3.94 \(\int \frac{f^{a+b x^2}}{x^{12}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0208784, 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} \[ -\frac{f^a \left (-b x^2 \log (f)\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-b x^2 \log (f)\right )}{2 x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^12,x]

[Out]

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

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 \frac{f^{a+b x^2}}{x^{12}} \, dx &=-\frac{f^a \Gamma \left (-\frac{11}{2},-b x^2 \log (f)\right ) \left (-b x^2 \log (f)\right )^{11/2}}{2 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0054607, size = 34, normalized size = 1. \[ -\frac{f^a \left (-b x^2 \log (f)\right )^{11/2} \text{Gamma}\left (-\frac{11}{2},-b x^2 \log (f)\right )}{2 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^12,x]

[Out]

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

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Maple [A]  time = 0.091, size = 155, normalized size = 4.6 \begin{align*} -{\frac{{f}^{a}{f}^{b{x}^{2}}}{11\,{x}^{11}}}-{\frac{2\,{f}^{a}\ln \left ( f \right ) b{f}^{b{x}^{2}}}{99\,{x}^{9}}}-{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{b{x}^{2}}}{693\,{x}^{7}}}-{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{f}^{b{x}^{2}}}{3465\,{x}^{5}}}-{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}{f}^{b{x}^{2}}}{10395\,{x}^{3}}}-{\frac{32\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}{f}^{b{x}^{2}}}{10395\,x}}+{\frac{32\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}\sqrt{\pi }}{10395}{\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^12,x)

[Out]

-1/11*f^a/x^11*f^(b*x^2)-2/99*f^a*ln(f)*b/x^9*f^(b*x^2)-4/693*f^a*ln(f)^2*b^2/x^7*f^(b*x^2)-8/3465*f^a*ln(f)^3
*b^3/x^5*f^(b*x^2)-16/10395*f^a*ln(f)^4*b^4/x^3*f^(b*x^2)-32/10395*f^a*ln(f)^5*b^5/x*f^(b*x^2)+32/10395*f^a*ln
(f)^6*b^6*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)*x)

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Maxima [A]  time = 1.30194, size = 38, normalized size = 1.12 \begin{align*} -\frac{\left (-b x^{2} \log \left (f\right )\right )^{\frac{11}{2}} f^{a} \Gamma \left (-\frac{11}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^12,x, algorithm="maxima")

[Out]

-1/2*(-b*x^2*log(f))^(11/2)*f^a*gamma(-11/2, -b*x^2*log(f))/x^11

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Fricas [A]  time = 1.82728, size = 297, normalized size = 8.74 \begin{align*} -\frac{32 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} b^{5} f^{a} x^{11} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) \log \left (f\right )^{5} +{\left (32 \, b^{5} x^{10} \log \left (f\right )^{5} + 16 \, b^{4} x^{8} \log \left (f\right )^{4} + 24 \, b^{3} x^{6} \log \left (f\right )^{3} + 60 \, b^{2} x^{4} \log \left (f\right )^{2} + 210 \, b x^{2} \log \left (f\right ) + 945\right )} f^{b x^{2} + a}}{10395 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^12,x, algorithm="fricas")

[Out]

-1/10395*(32*sqrt(pi)*sqrt(-b*log(f))*b^5*f^a*x^11*erf(sqrt(-b*log(f))*x)*log(f)^5 + (32*b^5*x^10*log(f)^5 + 1
6*b^4*x^8*log(f)^4 + 24*b^3*x^6*log(f)^3 + 60*b^2*x^4*log(f)^2 + 210*b*x^2*log(f) + 945)*f^(b*x^2 + a))/x^11

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)/x**12,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{2} + a}}{x^{12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^12,x, algorithm="giac")

[Out]

integrate(f^(b*x^2 + a)/x^12, x)