^{2} \log \left (f\right ) - 4 \, b c d e \log \left (f\right ) + b^{2} e^{2} \log \left (f\right ) - 2 \, c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{\sqrt {-c \log \left (f\right )} \log \left (f\right )} - \frac {2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right )\right )}}{\log \left (f\right )}}{8 \, c^{2}} \]
-1/8*(sqrt(pi)*(4*c^2*d^2*log(f) - 4*b*c*d*e*log(f) + b^2*e^2*log(f) - 2*c *e^2)*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*lo g(f))/c)/(sqrt(-c*log(f))*log(f)) - 2*(c*e^2*(2*x + b/c) + 4*c*d*e - 2*b*e ^2)*e^(c*x^2*log(f) + b*x*log(f) + a*log(f))/log(f))/c^2
Time = 0.49 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.81 \[ \int f^{a+b x+c x^2} (d+e x)^2 \, dx=f^a\,f^{c\,x^2}\,f^{b\,x}\,\left (\frac {d\,e}{c\,\ln \left (f\right )}-\frac {b\,e^2}{4\,c^2\,\ln \left (f\right )}\right )+\frac {e^2\,f^a\,f^{c\,x^2}\,f^{b\,x}\,x}{2\,c\,\ln \left (f\right )}-\frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \left (f\right )}{2}+c\,x\,\ln \left (f\right )}{\sqrt {c\,\ln \left (f\right )}}\right )\,\left (-\frac {\ln \left (f\right )\,b^2\,e^2}{8}+\frac {\ln \left (f\right )\,b\,c\,d\,e}{2}-\frac {\ln \left (f\right )\,c^2\,d^2}{2}+\frac {c\,e^2}{4}\right )}{c^2\,\ln \left (f\right )\,\sqrt {c\,\ln \left (f\right )}} \]
f^a*f^(c*x^2)*f^(b*x)*((d*e)/(c*log(f)) - (b*e^2)/(4*c^2*log(f))) + (e^2*f ^a*f^(c*x^2)*f^(b*x)*x)/(2*c*log(f)) - (f^(a - b^2/(4*c))*pi^(1/2)*erfi((( b*log(f))/2 + c*x*log(f))/(c*log(f))^(1/2))*((c*e^2)/4 - (b^2*e^2*log(f))/ 8 - (c^2*d^2*log(f))/2 + (b*c*d*e*log(f))/2))/(c^2*log(f)*(c*log(f))^(1/2) )