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

Optimal. Leaf size=34 \[ -\frac{f^a \left (-b x^2 \log (f)\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b x^2 \log (f)\right )}{2 x^9} \]

[Out]

-(f^a*Gamma[-9/2, -(b*x^2*Log[f])]*(-(b*x^2*Log[f]))^(9/2))/(2*x^9)

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Rubi [A]  time = 0.0206185, 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} \[ -\frac{f^a \left (-b x^2 \log (f)\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b x^2 \log (f)\right )}{2 x^9} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^10,x]

[Out]

-(f^a*Gamma[-9/2, -(b*x^2*Log[f])]*(-(b*x^2*Log[f]))^(9/2))/(2*x^9)

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^{10}} \, dx &=-\frac{f^a \Gamma \left (-\frac{9}{2},-b x^2 \log (f)\right ) \left (-b x^2 \log (f)\right )^{9/2}}{2 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0056244, size = 34, normalized size = 1. \[ -\frac{f^a \left (-b x^2 \log (f)\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b x^2 \log (f)\right )}{2 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^10,x]

[Out]

-(f^a*Gamma[-9/2, -(b*x^2*Log[f])]*(-(b*x^2*Log[f]))^(9/2))/(2*x^9)

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Maple [A]  time = 0.058, size = 133, normalized size = 3.9 \begin{align*} -{\frac{{f}^{a}{f}^{b{x}^{2}}}{9\,{x}^{9}}}-{\frac{2\,{f}^{a}\ln \left ( f \right ) b{f}^{b{x}^{2}}}{63\,{x}^{7}}}-{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{b{x}^{2}}}{315\,{x}^{5}}}-{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{f}^{b{x}^{2}}}{945\,{x}^{3}}}-{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}{f}^{b{x}^{2}}}{945\,x}}+{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}\sqrt{\pi }}{945}{\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]

-1/9*f^a/x^9*f^(b*x^2)-2/63*f^a*ln(f)*b/x^7*f^(b*x^2)-4/315*f^a*ln(f)^2*b^2/x^5*f^(b*x^2)-8/945*f^a*ln(f)^3*b^
3/x^3*f^(b*x^2)-16/945*f^a*ln(f)^4*b^4/x*f^(b*x^2)+16/945*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 = 1.19433, size = 38, normalized size = 1.12 \begin{align*} -\frac{\left (-b x^{2} \log \left (f\right )\right )^{\frac{9}{2}} f^{a} \Gamma \left (-\frac{9}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, x^{9}} \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]

-1/2*(-b*x^2*log(f))^(9/2)*f^a*gamma(-9/2, -b*x^2*log(f))/x^9

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Fricas [A]  time = 1.74763, size = 258, normalized size = 7.59 \begin{align*} -\frac{16 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} b^{4} f^{a} x^{9} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) \log \left (f\right )^{4} +{\left (16 \, b^{4} x^{8} \log \left (f\right )^{4} + 8 \, b^{3} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} x^{4} \log \left (f\right )^{2} + 30 \, b x^{2} \log \left (f\right ) + 105\right )} f^{b x^{2} + a}}{945 \, x^{9}} \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]

-1/945*(16*sqrt(pi)*sqrt(-b*log(f))*b^4*f^a*x^9*erf(sqrt(-b*log(f))*x)*log(f)^4 + (16*b^4*x^8*log(f)^4 + 8*b^3
*x^6*log(f)^3 + 12*b^2*x^4*log(f)^2 + 30*b*x^2*log(f) + 105)*f^(b*x^2 + a))/x^9

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

integrate(f^(b*x^2 + a)/x^10, x)