\(\int f^{a+b x^2} x^{10} \, dx\) [83]

Optimal result

Integrand size = 13, antiderivative size = 34 \[ \int f^{a+b x^2} x^{10} \, dx=-\frac {f^a x^{11} \Gamma \left (\frac {11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \]

[Out]

-1/2*f^a*x^11*(1048576/61836869254970658257624840625*GAMMA(51/2,-b*x^2*ln(f))-1048576/618368692549706582576248
40625*(-b*x^2*ln(f))^(49/2)*exp(b*x^2*ln(f))-524288/1261976923570829760359690625*(-b*x^2*ln(f))^(47/2)*exp(b*x
^2*ln(f))-262144/26850572841932548092759375*(-b*x^2*ln(f))^(45/2)*exp(b*x^2*ln(f))-131072/59667939648738995761
6875*(-b*x^2*ln(f))^(43/2)*exp(b*x^2*ln(f))-65536/13876265034590464130625*(-b*x^2*ln(f))^(41/2)*exp(b*x^2*ln(f
))-32768/338445488648547905625*(-b*x^2*ln(f))^(39/2)*exp(b*x^2*ln(f))-16384/8678089452526869375*(-b*x^2*ln(f))
^(37/2)*exp(b*x^2*ln(f))-8192/234542958176401875*(-b*x^2*ln(f))^(35/2)*exp(b*x^2*ln(f))-4096/6701227376468625*
(-b*x^2*ln(f))^(33/2)*exp(b*x^2*ln(f))-2048/203067496256625*(-b*x^2*ln(f))^(31/2)*exp(b*x^2*ln(f))-1024/655056
4395375*(-b*x^2*ln(f))^(29/2)*exp(b*x^2*ln(f))-512/225881530875*(-b*x^2*ln(f))^(27/2)*exp(b*x^2*ln(f))-256/836
5982625*(-b*x^2*ln(f))^(25/2)*exp(b*x^2*ln(f))-128/334639305*(-b*x^2*ln(f))^(23/2)*exp(b*x^2*ln(f))-64/1454953
5*(-b*x^2*ln(f))^(21/2)*exp(b*x^2*ln(f))-32/692835*(-b*x^2*ln(f))^(19/2)*exp(b*x^2*ln(f))-16/36465*(-b*x^2*ln(
f))^(17/2)*exp(b*x^2*ln(f))-8/2145*(-b*x^2*ln(f))^(15/2)*exp(b*x^2*ln(f))-4/143*(-b*x^2*ln(f))^(13/2)*exp(b*x^
2*ln(f))-2/11*(-b*x^2*ln(f))^(11/2)*exp(b*x^2*ln(f)))/(-b*x^2*ln(f))^(11/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, 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 = {2250} \[ \int f^{a+b x^2} x^{10} \, dx=-\frac {x^{11} f^a \Gamma \left (\frac {11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \]

[In]

Int[f^(a + b*x^2)*x^10,x]

[Out]

-1/2*(f^a*x^11*Gamma[11/2, -(b*x^2*Log[f])])/(-(b*x^2*Log[f]))^(11/2)

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {f^a x^{11} \Gamma \left (\frac {11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^2} x^{10} \, dx=-\frac {f^a x^{11} \Gamma \left (\frac {11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \]

[In]

Integrate[f^(a + b*x^2)*x^10,x]

[Out]

-1/2*(f^a*x^11*Gamma[11/2, -(b*x^2*Log[f])])/(-(b*x^2*Log[f]))^(11/2)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.26

method	result	size
meijerg	\(-\frac {f^{a} \left (\frac {x \left (-b \right )^{\frac {11}{2}} \sqrt {\ln \left (f \right )}\, \left (176 b^{4} x^{8} \ln \left (f \right )^{4}-792 b^{3} x^{6} \ln \left (f \right )^{3}+2772 b^{2} x^{4} \ln \left (f \right )^{2}-6930 b \,x^{2} \ln \left (f \right )+10395\right ) {\mathrm e}^{b \,x^{2} \ln \left (f \right )}}{176 b^{5}}-\frac {945 \left (-b \right )^{\frac {11}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{32 b^{\frac {11}{2}}}\right )}{2 b^{5} \ln \left (f \right )^{\frac {11}{2}} \sqrt {-b}}\)	\(111\)
risch	\(\frac {f^{a} x^{9} f^{b \,x^{2}}}{2 \ln \left (f \right ) b}-\frac {9 f^{a} x^{7} f^{b \,x^{2}}}{4 \ln \left (f \right )^{2} b^{2}}+\frac {63 f^{a} x^{5} f^{b \,x^{2}}}{8 \ln \left (f \right )^{3} b^{3}}-\frac {315 f^{a} x^{3} f^{b \,x^{2}}}{16 \ln \left (f \right )^{4} b^{4}}+\frac {945 f^{a} x \,f^{b \,x^{2}}}{32 \ln \left (f \right )^{5} b^{5}}-\frac {945 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x \right )}{64 \ln \left (f \right )^{5} b^{5} \sqrt {-b \ln \left (f \right )}}\)	\(142\)

[In]

int(f^(b*x^2+a)*x^10,x,method=_RETURNVERBOSE)

[Out]

-1/2*f^a/b^5/ln(f)^(11/2)/(-b)^(1/2)*(1/176*x*(-b)^(11/2)*ln(f)^(1/2)*(176*b^4*x^8*ln(f)^4-792*b^3*x^6*ln(f)^3
+2772*b^2*x^4*ln(f)^2-6930*b*x^2*ln(f)+10395)/b^5*exp(b*x^2*ln(f))-945/32*(-b)^(11/2)/b^(11/2)*Pi^(1/2)*erfi(x
*b^(1/2)*ln(f)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.97 \[ \int f^{a+b x^2} x^{10} \, dx=\frac {945 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right ) + 2 \, {\left (16 \, b^{5} x^{9} \log \left (f\right )^{5} - 72 \, b^{4} x^{7} \log \left (f\right )^{4} + 252 \, b^{3} x^{5} \log \left (f\right )^{3} - 630 \, b^{2} x^{3} \log \left (f\right )^{2} + 945 \, b x \log \left (f\right )\right )} f^{b x^{2} + a}}{64 \, b^{6} \log \left (f\right )^{6}} \]

[In]

integrate(f^(b*x^2+a)*x^10,x, algorithm="fricas")

[Out]

1/64*(945*sqrt(pi)*sqrt(-b*log(f))*f^a*erf(sqrt(-b*log(f))*x) + 2*(16*b^5*x^9*log(f)^5 - 72*b^4*x^7*log(f)^4 +
252*b^3*x^5*log(f)^3 - 630*b^2*x^3*log(f)^2 + 945*b*x*log(f))*f^(b*x^2 + a))/(b^6*log(f)^6)

Sympy [F]

\[ \int f^{a+b x^2} x^{10} \, dx=\int f^{a + b x^{2}} x^{10}\, dx \]

[In]

integrate(f**(b*x**2+a)*x**10,x)

[Out]

Integral(f**(a + b*x**2)*x**10, x)

Maxima [A] (verification not implemented)

none

Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.29 \[ \int f^{a+b x^2} x^{10} \, dx=\frac {{\left (16 \, b^{4} f^{a} x^{9} \log \left (f\right )^{4} - 72 \, b^{3} f^{a} x^{7} \log \left (f\right )^{3} + 252 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} - 630 \, b f^{a} x^{3} \log \left (f\right ) + 945 \, f^{a} x\right )} f^{b x^{2}}}{32 \, b^{5} \log \left (f\right )^{5}} - \frac {945 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right )}{64 \, \sqrt {-b \log \left (f\right )} b^{5} \log \left (f\right )^{5}} \]

[In]

integrate(f^(b*x^2+a)*x^10,x, algorithm="maxima")

[Out]

1/32*(16*b^4*f^a*x^9*log(f)^4 - 72*b^3*f^a*x^7*log(f)^3 + 252*b^2*f^a*x^5*log(f)^2 - 630*b*f^a*x^3*log(f) + 94
5*f^a*x)*f^(b*x^2)/(b^5*log(f)^5) - 945/64*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^5*log(f)^5)

Giac [A] (verification not implemented)

none

Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.06 \[ \int f^{a+b x^2} x^{10} \, dx=\frac {945 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (f\right )} x\right )}{64 \, \sqrt {-b \log \left (f\right )} b^{5} \log \left (f\right )^{5}} + \frac {{\left (16 \, b^{4} x^{9} \log \left (f\right )^{4} - 72 \, b^{3} x^{7} \log \left (f\right )^{3} + 252 \, b^{2} x^{5} \log \left (f\right )^{2} - 630 \, b x^{3} \log \left (f\right ) + 945 \, x\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{32 \, b^{5} \log \left (f\right )^{5}} \]

[In]

integrate(f^(b*x^2+a)*x^10,x, algorithm="giac")

[Out]

945/64*sqrt(pi)*f^a*erf(-sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^5*log(f)^5) + 1/32*(16*b^4*x^9*log(f)^4 - 72*b^
3*x^7*log(f)^3 + 252*b^2*x^5*log(f)^2 - 630*b*x^3*log(f) + 945*x)*e^(b*x^2*log(f) + a*log(f))/(b^5*log(f)^5)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.09 \[ \int f^{a+b x^2} x^{10} \, dx=-\frac {\frac {f^a\,\left (945\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (f\right )}{\sqrt {b\,\ln \left (f\right )}}\right )-1890\,f^{b\,x^2}\,x\,\sqrt {b\,\ln \left (f\right )}\right )}{64\,\sqrt {b\,\ln \left (f\right )}}-\frac {63\,b^2\,f^a\,f^{b\,x^2}\,x^5\,{\ln \left (f\right )}^2}{8}+\frac {9\,b^3\,f^a\,f^{b\,x^2}\,x^7\,{\ln \left (f\right )}^3}{4}-\frac {b^4\,f^a\,f^{b\,x^2}\,x^9\,{\ln \left (f\right )}^4}{2}+\frac {315\,b\,f^a\,f^{b\,x^2}\,x^3\,\ln \left (f\right )}{16}}{b^5\,{\ln \left (f\right )}^5} \]

[In]

int(f^(a + b*x^2)*x^10,x)

[Out]

-((f^a*(945*pi^(1/2)*erfi((b*x*log(f))/(b*log(f))^(1/2)) - 1890*f^(b*x^2)*x*(b*log(f))^(1/2)))/(64*(b*log(f))^
(1/2)) - (63*b^2*f^a*f^(b*x^2)*x^5*log(f)^2)/8 + (9*b^3*f^a*f^(b*x^2)*x^7*log(f)^3)/4 - (b^4*f^a*f^(b*x^2)*x^9
*log(f)^4)/2 + (315*b*f^a*f^(b*x^2)*x^3*log(f))/16)/(b^5*log(f)^5)