Integrand size = 20, antiderivative size = 266 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=-\frac {e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac {3 e^2 (2 c d-b e) f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)}+\frac {e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac {e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac {e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac {(2 c d-b e)^3 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{16 c^{7/2} \sqrt {\log (f)}} \]
-1/2*e^3*f^(c*x^2+b*x+a)/c^2/ln(f)^2+1/8*e*(-b*e+2*c*d)^2*f^(c*x^2+b*x+a)/ c^3/ln(f)+1/4*e*(-b*e+2*c*d)*f^(c*x^2+b*x+a)*(e*x+d)/c^2/ln(f)+1/2*e*f^(c* x^2+b*x+a)*(e*x+d)^2/c/ln(f)-3/8*e^2*(-b*e+2*c*d)*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)^(3/2)+1/16*(-b*e+2* c*d)^3*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^ (7/2)/ln(f)^(1/2)
Time = 0.44 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.64 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=\frac {f^{a-\frac {b^2}{4 c}} \left ((2 c d-b e) \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)} \left (-6 c e^2+(-2 c d+b e)^2 \log (f)\right )+2 \sqrt {c} e f^{\frac {(b+2 c x)^2}{4 c}} \left (-4 c e^2+\left (b^2 e^2-2 b c e (3 d+e x)+4 c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \log (f)\right )\right )}{16 c^{7/2} \log ^2(f)} \]
(f^(a - b^2/(4*c))*((2*c*d - b*e)*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]]) /(2*Sqrt[c])]*Sqrt[Log[f]]*(-6*c*e^2 + (-2*c*d + b*e)^2*Log[f]) + 2*Sqrt[c ]*e*f^((b + 2*c*x)^2/(4*c))*(-4*c*e^2 + (b^2*e^2 - 2*b*c*e*(3*d + e*x) + 4 *c^2*(3*d^2 + 3*d*e*x + e^2*x^2))*Log[f])))/(16*c^(7/2)*Log[f]^2)
Time = 0.96 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.30, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2671, 2670, 2664, 2633, 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.
| \(\displaystyle \int (d+e x)^3 f^{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 2671 |
| \(\displaystyle -\frac {e^2 \int f^{c x^2+b x+a} (d+e x)dx}{c \log (f)}+\frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)^2dx}{2 c}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2670 |
| \(\displaystyle -\frac {e^2 \left (\frac {(2 c d-b e) \int f^{c x^2+b x+a}dx}{2 c}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)^2dx}{2 c}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle -\frac {e^2 \left (\frac {f^{a-\frac {b^2}{4 c}} (2 c d-b e) \int f^{\frac {(b+2 c x)^2}{4 c}}dx}{2 c}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)^2dx}{2 c}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)^2dx}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2671 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)dx}{2 c}-\frac {e^2 \int f^{c x^2+b x+a}dx}{2 c \log (f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {(2 c d-b e) \left (-\frac {e^2 f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}}dx}{2 c \log (f)}+\frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)dx}{2 c}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \int f^{c x^2+b x+a} (d+e x)dx}{2 c}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2670 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {(2 c d-b e) \int f^{c x^2+b x+a}dx}{2 c}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2664 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {f^{a-\frac {b^2}{4 c}} (2 c d-b e) \int f^{\frac {(b+2 c x)^2}{4 c}}dx}{2 c}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {\sqrt {\pi } e^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {e (d+e x) f^{a+b x+c x^2}}{2 c \log (f)}\right )}{2 c}-\frac {e^2 \left (\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \sqrt {\log (f)}}+\frac {e f^{a+b x+c x^2}}{2 c \log (f)}\right )}{c \log (f)}+\frac {e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)}\) |
((2*c*d - b*e)*(((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])))/(2*c) + (e*f^(a + b*x + c*x^2)*(d + e*x)^2)/(2*c*Log[f]) - (e^2*((e*f^(a + b*x + c*x^2))/(2*c*Lo g[f]) + ((2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[L og[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Sqrt[Log[f]])))/(c*Log[f])
3.5.44.3.1 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]
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 )}}\) | \(550\) |
-1/2*f^a*d^3*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^3*f^a/ln(f)/c*x^2*f^(b*x)*f^(c*x^2 )-1/4*e^3*f^a*b/c^2/ln(f)*x*f^(b*x)*f^(c*x^2)+1/8*e^3*f^a*b^2/c^3/ln(f)*f^ (b*x)*f^(c*x^2)+1/16*e^3*f^a*b^3/c^3*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))-3/8*e^3*f^a*b/c^ 2/ln(f)*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^3*f^a/c^2/ln(f)^2*f^(b*x)*f^(c*x^2)+3/2 *e^2*d*f^a/ln(f)/c*x*f^(b*x)*f^(c*x^2)-3/4*e^2*d*f^a*b/c^2/ln(f)*f^(b*x)*f ^(c*x^2)-3/8*e^2*d*f^a*b^2/c^2*Pi^(1/2)*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*er f(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))+3/4*e^2*d*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))+3/2*f^a*d^2*e/ln(f)/c*f^(b*x)*f^(c*x^2)+3/4*f^a*d^2*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 = 0.36 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.77 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=-\frac {2 \, {\left (4 \, c^{2} e^{3} - {\left (4 \, c^{3} e^{3} x^{2} + 12 \, c^{3} d^{2} e - 6 \, b c^{2} d e^{2} + b^{2} c e^{3} + 2 \, {\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x\right )} \log \left (f\right )\right )} f^{c x^{2} + b x + a} - \frac {\sqrt {\pi } {\left (12 \, c^{2} d e^{2} - 6 \, b c e^{3} - {\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{16 \, c^{4} \log \left (f\right )^{2}} \]
-1/16*(2*(4*c^2*e^3 - (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*(6*c^3*d*e^2 - b*c^2*e^3)*x)*log(f))*f^(c*x^2 + b*x + a) - sqrt( pi)*(12*c^2*d*e^2 - 6*b*c*e^3 - (8*c^3*d^3 - 12*b*c^2*d^2*e + 6*b^2*c*d*e^ 2 - b^3*e^3)*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^4*log(f)^2)
\[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=\int f^{a + b x + c x^{2}} \left (d + e x\right )^{3}\, dx \]
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}}} \]
-3/4*(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^2*e*f^(a - 1/4*b^2/c)/sqrt(c*log( f)) + 3/8*(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 *log(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))*d*e^2*f^(a - 1/4*b^2/c)/sqrt(c*log(f)) - 1/16*(sqrt(pi )*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^4/(s qrt(-(2*c*x + b)^2*log(f)/c)*(c*log(f))^(7/2)) - 12*(2*c*x + b)^3*b*gamma( 3/2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^4/((-(2*c*x + b)^2*log(f)/c)^(3/2 )*(c*log(f))^(7/2)) - 6*b^2*c*f^(1/4*(2*c*x + b)^2/c)*log(f)^3/(c*log(f))^ (7/2) + 8*c^2*gamma(2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^2/(c*log(f))^(7 /2))*e^3*f^(a - 1/4*b^2/c)/sqrt(c*log(f)) + 1/2*sqrt(pi)*d^3*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 = 0.33 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.90 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=-\frac {\frac {\sqrt {\pi } {\left (8 \, c^{3} d^{3} \log \left (f\right ) - 12 \, b c^{2} d^{2} e \log \left (f\right ) + 6 \, b^{2} c d e^{2} \log \left (f\right ) - b^{3} e^{3} \log \left (f\right ) - 12 \, c^{2} d e^{2} + 6 \, b c e^{3}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{\sqrt {-c \log \left (f\right )} \log \left (f\right )} - \frac {2 \, {\left (c^{2} e^{3} {\left (2 \, x + \frac {b}{c}\right )}^{2} \log \left (f\right ) + 6 \, c^{2} d e^{2} {\left (2 \, x + \frac {b}{c}\right )} \log \left (f\right ) - 3 \, b c e^{3} {\left (2 \, x + \frac {b}{c}\right )} \log \left (f\right ) + 12 \, c^{2} d^{2} e \log \left (f\right ) - 12 \, b c d e^{2} \log \left (f\right ) + 3 \, b^{2} e^{3} \log \left (f\right ) - 4 \, c e^{3}\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right )\right )}}{\log \left (f\right )^{2}}}{16 \, c^{3}} \]
-1/16*(sqrt(pi)*(8*c^3*d^3*log(f) - 12*b*c^2*d^2*e*log(f) + 6*b^2*c*d*e^2* log(f) - b^3*e^3*log(f) - 12*c^2*d*e^2 + 6*b*c*e^3)*erf(-1/2*sqrt(-c*log(f ))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f))/c)/(sqrt(-c*log(f))*lo g(f)) - 2*(c^2*e^3*(2*x + b/c)^2*log(f) + 6*c^2*d*e^2*(2*x + b/c)*log(f) - 3*b*c*e^3*(2*x + b/c)*log(f) + 12*c^2*d^2*e*log(f) - 12*b*c*d*e^2*log(f) + 3*b^2*e^3*log(f) - 4*c*e^3)*e^(c*x^2*log(f) + b*x*log(f) + a*log(f))/log (f)^2)/c^3
Time = 0.60 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.94 \[ \int f^{a+b x+c x^2} (d+e x)^3 \, dx=\frac {e^3\,f^a\,f^{c\,x^2}\,f^{b\,x}\,x^2}{2\,c\,\ln \left (f\right )}-\frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \left (f\right )}{2}+c\,x\,\ln \left (f\right )}{\sqrt {c\,\ln \left (f\right )}}\right )\,\left (\frac {\ln \left (f\right )\,b^3\,e^3}{16}-\frac {3\,\ln \left (f\right )\,b^2\,c\,d\,e^2}{8}+\frac {3\,\ln \left (f\right )\,b\,c^2\,d^2\,e}{4}-\frac {3\,b\,c\,e^3}{8}-\frac {\ln \left (f\right )\,c^3\,d^3}{2}+\frac {3\,c^2\,d\,e^2}{4}\right )}{c^3\,\ln \left (f\right )\,\sqrt {c\,\ln \left (f\right )}}-\frac {f^a\,f^{c\,x^2}\,f^{b\,x}\,x\,\left (b\,e^3-6\,c\,d\,e^2\right )}{4\,c^2\,\ln \left (f\right )}-f^a\,f^{c\,x^2}\,f^{b\,x}\,\left (\frac {e^3}{2\,c^2\,{\ln \left (f\right )}^2}-\frac {3\,d^2\,e}{2\,c\,\ln \left (f\right )}-\frac {b^2\,e^3}{8\,c^3\,\ln \left (f\right )}+\frac {3\,b\,d\,e^2}{4\,c^2\,\ln \left (f\right )}\right ) \]
(e^3*f^a*f^(c*x^2)*f^(b*x)*x^2)/(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))*((3*c^2*d*e^2)/4 + (b^ 3*e^3*log(f))/16 - (c^3*d^3*log(f))/2 - (3*b*c*e^3)/8 + (3*b*c^2*d^2*e*log (f))/4 - (3*b^2*c*d*e^2*log(f))/8))/(c^3*log(f)*(c*log(f))^(1/2)) - (f^a*f ^(c*x^2)*f^(b*x)*x*(b*e^3 - 6*c*d*e^2))/(4*c^2*log(f)) - f^a*f^(c*x^2)*f^( b*x)*(e^3/(2*c^2*log(f)^2) - (3*d^2*e)/(2*c*log(f)) - (b^2*e^3)/(8*c^3*log (f)) + (3*b*d*e^2)/(4*c^2*log(f)))