Integrand size = 21, antiderivative size = 258 \[ \int F^{a+b (c+d x)^2} (e+f x)^3 \, dx=-\frac {f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}-\frac {3 f^2 (d e-c f) F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{4 b^{3/2} d^4 \log ^{\frac {3}{2}}(F)}+\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {3 f^2 (d e-c f) F^{a+b (c+d x)^2} (c+d x)}{2 b d^4 \log (F)}+\frac {f^3 F^{a+b (c+d x)^2} (c+d x)^2}{2 b d^4 \log (F)}+\frac {(d e-c f)^3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d^4 \sqrt {\log (F)}} \]
-1/2*f^3*F^(a+b*(d*x+c)^2)/b^2/d^4/ln(F)^2+3/2*f*(-c*f+d*e)^2*F^(a+b*(d*x+ c)^2)/b/d^4/ln(F)+3/2*f^2*(-c*f+d*e)*F^(a+b*(d*x+c)^2)*(d*x+c)/b/d^4/ln(F) +1/2*f^3*F^(a+b*(d*x+c)^2)*(d*x+c)^2/b/d^4/ln(F)-3/4*f^2*(-c*f+d*e)*F^a*er fi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(3/2)/d^4/ln(F)^(3/2)+1/2*(-c*f +d*e)^3*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/d^4/b^(1/2)/ln(F)^( 1/2)
Time = 1.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.57 \[ \int F^{a+b (c+d x)^2} (e+f x)^3 \, dx=\frac {F^a \left (\sqrt {b} (d e-c f) \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \sqrt {\log (F)} \left (-3 f^2+2 b (d e-c f)^2 \log (F)\right )+2 f F^{b (c+d x)^2} \left (-f^2+b \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right ) \log (F)\right )\right )}{4 b^2 d^4 \log ^2(F)} \]
(F^a*(Sqrt[b]*(d*e - c*f)*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Sq rt[Log[F]]*(-3*f^2 + 2*b*(d*e - c*f)^2*Log[F]) + 2*f*F^(b*(c + d*x)^2)*(-f ^2 + b*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))*Log [F])))/(4*b^2*d^4*Log[F]^2)
Time = 0.62 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2656, 2009}
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.
| \(\displaystyle \int (e+f x)^3 F^{a+b (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \int \left (\frac {3 f^2 (c+d x)^2 (d e-c f) F^{a+b (c+d x)^2}}{d^3}+\frac {(d e-c f)^3 F^{a+b (c+d x)^2}}{d^3}+\frac {3 f (c+d x) (d e-c f)^2 F^{a+b (c+d x)^2}}{d^3}+\frac {f^3 (c+d x)^3 F^{a+b (c+d x)^2}}{d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt {\pi } f^2 F^a (d e-c f) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac {3}{2}}(F)}-\frac {f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^3 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^4 \sqrt {\log (F)}}+\frac {3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}\) |
-1/2*(f^3*F^(a + b*(c + d*x)^2))/(b^2*d^4*Log[F]^2) - (3*f^2*(d*e - c*f)*F ^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(4*b^(3/2)*d^4*Log[F]^(3 /2)) + (3*f*(d*e - c*f)^2*F^(a + b*(c + d*x)^2))/(2*b*d^4*Log[F]) + (3*f^2 *(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x))/(2*b*d^4*Log[F]) + (f^3*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(2*b*d^4*Log[F]) + ((d*e - c*f)^3*F^a*Sqrt[ Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^4*Sqrt[Log[F]])
3.4.84.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[Px, x]
Leaf count of result is larger than twice the leaf count of optimal. \(706\) vs. \(2(226)=452\).
Time = 0.36 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.74
| method | result | size |
| risch | \(-\frac {F^{b \,c^{2}} F^{a} e^{3} \sqrt {\pi }\, F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{2 d \sqrt {-b \ln \left (F \right )}}+\frac {f^{3} F^{a} F^{b \,c^{2}} x^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b \,d^{2}}-\frac {f^{3} F^{a} F^{b \,c^{2}} c x \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 d^{3} \ln \left (F \right ) b}+\frac {f^{3} F^{a} F^{b \,c^{2}} c^{2} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 d^{4} \ln \left (F \right ) b}+\frac {f^{3} F^{a} F^{b \,c^{2}} c^{3} \sqrt {\pi }\, F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{2 d^{4} \sqrt {-b \ln \left (F \right )}}-\frac {3 f^{3} F^{a} F^{b \,c^{2}} c \sqrt {\pi }\, F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{4 d^{4} \ln \left (F \right ) b \sqrt {-b \ln \left (F \right )}}-\frac {f^{3} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 d^{4} b^{2} \ln \left (F \right )^{2}}+\frac {3 f^{2} e \,F^{a} F^{b \,c^{2}} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b \,d^{2}}-\frac {3 f^{2} e \,F^{a} F^{b \,c^{2}} c \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 d^{3} \ln \left (F \right ) b}-\frac {3 f^{2} e \,F^{a} F^{b \,c^{2}} c^{2} \sqrt {\pi }\, F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{2 d^{3} \sqrt {-b \ln \left (F \right )}}+\frac {3 f^{2} e \,F^{a} F^{b \,c^{2}} \sqrt {\pi }\, F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{4 \ln \left (F \right ) b \,d^{3} \sqrt {-b \ln \left (F \right )}}+\frac {3 F^{b \,c^{2}} F^{a} e^{2} f \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b \,d^{2}}+\frac {3 F^{b \,c^{2}} F^{a} e^{2} f c \sqrt {\pi }\, F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{2 d^{2} \sqrt {-b \ln \left (F \right )}}\) | \(707\) |
-1/2*F^(b*c^2)*F^a*e^3*Pi^(1/2)*F^(-b*c^2)/d/(-b*ln(F))^(1/2)*erf(-d*(-b*l n(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+1/2*f^3*F^a*F^(b*c^2)/ln(F)/b/d^ 2*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)-1/2*f^3*F^a*F^(b*c^2)/d^3*c/ln(F)/b*x*F^ (b*d^2*x^2)*F^(2*b*c*d*x)+1/2*f^3*F^a*F^(b*c^2)/d^4*c^2/ln(F)/b*F^(b*d^2*x ^2)*F^(2*b*c*d*x)+1/2*f^3*F^a*F^(b*c^2)/d^4*c^3*Pi^(1/2)*F^(-b*c^2)/(-b*ln (F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))-3/4*f^3*F ^a*F^(b*c^2)/d^4*c/ln(F)/b*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b *ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))-1/2*f^3*F^a*F^(b*c^2)/d^4/b^2/ ln(F)^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)+3/2*f^2*e*F^a*F^(b*c^2)/ln(F)/b/d^2*x* F^(b*d^2*x^2)*F^(2*b*c*d*x)-3/2*f^2*e*F^a*F^(b*c^2)/d^3*c/ln(F)/b*F^(b*d^2 *x^2)*F^(2*b*c*d*x)-3/2*f^2*e*F^a*F^(b*c^2)/d^3*c^2*Pi^(1/2)*F^(-b*c^2)/(- b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+3/4*f ^2*e*F^a*F^(b*c^2)/ln(F)/b/d^3*Pi^(1/2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d *(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln(F))^(1/2))+3/2*F^(b*c^2)*F^a*e^2*f/ln (F)/b/d^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)+3/2*F^(b*c^2)*F^a*e^2*f*c/d^2*Pi^(1/ 2)*F^(-b*c^2)/(-b*ln(F))^(1/2)*erf(-d*(-b*ln(F))^(1/2)*x+b*c*ln(F)/(-b*ln( F))^(1/2))
Time = 0.32 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.81 \[ \int F^{a+b (c+d x)^2} (e+f x)^3 \, dx=\frac {\sqrt {\pi } {\left (3 \, d e f^{2} - 3 \, c f^{3} - 2 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (F\right )\right )} \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (d f^{3} - {\left (b d^{3} f^{3} x^{2} + 3 \, b d^{3} e^{2} f - 3 \, b c d^{2} e f^{2} + b c^{2} d f^{3} + {\left (3 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{4 \, b^{2} d^{5} \log \left (F\right )^{2}} \]
1/4*(sqrt(pi)*(3*d*e*f^2 - 3*c*f^3 - 2*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b* c^2*d*e*f^2 - b*c^3*f^3)*log(F))*sqrt(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*l og(F))*(d*x + c)/d) - 2*(d*f^3 - (b*d^3*f^3*x^2 + 3*b*d^3*e^2*f - 3*b*c*d^ 2*e*f^2 + b*c^2*d*f^3 + (3*b*d^3*e*f^2 - b*c*d^2*f^3)*x)*log(F))*F^(b*d^2* x^2 + 2*b*c*d*x + b*c^2 + a))/(b^2*d^5*log(F)^2)
\[ \int F^{a+b (c+d x)^2} (e+f x)^3 \, dx=\int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )^{3}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (226) = 452\).
Time = 0.56 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.69 \[ \int F^{a+b (c+d x)^2} (e+f x)^3 \, dx=-\frac {3 \, {\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b c {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {3}{2}} d^{2} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b \log \left (F\right )}{\left (b \log \left (F\right )\right )^{\frac {3}{2}} d}\right )} F^{a} e^{2} f}{2 \, \sqrt {b \log \left (F\right )} d} + \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{2} c^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{3} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {2 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{2} c \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{2}} - \frac {{\left (b d^{2} x + b c d\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{5} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right )^{\frac {3}{2}}}\right )} F^{a} e f^{2}}{2 \, \sqrt {b \log \left (F\right )} d} - \frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{3} c^{3} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{4}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{4} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {3 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{3} c^{2} \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{3}} - \frac {3 \, {\left (b d^{2} x + b c d\right )}^{3} b c \Gamma \left (\frac {3}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{4}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{6} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right )^{\frac {3}{2}}} + \frac {b^{2} \Gamma \left (2, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{3}}\right )} F^{a} f^{3}}{2 \, \sqrt {b \log \left (F\right )} d} + \frac {\sqrt {\pi } F^{b c^{2} + a} e^{3} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} d x - \frac {b c \log \left (F\right )}{\sqrt {-b \log \left (F\right )}}\right )}{2 \, \sqrt {-b \log \left (F\right )} F^{b c^{2}} d} \]
-3/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b*c*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F) /(b*d^2))) - 1)*log(F)^2/((b*log(F))^(3/2)*d^2*sqrt(-(b*d^2*x + b*c*d)^2*l og(F)/(b*d^2))) - F^((b*d^2*x + b*c*d)^2/(b*d^2))*b*log(F)/((b*log(F))^(3/ 2)*d))*F^a*e^2*f/(sqrt(b*log(F))*d) + 3/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2* c^2*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^3/((b*log( F))^(5/2)*d^3*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 2*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^2*c*log(F)^2/((b*log(F))^(5/2)*d^2) - (b*d^2*x + b*c* d)^3*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*log(F))^ (5/2)*d^5*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^a*e*f^2/(sqrt(b* log(F))*d) - 1/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^4/((b*log(F))^(7/2)*d^4*sqrt(-(b*d^ 2*x + b*c*d)^2*log(F)/(b*d^2))) - 3*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^3*c^ 2*log(F)^3/((b*log(F))^(7/2)*d^3) - 3*(b*d^2*x + b*c*d)^3*b*c*gamma(3/2, - (b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*log(F))^(7/2)*d^6*(-(b*d^ 2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + b^2*gamma(2, -(b*d^2*x + b*c*d)^2* log(F)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*f^3/(sqrt(b*log(F))*d ) + 1/2*sqrt(pi)*F^(b*c^2 + a)*e^3*erf(sqrt(-b*log(F))*d*x - b*c*log(F)/sq rt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)*d)
Time = 0.36 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.96 \[ \int F^{a+b (c+d x)^2} (e+f x)^3 \, dx=-\frac {\frac {\sqrt {\pi } {\left (2 \, b d^{3} e^{3} \log \left (F\right ) - 6 \, b c d^{2} e^{2} f \log \left (F\right ) + 6 \, b c^{2} d e f^{2} \log \left (F\right ) - 2 \, b c^{3} f^{3} \log \left (F\right ) - 3 \, d e f^{2} + 3 \, c f^{3}\right )} F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {-b \log \left (F\right )} b d \log \left (F\right )} - \frac {2 \, {\left (b d^{2} f^{3} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right ) + 3 \, b d^{2} e f^{2} {\left (x + \frac {c}{d}\right )} \log \left (F\right ) - 3 \, b c d f^{3} {\left (x + \frac {c}{d}\right )} \log \left (F\right ) + 3 \, b d^{2} e^{2} f \log \left (F\right ) - 6 \, b c d e f^{2} \log \left (F\right ) + 3 \, b c^{2} f^{3} \log \left (F\right ) - f^{3}\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{2} d \log \left (F\right )^{2}}}{4 \, d^{3}} \]
-1/4*(sqrt(pi)*(2*b*d^3*e^3*log(F) - 6*b*c*d^2*e^2*f*log(F) + 6*b*c^2*d*e* f^2*log(F) - 2*b*c^3*f^3*log(F) - 3*d*e*f^2 + 3*c*f^3)*F^a*erf(-sqrt(-b*lo g(F))*d*(x + c/d))/(sqrt(-b*log(F))*b*d*log(F)) - 2*(b*d^2*f^3*(x + c/d)^2 *log(F) + 3*b*d^2*e*f^2*(x + c/d)*log(F) - 3*b*c*d*f^3*(x + c/d)*log(F) + 3*b*d^2*e^2*f*log(F) - 6*b*c*d*e*f^2*log(F) + 3*b*c^2*f^3*log(F) - f^3)*e^ (b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) + b*c^2*log(F) + a*log(F))/(b^2*d*log (F)^2))/d^3
Time = 0.48 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.21 \[ \int F^{a+b (c+d x)^2} (e+f x)^3 \, dx=\frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (F\right )\,d^2+b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )\,\left (-2\,b\,\ln \left (F\right )\,c^3\,f^3+6\,b\,\ln \left (F\right )\,c^2\,d\,e\,f^2-6\,b\,\ln \left (F\right )\,c\,d^2\,e^2\,f+3\,c\,f^3+2\,b\,\ln \left (F\right )\,d^3\,e^3-3\,d\,e\,f^2\right )}{4\,b\,d^3\,\ln \left (F\right )\,\sqrt {b\,d^2\,\ln \left (F\right )}}-F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (\frac {f^3}{2\,b^2\,d^4\,{\ln \left (F\right )}^2}-\frac {3\,e^2\,f}{2\,b\,d^2\,\ln \left (F\right )}-\frac {c^2\,f^3}{2\,b\,d^4\,\ln \left (F\right )}+\frac {3\,c\,e\,f^2}{2\,b\,d^3\,\ln \left (F\right )}\right )-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (c\,f^3-3\,d\,e\,f^2\right )}{2\,b\,d^3\,\ln \left (F\right )}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f^3\,x^2}{2\,b\,d^2\,\ln \left (F\right )} \]
(F^a*pi^(1/2)*erfi((b*c*d*log(F) + b*d^2*x*log(F))/(b*d^2*log(F))^(1/2))*( 3*c*f^3 - 3*d*e*f^2 - 2*b*c^3*f^3*log(F) + 2*b*d^3*e^3*log(F) - 6*b*c*d^2* e^2*f*log(F) + 6*b*c^2*d*e*f^2*log(F)))/(4*b*d^3*log(F)*(b*d^2*log(F))^(1/ 2)) - F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(f^3/(2*b^2*d^4*log(F)^2) - (3*e^2*f)/(2*b*d^2*log(F)) - (c^2*f^3)/(2*b*d^4*log(F)) + (3*c*e*f^2)/(2 *b*d^3*log(F))) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x*(c*f^3 - 3* d*e*f^2))/(2*b*d^3*log(F)) + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*f^ 3*x^2)/(2*b*d^2*log(F))