\(\int f^{a+b x^2} x^{11} \, dx\) [70]

Optimal result

Integrand size = 13, antiderivative size = 78 \[ \int f^{a+b x^2} x^{11} \, dx=-\frac {f^{a+b x^2} \left (120-120 b x^2 \log (f)+60 b^2 x^4 \log ^2(f)-20 b^3 x^6 \log ^3(f)+5 b^4 x^8 \log ^4(f)-b^5 x^{10} \log ^5(f)\right )}{2 b^6 \log ^6(f)} \]

[Out]

-1/2*f^(b*x^2+a)*(120-120*b*x^2*ln(f)+60*b^2*x^4*ln(f)^2-20*b^3*x^6*ln(f)^3+5*b^4*x^8*ln(f)^4-b^5*x^10*ln(f)^5
)/b^6/ln(f)^6

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2249} \[ \int f^{a+b x^2} x^{11} \, dx=-\frac {f^{a+b x^2} \left (-b^5 x^{10} \log ^5(f)+5 b^4 x^8 \log ^4(f)-20 b^3 x^6 \log ^3(f)+60 b^2 x^4 \log ^2(f)-120 b x^2 \log (f)+120\right )}{2 b^6 \log ^6(f)} \]

[In]

Int[f^(a + b*x^2)*x^11,x]

[Out]

-1/2*(f^(a + b*x^2)*(120 - 120*b*x^2*Log[f] + 60*b^2*x^4*Log[f]^2 - 20*b^3*x^6*Log[f]^3 + 5*b^4*x^8*Log[f]^4 -
b^5*x^10*Log[f]^5))/(b^6*Log[f]^6)

Rule 2249

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{p = Simplify
[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n*((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Lo
g[F]]]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+b x^2} \left (120-120 b x^2 \log (f)+60 b^2 x^4 \log ^2(f)-20 b^3 x^6 \log ^3(f)+5 b^4 x^8 \log ^4(f)-b^5 x^{10} \log ^5(f)\right )}{2 b^6 \log ^6(f)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int f^{a+b x^2} x^{11} \, dx=-\frac {f^a \Gamma \left (6,-b x^2 \log (f)\right )}{2 b^6 \log ^6(f)} \]

[In]

Integrate[f^(a + b*x^2)*x^11,x]

[Out]

-1/2*(f^a*Gamma[6, -(b*x^2*Log[f])])/(b^6*Log[f]^6)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97

method	result	size
gosper	\(\frac {\left (b^{5} x^{10} \ln \left (f \right )^{5}-5 b^{4} x^{8} \ln \left (f \right )^{4}+20 b^{3} x^{6} \ln \left (f \right )^{3}-60 b^{2} x^{4} \ln \left (f \right )^{2}+120 b \,x^{2} \ln \left (f \right )-120\right ) f^{b \,x^{2}+a}}{2 \ln \left (f \right )^{6} b^{6}}\)	\(76\)
risch	\(\frac {\left (b^{5} x^{10} \ln \left (f \right )^{5}-5 b^{4} x^{8} \ln \left (f \right )^{4}+20 b^{3} x^{6} \ln \left (f \right )^{3}-60 b^{2} x^{4} \ln \left (f \right )^{2}+120 b \,x^{2} \ln \left (f \right )-120\right ) f^{b \,x^{2}+a}}{2 \ln \left (f \right )^{6} b^{6}}\)	\(76\)
meijerg	\(\frac {f^{a} \left (120-\frac {\left (-6 b^{5} x^{10} \ln \left (f \right )^{5}+30 b^{4} x^{8} \ln \left (f \right )^{4}-120 b^{3} x^{6} \ln \left (f \right )^{3}+360 b^{2} x^{4} \ln \left (f \right )^{2}-720 b \,x^{2} \ln \left (f \right )+720\right ) {\mathrm e}^{b \,x^{2} \ln \left (f \right )}}{6}\right )}{2 b^{6} \ln \left (f \right )^{6}}\)	\(83\)
parallelrisch	\(\frac {f^{b \,x^{2}+a} x^{10} \ln \left (f \right )^{5} b^{5}-5 f^{b \,x^{2}+a} x^{8} \ln \left (f \right )^{4} b^{4}+20 f^{b \,x^{2}+a} x^{6} \ln \left (f \right )^{3} b^{3}-60 f^{b \,x^{2}+a} x^{4} \ln \left (f \right )^{2} b^{2}+120 f^{b \,x^{2}+a} x^{2} \ln \left (f \right ) b -120 f^{b \,x^{2}+a}}{2 \ln \left (f \right )^{6} b^{6}}\)	\(122\)
norman	\(-\frac {60 \,{\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{b^{6} \ln \left (f \right )^{6}}+\frac {60 x^{2} {\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{b^{5} \ln \left (f \right )^{5}}+\frac {x^{10} {\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{2 b \ln \left (f \right )}-\frac {30 x^{4} {\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{\ln \left (f \right )^{4} b^{4}}+\frac {10 x^{6} {\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{\ln \left (f \right )^{3} b^{3}}-\frac {5 x^{8} {\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{2 \ln \left (f \right )^{2} b^{2}}\)	\(137\)

[In]

int(f^(b*x^2+a)*x^11,x,method=_RETURNVERBOSE)

[Out]

1/2*(b^5*x^10*ln(f)^5-5*b^4*x^8*ln(f)^4+20*b^3*x^6*ln(f)^3-60*b^2*x^4*ln(f)^2+120*b*x^2*ln(f)-120)*f^(b*x^2+a)
/ln(f)^6/b^6

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96 \[ \int f^{a+b x^2} x^{11} \, dx=\frac {{\left (b^{5} x^{10} \log \left (f\right )^{5} - 5 \, b^{4} x^{8} \log \left (f\right )^{4} + 20 \, b^{3} x^{6} \log \left (f\right )^{3} - 60 \, b^{2} x^{4} \log \left (f\right )^{2} + 120 \, b x^{2} \log \left (f\right ) - 120\right )} f^{b x^{2} + a}}{2 \, b^{6} \log \left (f\right )^{6}} \]

[In]

integrate(f^(b*x^2+a)*x^11,x, algorithm="fricas")

[Out]

1/2*(b^5*x^10*log(f)^5 - 5*b^4*x^8*log(f)^4 + 20*b^3*x^6*log(f)^3 - 60*b^2*x^4*log(f)^2 + 120*b*x^2*log(f) - 1
20)*f^(b*x^2 + a)/(b^6*log(f)^6)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.21 \[ \int f^{a+b x^2} x^{11} \, dx=\begin {cases} \frac {f^{a + b x^{2}} \left (b^{5} x^{10} \log {\left (f \right )}^{5} - 5 b^{4} x^{8} \log {\left (f \right )}^{4} + 20 b^{3} x^{6} \log {\left (f \right )}^{3} - 60 b^{2} x^{4} \log {\left (f \right )}^{2} + 120 b x^{2} \log {\left (f \right )} - 120\right )}{2 b^{6} \log {\left (f \right )}^{6}} & \text {for}\: b^{6} \log {\left (f \right )}^{6} \neq 0 \\\frac {x^{12}}{12} & \text {otherwise} \end {cases} \]

[In]

integrate(f**(b*x**2+a)*x**11,x)

[Out]

Piecewise((f**(a + b*x**2)*(b**5*x**10*log(f)**5 - 5*b**4*x**8*log(f)**4 + 20*b**3*x**6*log(f)**3 - 60*b**2*x*
*4*log(f)**2 + 120*b*x**2*log(f) - 120)/(2*b**6*log(f)**6), Ne(b**6*log(f)**6, 0)), (x**12/12, True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.18 \[ \int f^{a+b x^2} x^{11} \, dx=\frac {{\left (b^{5} f^{a} x^{10} \log \left (f\right )^{5} - 5 \, b^{4} f^{a} x^{8} \log \left (f\right )^{4} + 20 \, b^{3} f^{a} x^{6} \log \left (f\right )^{3} - 60 \, b^{2} f^{a} x^{4} \log \left (f\right )^{2} + 120 \, b f^{a} x^{2} \log \left (f\right ) - 120 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{6} \log \left (f\right )^{6}} \]

[In]

integrate(f^(b*x^2+a)*x^11,x, algorithm="maxima")

[Out]

1/2*(b^5*f^a*x^10*log(f)^5 - 5*b^4*f^a*x^8*log(f)^4 + 20*b^3*f^a*x^6*log(f)^3 - 60*b^2*f^a*x^4*log(f)^2 + 120*
b*f^a*x^2*log(f) - 120*f^a)*f^(b*x^2)/(b^6*log(f)^6)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.01 \[ \int f^{a+b x^2} x^{11} \, dx=\frac {{\left (b^{5} x^{10} \log \left (f\right )^{5} - 5 \, b^{4} x^{8} \log \left (f\right )^{4} + 20 \, b^{3} x^{6} \log \left (f\right )^{3} - 60 \, b^{2} x^{4} \log \left (f\right )^{2} + 120 \, b x^{2} \log \left (f\right ) - 120\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, b^{6} \log \left (f\right )^{6}} \]

[In]

integrate(f^(b*x^2+a)*x^11,x, algorithm="giac")

[Out]

1/2*(b^5*x^10*log(f)^5 - 5*b^4*x^8*log(f)^4 + 20*b^3*x^6*log(f)^3 - 60*b^2*x^4*log(f)^2 + 120*b*x^2*log(f) - 1
20)*e^(b*x^2*log(f) + a*log(f))/(b^6*log(f)^6)

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97 \[ \int f^{a+b x^2} x^{11} \, dx=-\frac {f^{b\,x^2+a}\,\left (-\frac {b^5\,x^{10}\,{\ln \left (f\right )}^5}{2}+\frac {5\,b^4\,x^8\,{\ln \left (f\right )}^4}{2}-10\,b^3\,x^6\,{\ln \left (f\right )}^3+30\,b^2\,x^4\,{\ln \left (f\right )}^2-60\,b\,x^2\,\ln \left (f\right )+60\right )}{b^6\,{\ln \left (f\right )}^6} \]

[In]

int(f^(a + b*x^2)*x^11,x)

[Out]

-(f^(a + b*x^2)*(30*b^2*x^4*log(f)^2 - 10*b^3*x^6*log(f)^3 + (5*b^4*x^8*log(f)^4)/2 - (b^5*x^10*log(f)^5)/2 -
60*b*x^2*log(f) + 60))/(b^6*log(f)^6)