Integrand size = 20, antiderivative size = 43 \[ \int f^{b x+c x^2} (b+2 c x)^3 \, dx=-\frac {4 c f^{b x+c x^2}}{\log ^2(f)}+\frac {f^{b x+c x^2} (b+2 c x)^2}{\log (f)} \]
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.67 \[ \int f^{b x+c x^2} (b+2 c x)^3 \, dx=\frac {f^{x (b+c x)} \left (-4 c+(b+2 c x)^2 \log (f)\right )}{\log ^2(f)} \]
Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2667, 2666}
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.
\(\displaystyle \int (b+2 c x)^3 f^{b x+c x^2} \, dx\) |
\(\Big \downarrow \) 2667 |
\(\displaystyle \frac {(b+2 c x)^2 f^{b x+c x^2}}{\log (f)}-\frac {4 c \int f^{c x^2+b x} (b+2 c x)dx}{\log (f)}\) |
\(\Big \downarrow \) 2666 |
\(\displaystyle \frac {(b+2 c x)^2 f^{b x+c x^2}}{\log (f)}-\frac {4 c f^{b x+c x^2}}{\log ^2(f)}\) |
3.5.56.3.1 Defintions of rubi rules used
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] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[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[(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] && EqQ[b*e - 2*c*d, 0] && GtQ[m, 1]
Time = 0.35 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {\left (4 \ln \left (f \right ) c^{2} x^{2}+4 b c x \ln \left (f \right )+\ln \left (f \right ) b^{2}-4 c \right ) f^{x \left (x c +b \right )}}{\ln \left (f \right )^{2}}\) | \(42\) |
gosper | \(\frac {\left (4 \ln \left (f \right ) c^{2} x^{2}+4 b c x \ln \left (f \right )+\ln \left (f \right ) b^{2}-4 c \right ) f^{c \,x^{2}+b x}}{\ln \left (f \right )^{2}}\) | \(44\) |
norman | \(\frac {\left (\ln \left (f \right ) b^{2}-4 c \right ) {\mathrm e}^{\left (c \,x^{2}+b x \right ) \ln \left (f \right )}}{\ln \left (f \right )^{2}}+\frac {4 c^{2} x^{2} {\mathrm e}^{\left (c \,x^{2}+b x \right ) \ln \left (f \right )}}{\ln \left (f \right )}+\frac {4 c b x \,{\mathrm e}^{\left (c \,x^{2}+b x \right ) \ln \left (f \right )}}{\ln \left (f \right )}\) | \(77\) |
parallelrisch | \(\frac {4 x^{2} f^{c \,x^{2}+b x} c^{2} \ln \left (f \right )+4 x \,f^{c \,x^{2}+b x} c b \ln \left (f \right )+\ln \left (f \right ) f^{c \,x^{2}+b x} b^{2}-4 f^{c \,x^{2}+b x} c}{\ln \left (f \right )^{2}}\) | \(77\) |
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.93 \[ \int f^{b x+c x^2} (b+2 c x)^3 \, dx=\frac {{\left ({\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (f\right ) - 4 \, c\right )} f^{c x^{2} + b x}}{\log \left (f\right )^{2}} \]
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (39) = 78\).
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.93 \[ \int f^{b x+c x^2} (b+2 c x)^3 \, dx=\begin {cases} \frac {f^{b x + c x^{2}} \left (b^{2} \log {\left (f \right )} + 4 b c x \log {\left (f \right )} + 4 c^{2} x^{2} \log {\left (f \right )} - 4 c\right )}{\log {\left (f \right )}^{2}} & \text {for}\: \log {\left (f \right )}^{2} \neq 0 \\b^{3} x + 3 b^{2} c x^{2} + 4 b c^{2} x^{3} + 2 c^{3} x^{4} & \text {otherwise} \end {cases} \]
Piecewise((f**(b*x + c*x**2)*(b**2*log(f) + 4*b*c*x*log(f) + 4*c**2*x**2*l og(f) - 4*c)/log(f)**2, Ne(log(f)**2, 0)), (b**3*x + 3*b**2*c*x**2 + 4*b*c **2*x**3 + 2*c**3*x**4, True))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.50 (sec) , antiderivative size = 536, normalized size of antiderivative = 12.47 \[ \int f^{b x+c x^2} (b+2 c x)^3 \, dx=\frac {\sqrt {\pi } b^{3} \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}}} - \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 )} b^{2} c}{2 \, \sqrt {c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} + \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 )} b c^{2}}{2 \, \sqrt {c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} - \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 )} c^{3}}{2 \, \sqrt {c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
1/2*sqrt(pi)*b^3*erf(sqrt(-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log(f)))/(sq rt(-c*log(f))*f^(1/4*b^2/c)) - 3/2*(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*l og(f))^(3/2)) - 2*c*f^(1/4*(2*c*x + b)^2/c)*log(f)/(c*log(f))^(3/2))*b^2*c /(sqrt(c*log(f))*f^(1/4*b^2/c)) + 3/2*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*s qrt(-(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))*b*c^2/(sqrt(c*log(f))*f^(1 /4*b^2/c)) - 1/2*(sqrt(pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2*lo g(f)/c)) - 1)*log(f)^4/(sqrt(-(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))*c^3/(sqrt(c*log(f))*f^(1/4*b^2/c))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 742, normalized size of antiderivative = 17.26 \[ \int f^{b x+c x^2} (b+2 c x)^3 \, dx=\text {Too large to display} \]
(2*((b^2*log(abs(f)) + 4*(c*x^2 + b*x)*c*log(abs(f)) - 4*c)*(pi^2*sgn(f) - pi^2 + 2*log(abs(f))^2)/((pi^2*sgn(f) - pi^2 + 2*log(abs(f))^2)^2 + 4*(pi *log(abs(f))*sgn(f) - pi*log(abs(f)))^2) + (pi*b^2*sgn(f) + 4*pi*(c*x^2 + b*x)*c*sgn(f) - pi*b^2 - 4*pi*(c*x^2 + b*x)*c)*(pi*log(abs(f))*sgn(f) - pi *log(abs(f)))/((pi^2*sgn(f) - pi^2 + 2*log(abs(f))^2)^2 + 4*(pi*log(abs(f) )*sgn(f) - pi*log(abs(f)))^2))*cos(-1/2*pi*c*x^2*sgn(f) + 1/2*pi*c*x^2 - 1 /2*pi*b*x*sgn(f) + 1/2*pi*b*x) + ((pi*b^2*sgn(f) + 4*pi*(c*x^2 + b*x)*c*sg n(f) - pi*b^2 - 4*pi*(c*x^2 + b*x)*c)*(pi^2*sgn(f) - pi^2 + 2*log(abs(f))^ 2)/((pi^2*sgn(f) - pi^2 + 2*log(abs(f))^2)^2 + 4*(pi*log(abs(f))*sgn(f) - pi*log(abs(f)))^2) - 4*(b^2*log(abs(f)) + 4*(c*x^2 + b*x)*c*log(abs(f)) - 4*c)*(pi*log(abs(f))*sgn(f) - pi*log(abs(f)))/((pi^2*sgn(f) - pi^2 + 2*log (abs(f))^2)^2 + 4*(pi*log(abs(f))*sgn(f) - pi*log(abs(f)))^2))*sin(-1/2*pi *c*x^2*sgn(f) + 1/2*pi*c*x^2 - 1/2*pi*b*x*sgn(f) + 1/2*pi*b*x))*abs(f)^(c* x^2 + b*x) - 1/2*I*abs(f)^(c*x^2 + b*x)*((pi*b^2*sgn(f) + 4*pi*(c*x^2 + b* x)*c*sgn(f) - pi*b^2 - 4*pi*(c*x^2 + b*x)*c - 2*I*b^2*log(abs(f)) + 8*(-I* c*x^2 - I*b*x)*c*log(abs(f)) + 8*I*c)*e^(1/2*I*pi*c*x^2*sgn(f) - 1/2*I*pi* c*x^2 + 1/2*I*pi*b*x*sgn(f) - 1/2*I*pi*b*x)/(pi^2*sgn(f) + 2*I*pi*log(abs( f))*sgn(f) - pi^2 - 2*I*pi*log(abs(f)) + 2*log(abs(f))^2) + (pi*b^2*sgn(f) + 4*pi*(c*x^2 + b*x)*c*sgn(f) - pi*b^2 - 4*pi*(c*x^2 + b*x)*c + 2*I*b^2*l og(abs(f)) - 8*(-I*c*x^2 - I*b*x)*c*log(abs(f)) - 8*I*c)*e^(-1/2*I*pi*c...
Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int f^{b x+c x^2} (b+2 c x)^3 \, dx=\frac {f^{c\,x^2+b\,x}\,\left (\ln \left (f\right )\,b^2+4\,\ln \left (f\right )\,b\,c\,x+4\,\ln \left (f\right )\,c^2\,x^2-4\,c\right )}{{\ln \left (f\right )}^2} \]