Integrand size = 15, antiderivative size = 140 \[ \int f^{c (a+b x)^2} x^2 \, dx=-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{4 b^3 c^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {a f^{c (a+b x)^2}}{b^3 c \log (f)}+\frac {f^{c (a+b x)^2} (a+b x)}{2 b^3 c \log (f)}+\frac {a^2 \sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b^3 \sqrt {c} \sqrt {\log (f)}} \] Output:
-1/4*Pi^(1/2)*erfi(c^(1/2)*(b*x+a)*ln(f)^(1/2))/b^3/c^(3/2)/ln(f)^(3/2)-a* f^(c*(b*x+a)^2)/b^3/c/ln(f)+1/2*f^(c*(b*x+a)^2)*(b*x+a)/b^3/c/ln(f)+1/2*a^ 2*Pi^(1/2)*erfi(c^(1/2)*(b*x+a)*ln(f)^(1/2))/b^3/c^(1/2)/ln(f)^(1/2)
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.59 \[ \int f^{c (a+b x)^2} x^2 \, dx=\frac {-2 \sqrt {c} f^{c (a+b x)^2} (a-b x) \sqrt {\log (f)}+\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right ) \left (-1+2 a^2 c \log (f)\right )}{4 b^3 c^{3/2} \log ^{\frac {3}{2}}(f)} \] Input:
Integrate[f^(c*(a + b*x)^2)*x^2,x]
Output:
(-2*Sqrt[c]*f^(c*(a + b*x)^2)*(a - b*x)*Sqrt[Log[f]] + Sqrt[Pi]*Erfi[Sqrt[ c]*(a + b*x)*Sqrt[Log[f]]]*(-1 + 2*a^2*c*Log[f]))/(4*b^3*c^(3/2)*Log[f]^(3 /2))
Time = 0.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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.
\(\displaystyle \int x^2 f^{c (a+b x)^2} \, dx\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle \int \left (\frac {a^2 f^{c (a+b x)^2}}{b^2}+\frac {(a+b x)^2 f^{c (a+b x)^2}}{b^2}-\frac {2 a (a+b x) f^{c (a+b x)^2}}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\pi } a^2 \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b^3 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{4 b^3 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {(a+b x) f^{c (a+b x)^2}}{2 b^3 c \log (f)}-\frac {a f^{c (a+b x)^2}}{b^3 c \log (f)}\) |
Input:
Int[f^(c*(a + b*x)^2)*x^2,x]
Output:
-1/4*(Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(b^3*c^(3/2)*Log[f]^( 3/2)) - (a*f^(c*(a + b*x)^2))/(b^3*c*Log[f]) + (f^(c*(a + b*x)^2)*(a + b*x ))/(2*b^3*c*Log[f]) + (a^2*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/ (2*b^3*Sqrt[c]*Sqrt[Log[f]])
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]
Time = 0.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {x \,f^{b^{2} c \,x^{2}} f^{2 a b c x} f^{a^{2} c}}{2 c \,b^{2} \ln \left (f \right )}-\frac {a \,f^{b^{2} c \,x^{2}} f^{2 a b c x} f^{a^{2} c}}{2 b^{3} c \ln \left (f \right )}-\frac {a^{2} \sqrt {\pi }\, \operatorname {erf}\left (-b \sqrt {-c \ln \left (f \right )}\, x +\frac {a c \ln \left (f \right )}{\sqrt {-c \ln \left (f \right )}}\right )}{2 b^{3} \sqrt {-c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (-b \sqrt {-c \ln \left (f \right )}\, x +\frac {a c \ln \left (f \right )}{\sqrt {-c \ln \left (f \right )}}\right )}{4 c \,b^{3} \ln \left (f \right ) \sqrt {-c \ln \left (f \right )}}\) | \(168\) |
Input:
int(f^(c*(b*x+a)^2)*x^2,x,method=_RETURNVERBOSE)
Output:
1/2/c/b^2/ln(f)*x*f^(b^2*c*x^2)*f^(2*a*b*c*x)*f^(a^2*c)-1/2*a/b^3/c/ln(f)* f^(b^2*c*x^2)*f^(2*a*b*c*x)*f^(a^2*c)-1/2*a^2/b^3*Pi^(1/2)/(-c*ln(f))^(1/2 )*erf(-b*(-c*ln(f))^(1/2)*x+a*c*ln(f)/(-c*ln(f))^(1/2))+1/4/c/b^3/ln(f)*Pi ^(1/2)/(-c*ln(f))^(1/2)*erf(-b*(-c*ln(f))^(1/2)*x+a*c*ln(f)/(-c*ln(f))^(1/ 2))
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.68 \[ \int f^{c (a+b x)^2} x^2 \, dx=-\frac {\sqrt {\pi } {\left (2 \, a^{2} c \log \left (f\right ) - 1\right )} \sqrt {-b^{2} c \log \left (f\right )} \operatorname {erf}\left (\frac {\sqrt {-b^{2} c \log \left (f\right )} {\left (b x + a\right )}}{b}\right ) - 2 \, {\left (b^{2} c x - a b c\right )} f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c} \log \left (f\right )}{4 \, b^{4} c^{2} \log \left (f\right )^{2}} \] Input:
integrate(f^(c*(b*x+a)^2)*x^2,x, algorithm="fricas")
Output:
-1/4*(sqrt(pi)*(2*a^2*c*log(f) - 1)*sqrt(-b^2*c*log(f))*erf(sqrt(-b^2*c*lo g(f))*(b*x + a)/b) - 2*(b^2*c*x - a*b*c)*f^(b^2*c*x^2 + 2*a*b*c*x + a^2*c) *log(f))/(b^4*c^2*log(f)^2)
\[ \int f^{c (a+b x)^2} x^2 \, dx=\int f^{c \left (a + b x\right )^{2}} x^{2}\, dx \] Input:
integrate(f**(c*(b*x+a)**2)*x**2,x)
Output:
Integral(f**(c*(a + b*x)**2)*x**2, x)
Time = 0.14 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.56 \[ \int f^{c (a+b x)^2} x^2 \, dx=\frac {\frac {\sqrt {\pi } {\left (b^{2} c x + a b c\right )} a^{2} c^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}} b^{3} \sqrt {-\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}}} - \frac {2 \, a c^{2} f^{\frac {{\left (b^{2} c x + a b c\right )}^{2}}{b^{2} c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}} b^{2}} - \frac {{\left (b^{2} c x + a b c\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}\right ) \log \left (f\right )^{3}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}} b^{5} \left (-\frac {{\left (b^{2} c x + a b c\right )}^{2} \log \left (f\right )}{b^{2} c}\right )^{\frac {3}{2}}}}{2 \, \sqrt {c \log \left (f\right )} b} \] Input:
integrate(f^(c*(b*x+a)^2)*x^2,x, algorithm="maxima")
Output:
1/2*(sqrt(pi)*(b^2*c*x + a*b*c)*a^2*c^2*(erf(sqrt(-(b^2*c*x + a*b*c)^2*log (f)/(b^2*c))) - 1)*log(f)^3/((c*log(f))^(5/2)*b^3*sqrt(-(b^2*c*x + a*b*c)^ 2*log(f)/(b^2*c))) - 2*a*c^2*f^((b^2*c*x + a*b*c)^2/(b^2*c))*log(f)^2/((c* log(f))^(5/2)*b^2) - (b^2*c*x + a*b*c)^3*gamma(3/2, -(b^2*c*x + a*b*c)^2*l og(f)/(b^2*c))*log(f)^3/((c*log(f))^(5/2)*b^5*(-(b^2*c*x + a*b*c)^2*log(f) /(b^2*c))^(3/2)))/(sqrt(c*log(f))*b)
Time = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int f^{c (a+b x)^2} x^2 \, dx=-\frac {\frac {\sqrt {\pi } {\left (2 \, a^{2} c \log \left (f\right ) - 1\right )} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} b {\left (x + \frac {a}{b}\right )}\right )}{\sqrt {-c \log \left (f\right )} b c \log \left (f\right )} - \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (b^{2} c x^{2} \log \left (f\right ) + 2 \, a b c x \log \left (f\right ) + a^{2} c \log \left (f\right )\right )}}{b c \log \left (f\right )}}{4 \, b^{2}} \] Input:
integrate(f^(c*(b*x+a)^2)*x^2,x, algorithm="giac")
Output:
-1/4*(sqrt(pi)*(2*a^2*c*log(f) - 1)*erf(-sqrt(-c*log(f))*b*(x + a/b))/(sqr t(-c*log(f))*b*c*log(f)) - 2*(b*(x + a/b) - 2*a)*e^(b^2*c*x^2*log(f) + 2*a *b*c*x*log(f) + a^2*c*log(f))/(b*c*log(f)))/b^2
Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int f^{c (a+b x)^2} x^2 \, dx=\frac {\sqrt {\pi }\,\mathrm {erfi}\left (\sqrt {c\,\ln \left (f\right )}\,\left (a+b\,x\right )\right )\,\left (\frac {a^2}{b^3}-\frac {1}{2\,b^3\,c\,\ln \left (f\right )}\right )}{2\,\sqrt {c\,\ln \left (f\right )}}-\frac {a\,f^{b^2\,c\,x^2}\,f^{a^2\,c}\,f^{2\,a\,b\,c\,x}}{2\,b^3\,c\,\ln \left (f\right )}+\frac {f^{b^2\,c\,x^2}\,f^{a^2\,c}\,f^{2\,a\,b\,c\,x}\,x}{2\,b^2\,c\,\ln \left (f\right )} \] Input:
int(f^(c*(a + b*x)^2)*x^2,x)
Output:
(pi^(1/2)*erfi((c*log(f))^(1/2)*(a + b*x))*(a^2/b^3 - 1/(2*b^3*c*log(f)))) /(2*(c*log(f))^(1/2)) - (a*f^(b^2*c*x^2)*f^(a^2*c)*f^(2*a*b*c*x))/(2*b^3*c *log(f)) + (f^(b^2*c*x^2)*f^(a^2*c)*f^(2*a*b*c*x)*x)/(2*b^2*c*log(f))
Time = 0.18 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06 \[ \int f^{c (a+b x)^2} x^2 \, dx=\frac {-2 \sqrt {\pi }\, \mathrm {erf}\left (\frac {\mathrm {log}\left (f \right ) a c i +\mathrm {log}\left (f \right ) b c i x}{\sqrt {c}\, \sqrt {\mathrm {log}\left (f \right )}}\right ) \mathrm {log}\left (f \right ) a^{2} c i +\sqrt {\pi }\, \mathrm {erf}\left (\frac {\mathrm {log}\left (f \right ) a c i +\mathrm {log}\left (f \right ) b c i x}{\sqrt {c}\, \sqrt {\mathrm {log}\left (f \right )}}\right ) i -2 f^{b^{2} c \,x^{2}+2 a b c x +a^{2} c} \sqrt {c}\, \sqrt {\mathrm {log}\left (f \right )}\, a +2 f^{b^{2} c \,x^{2}+2 a b c x +a^{2} c} \sqrt {c}\, \sqrt {\mathrm {log}\left (f \right )}\, b x}{4 \sqrt {c}\, \sqrt {\mathrm {log}\left (f \right )}\, \mathrm {log}\left (f \right ) b^{3} c} \] Input:
int(f^(c*(b*x+a)^2)*x^2,x)
Output:
( - 2*sqrt(pi)*erf((log(f)*a*c*i + log(f)*b*c*i*x)/(sqrt(c)*sqrt(log(f)))) *log(f)*a**2*c*i + sqrt(pi)*erf((log(f)*a*c*i + log(f)*b*c*i*x)/(sqrt(c)*s qrt(log(f))))*i - 2*f**(a**2*c + 2*a*b*c*x + b**2*c*x**2)*sqrt(c)*sqrt(log (f))*a + 2*f**(a**2*c + 2*a*b*c*x + b**2*c*x**2)*sqrt(c)*sqrt(log(f))*b*x) /(4*sqrt(c)*sqrt(log(f))*log(f)*b**3*c)