3.78 \(\int \frac {f^{a+b x^2}}{x^5} \, dx\)

Optimal. Leaf size=58 \[ \frac {1}{4} b^2 f^a \log ^2(f) \text {Ei}\left (b x^2 \log (f)\right )-\frac {b \log (f) f^{a+b x^2}}{4 x^2}-\frac {f^{a+b x^2}}{4 x^4} \]

[Out]

-1/4*f^(b*x^2+a)/x^4-1/4*b*f^(b*x^2+a)*ln(f)/x^2+1/4*b^2*f^a*Ei(b*x^2*ln(f))*ln(f)^2

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2214, 2210} \[ \frac {1}{4} b^2 f^a \log ^2(f) \text {Ei}\left (b x^2 \log (f)\right )-\frac {f^{a+b x^2}}{4 x^4}-\frac {b \log (f) f^{a+b x^2}}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^5,x]

[Out]

-f^(a + b*x^2)/(4*x^4) - (b*f^(a + b*x^2)*Log[f])/(4*x^2) + (b^2*f^a*ExpIntegralEi[b*x^2*Log[f]]*Log[f]^2)/4

Rule 2210

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

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rubi steps

\begin {align*} \int \frac {f^{a+b x^2}}{x^5} \, dx &=-\frac {f^{a+b x^2}}{4 x^4}+\frac {1}{2} (b \log (f)) \int \frac {f^{a+b x^2}}{x^3} \, dx\\ &=-\frac {f^{a+b x^2}}{4 x^4}-\frac {b f^{a+b x^2} \log (f)}{4 x^2}+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+b x^2}}{x} \, dx\\ &=-\frac {f^{a+b x^2}}{4 x^4}-\frac {b f^{a+b x^2} \log (f)}{4 x^2}+\frac {1}{4} b^2 f^a \text {Ei}\left (b x^2 \log (f)\right ) \log ^2(f)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 48, normalized size = 0.83 \[ \frac {f^a \left (b^2 x^4 \log ^2(f) \text {Ei}\left (b x^2 \log (f)\right )-f^{b x^2} \left (b x^2 \log (f)+1\right )\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^5,x]

[Out]

(f^a*(b^2*x^4*ExpIntegralEi[b*x^2*Log[f]]*Log[f]^2 - f^(b*x^2)*(1 + b*x^2*Log[f])))/(4*x^4)

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 48, normalized size = 0.83 \[ \frac {b^{2} f^{a} x^{4} {\rm Ei}\left (b x^{2} \log \relax (f)\right ) \log \relax (f)^{2} - {\left (b x^{2} \log \relax (f) + 1\right )} f^{b x^{2} + a}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^5,x, algorithm="fricas")

[Out]

1/4*(b^2*f^a*x^4*Ei(b*x^2*log(f))*log(f)^2 - (b*x^2*log(f) + 1)*f^(b*x^2 + a))/x^4

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{b x^{2} + a}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^5,x, algorithm="giac")

[Out]

integrate(f^(b*x^2 + a)/x^5, x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 57, normalized size = 0.98 \[ -\frac {b^{2} f^{a} \Ei \left (1, -b \,x^{2} \ln \relax (f )\right ) \ln \relax (f )^{2}}{4}-\frac {b \,f^{a} f^{b \,x^{2}} \ln \relax (f )}{4 x^{2}}-\frac {f^{a} f^{b \,x^{2}}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)/x^5,x)

[Out]

-1/4*f^a/x^4*f^(b*x^2)-1/4*f^a*ln(f)*b/x^2*f^(b*x^2)-1/4*f^a*ln(f)^2*b^2*Ei(1,-b*x^2*ln(f))

________________________________________________________________________________________

maxima [A]  time = 0.62, size = 22, normalized size = 0.38 \[ -\frac {1}{2} \, b^{2} f^{a} \Gamma \left (-2, -b x^{2} \log \relax (f)\right ) \log \relax (f)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^5,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.53, size = 57, normalized size = 0.98 \[ -\frac {b^2\,f^a\,{\ln \relax (f)}^2\,\left (f^{b\,x^2}\,\left (\frac {1}{2\,b\,x^2\,\ln \relax (f)}+\frac {1}{2\,b^2\,x^4\,{\ln \relax (f)}^2}\right )+\frac {\mathrm {expint}\left (-b\,x^2\,\ln \relax (f)\right )}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^2)/x^5,x)

[Out]

-(b^2*f^a*log(f)^2*(f^(b*x^2)*(1/(2*b*x^2*log(f)) + 1/(2*b^2*x^4*log(f)^2)) + expint(-b*x^2*log(f))/2))/2

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x^{2}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)/x**5,x)

[Out]

Integral(f**(a + b*x**2)/x**5, x)

________________________________________________________________________________________