3.81 \(\int \frac {f^{a+b x^2}}{x^{11}} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{2} b^5 f^a \log ^5(f) \Gamma \left (-5,-b x^2 \log (f)\right ) \]

[Out]

-1/2*f^a/x^10*Ei(6,-b*x^2*ln(f))

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{2} b^5 f^a \log ^5(f) \text {Gamma}\left (-5,-b x^2 \log (f)\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^11,x]

[Out]

(b^5*f^a*Gamma[-5, -(b*x^2*Log[f])]*Log[f]^5)/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 \frac {f^{a+b x^2}}{x^{11}} \, dx &=\frac {1}{2} b^5 f^a \Gamma \left (-5,-b x^2 \log (f)\right ) \log ^5(f)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 1.00 \[ \frac {1}{2} b^5 f^a \log ^5(f) \Gamma \left (-5,-b x^2 \log (f)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^11,x]

[Out]

(b^5*f^a*Gamma[-5, -(b*x^2*Log[f])]*Log[f]^5)/2

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fricas [B]  time = 0.41, size = 83, normalized size = 3.46 \[ \frac {b^{5} f^{a} x^{10} {\rm Ei}\left (b x^{2} \log \relax (f)\right ) \log \relax (f)^{5} - {\left (b^{4} x^{8} \log \relax (f)^{4} + b^{3} x^{6} \log \relax (f)^{3} + 2 \, b^{2} x^{4} \log \relax (f)^{2} + 6 \, b x^{2} \log \relax (f) + 24\right )} f^{b x^{2} + a}}{240 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(b^5*f^a*x^10*Ei(b*x^2*log(f))*log(f)^5 - (b^4*x^8*log(f)^4 + b^3*x^6*log(f)^3 + 2*b^2*x^4*log(f)^2 + 6*
b*x^2*log(f) + 24)*f^(b*x^2 + a))/x^10

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{b x^{2} + a}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^2 + a)/x^11, x)

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maple [B]  time = 0.09, size = 123, normalized size = 5.12 \[ -\frac {b^{5} f^{a} \Ei \left (1, -b \,x^{2} \ln \relax (f )\right ) \ln \relax (f )^{5}}{240}-\frac {b^{4} f^{a} f^{b \,x^{2}} \ln \relax (f )^{4}}{240 x^{2}}-\frac {b^{3} f^{a} f^{b \,x^{2}} \ln \relax (f )^{3}}{240 x^{4}}-\frac {b^{2} f^{a} f^{b \,x^{2}} \ln \relax (f )^{2}}{120 x^{6}}-\frac {b \,f^{a} f^{b \,x^{2}} \ln \relax (f )}{40 x^{8}}-\frac {f^{a} f^{b \,x^{2}}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)/x^11,x)

[Out]

-1/10*f^a/x^10*f^(b*x^2)-1/40*f^a*ln(f)*b/x^8*f^(b*x^2)-1/120*f^a*ln(f)^2*b^2/x^6*f^(b*x^2)-1/240*f^a*ln(f)^3*
b^3/x^4*f^(b*x^2)-1/240*f^a*ln(f)^4*b^4/x^2*f^(b*x^2)-1/240*f^a*ln(f)^5*b^5*Ei(1,-b*x^2*ln(f))

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maxima [B]  time = 0.93, size = 22, normalized size = 0.92 \[ \frac {1}{2} \, b^{5} f^{a} \Gamma \left (-5, -b x^{2} \log \relax (f)\right ) \log \relax (f)^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^5*f^a*gamma(-5, -b*x^2*log(f))*log(f)^5

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mupad [B]  time = 3.51, size = 102, normalized size = 4.25 \[ -\frac {b^5\,f^a\,{\ln \relax (f)}^5\,\mathrm {expint}\left (-b\,x^2\,\ln \relax (f)\right )}{240}-\frac {b^5\,f^a\,f^{b\,x^2}\,{\ln \relax (f)}^5\,\left (\frac {1}{120\,b\,x^2\,\ln \relax (f)}+\frac {1}{120\,b^2\,x^4\,{\ln \relax (f)}^2}+\frac {1}{60\,b^3\,x^6\,{\ln \relax (f)}^3}+\frac {1}{20\,b^4\,x^8\,{\ln \relax (f)}^4}+\frac {1}{5\,b^5\,x^{10}\,{\ln \relax (f)}^5}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^2)/x^11,x)

[Out]

- (b^5*f^a*log(f)^5*expint(-b*x^2*log(f)))/240 - (b^5*f^a*f^(b*x^2)*log(f)^5*(1/(120*b*x^2*log(f)) + 1/(120*b^
2*x^4*log(f)^2) + 1/(60*b^3*x^6*log(f)^3) + 1/(20*b^4*x^8*log(f)^4) + 1/(5*b^5*x^10*log(f)^5)))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x^{2}}}{x^{11}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**2)/x**11, x)

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