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

Optimal. Leaf size=34 \[ -\frac{x^{11} f^a \text{Gamma}\left (\frac{11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \]

[Out]

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

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Rubi [A]  time = 0.0225988, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ -\frac{x^{11} f^a \text{Gamma}\left (\frac{11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)*x^10,x]

[Out]

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

Rule 2218

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

Rubi steps

\begin{align*} \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}}\\ \end{align*}

Mathematica [A]  time = 0.0057456, size = 34, normalized size = 1. \[ -\frac{x^{11} f^a \text{Gamma}\left (\frac{11}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)*x^10,x]

[Out]

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

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Maple [A]  time = 0.054, size = 142, normalized size = 4.2 \begin{align*}{\frac{{f}^{a}{x}^{9}{f}^{b{x}^{2}}}{2\,b\ln \left ( f \right ) }}-{\frac{9\,{f}^{a}{x}^{7}{f}^{b{x}^{2}}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}}+{\frac{63\,{f}^{a}{x}^{5}{f}^{b{x}^{2}}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}}-{\frac{315\,{f}^{a}{x}^{3}{f}^{b{x}^{2}}}{16\,{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}}+{\frac{945\,{f}^{a}x{f}^{b{x}^{2}}}{32\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}}-{\frac{945\,{f}^{a}\sqrt{\pi }}{64\,{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f^a/ln(f)/b*x^9*f^(b*x^2)-9/4*f^a/ln(f)^2/b^2*x^7*f^(b*x^2)+63/8*f^a/ln(f)^3/b^3*x^5*f^(b*x^2)-315/16*f^a/
ln(f)^4/b^4*x^3*f^(b*x^2)+945/32*f^a/ln(f)^5/b^5*x*f^(b*x^2)-945/64*f^a/ln(f)^5/b^5*Pi^(1/2)/(-b*ln(f))^(1/2)*
erf((-b*ln(f))^(1/2)*x)

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Maxima [A]  time = 1.0775, size = 151, normalized size = 4.44 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.73718, size = 275, normalized size = 8.09 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x^{2}} x^{10}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**2)*x**10, x)

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Giac [A]  time = 1.24632, size = 140, normalized size = 4.12 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)