Integrand size = 25, antiderivative size = 95 \[ \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx=\frac {e^{-\frac {(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}} f^a g^d \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+e \log (g)+2 x (c \log (f)+f \log (g))}{2 \sqrt {c \log (f)+f \log (g)}}\right )}{2 \sqrt {c \log (f)+f \log (g)}} \]
1/2*f^a*g^d*erfi(1/2*(b*ln(f)+e*ln(g)+2*x*(c*ln(f)+f*ln(g)))/(c*ln(f)+f*ln (g))^(1/2))*Pi^(1/2)/exp(1/4*(b*ln(f)+e*ln(g))^2/(c*ln(f)+f*ln(g)))/(c*ln( f)+f*ln(g))^(1/2)
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.98 \[ \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx=\frac {e^{-\frac {(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}} f^a g^d \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \log (f)+(e+2 f x) \log (g)}{2 \sqrt {c \log (f)+f \log (g)}}\right )}{2 \sqrt {c \log (f)+f \log (g)}} \]
(f^a*g^d*Sqrt[Pi]*Erfi[((b + 2*c*x)*Log[f] + (e + 2*f*x)*Log[g])/(2*Sqrt[c *Log[f] + f*Log[g]])])/(2*E^((b*Log[f] + e*Log[g])^2/(4*(c*Log[f] + f*Log[ g])))*Sqrt[c*Log[f] + f*Log[g]])
Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2725, 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.
\(\displaystyle \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx\) |
\(\Big \downarrow \) 2725 |
\(\displaystyle \int \exp \left (a \log (f)+x (b \log (f)+e \log (g))+x^2 (c \log (f)+f \log (g))+d \log (g)\right )dx\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle f^a g^d \exp \left (-\frac {(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}\right ) \int \exp \left (\frac {(b \log (f)+e \log (g)+2 x (c \log (f)+f \log (g)))^2}{4 (c \log (f)+f \log (g))}\right )dx\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {\sqrt {\pi } f^a g^d \exp \left (-\frac {(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+f \log (g))+e \log (g)}{2 \sqrt {c \log (f)+f \log (g)}}\right )}{2 \sqrt {c \log (f)+f \log (g)}}\) |
(f^a*g^d*Sqrt[Pi]*Erfi[(b*Log[f] + e*Log[g] + 2*x*(c*Log[f] + f*Log[g]))/( 2*Sqrt[c*Log[f] + f*Log[g]])])/(2*E^((b*Log[f] + e*Log[g])^2/(4*(c*Log[f] + f*Log[g])))*Sqrt[c*Log[f] + f*Log[g]])
3.1.65.3.1 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[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x], x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]
\[\int f^{c \,x^{2}+b x +a} g^{f \,x^{2}+e x +d}d x\]
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.42 \[ \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx=-\frac {\sqrt {\pi } \sqrt {-c \log \left (f\right ) - f \log \left (g\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + {\left (2 \, f x + e\right )} \log \left (g\right )\right )} \sqrt {-c \log \left (f\right ) - f \log \left (g\right )}}{2 \, {\left (c \log \left (f\right ) + f \log \left (g\right )\right )}}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 2 \, {\left (2 \, c d - b e + 2 \, a f\right )} \log \left (f\right ) \log \left (g\right ) + {\left (e^{2} - 4 \, d f\right )} \log \left (g\right )^{2}}{4 \, {\left (c \log \left (f\right ) + f \log \left (g\right )\right )}}\right )}}{2 \, {\left (c \log \left (f\right ) + f \log \left (g\right )\right )}} \]
-1/2*sqrt(pi)*sqrt(-c*log(f) - f*log(g))*erf(1/2*((2*c*x + b)*log(f) + (2* f*x + e)*log(g))*sqrt(-c*log(f) - f*log(g))/(c*log(f) + f*log(g)))*e^(-1/4 *((b^2 - 4*a*c)*log(f)^2 - 2*(2*c*d - b*e + 2*a*f)*log(f)*log(g) + (e^2 - 4*d*f)*log(g)^2)/(c*log(f) + f*log(g)))/(c*log(f) + f*log(g))
\[ \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx=\int f^{a + b x + c x^{2}} g^{d + e x + f x^{2}}\, dx \]
Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95 \[ \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx=\frac {\sqrt {\pi } f^{a} g^{d} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f \log \left (g\right )} x - \frac {b \log \left (f\right ) + e \log \left (g\right )}{2 \, \sqrt {-c \log \left (f\right ) - f \log \left (g\right )}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) + e \log \left (g\right )\right )}^{2}}{4 \, {\left (c \log \left (f\right ) + f \log \left (g\right )\right )}}\right )}}{2 \, \sqrt {-c \log \left (f\right ) - f \log \left (g\right )}} \]
1/2*sqrt(pi)*f^a*g^d*erf(sqrt(-c*log(f) - f*log(g))*x - 1/2*(b*log(f) + e* log(g))/sqrt(-c*log(f) - f*log(g)))*e^(-1/4*(b*log(f) + e*log(g))^2/(c*log (f) + f*log(g)))/sqrt(-c*log(f) - f*log(g))
Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.37 \[ \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - f \log \left (g\right )} {\left (2 \, x + \frac {b \log \left (f\right ) + e \log \left (g\right )}{c \log \left (f\right ) + f \log \left (g\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) \log \left (g\right ) + 2 \, b e \log \left (f\right ) \log \left (g\right ) - 4 \, a f \log \left (f\right ) \log \left (g\right ) + e^{2} \log \left (g\right )^{2} - 4 \, d f \log \left (g\right )^{2}}{4 \, {\left (c \log \left (f\right ) + f \log \left (g\right )\right )}}\right )}}{2 \, \sqrt {-c \log \left (f\right ) - f \log \left (g\right )}} \]
-1/2*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) - f*log(g))*(2*x + (b*log(f) + e*log (g))/(c*log(f) + f*log(g))))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*log(f)^2 - 4*c* d*log(f)*log(g) + 2*b*e*log(f)*log(g) - 4*a*f*log(f)*log(g) + e^2*log(g)^2 - 4*d*f*log(g)^2)/(c*log(f) + f*log(g)))/sqrt(-c*log(f) - f*log(g))
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.37 \[ \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx=-\frac {f^a\,g^d\,\sqrt {\pi }\,{\mathrm {e}}^{-\frac {b^2\,{\ln \left (f\right )}^2}{4\,\left (c\,\ln \left (f\right )+f\,\ln \left (g\right )\right )}-\frac {e^2\,{\ln \left (g\right )}^2}{4\,\left (c\,\ln \left (f\right )+f\,\ln \left (g\right )\right )}-\frac {b\,e\,\ln \left (f\right )\,\ln \left (g\right )}{2\,\left (c\,\ln \left (f\right )+f\,\ln \left (g\right )\right )}}\,\mathrm {erf}\left (\frac {x\,\left (c\,\ln \left (f\right )+f\,\ln \left (g\right )\right )\,2{}\mathrm {i}+b\,\ln \left (f\right )\,1{}\mathrm {i}+e\,\ln \left (g\right )\,1{}\mathrm {i}}{2\,\sqrt {c\,\ln \left (f\right )+f\,\ln \left (g\right )}}\right )\,1{}\mathrm {i}}{2\,\sqrt {c\,\ln \left (f\right )+f\,\ln \left (g\right )}} \]