3.79 \(\int \frac{f^{a+b x^2}}{x^7} \, dx\)

Optimal. Leaf size=81 \[ \frac{1}{12} b^3 f^a \log ^3(f) \text{Ei}\left (b x^2 \log (f)\right )-\frac{b^2 \log ^2(f) f^{a+b x^2}}{12 x^2}-\frac{f^{a+b x^2}}{6 x^6}-\frac{b \log (f) f^{a+b x^2}}{12 x^4} \]

[Out]

-f^(a + b*x^2)/(6*x^6) - (b*f^(a + b*x^2)*Log[f])/(12*x^4) - (b^2*f^(a + b*x^2)*Log[f]^2)/(12*x^2) + (b^3*f^a*
ExpIntegralEi[b*x^2*Log[f]]*Log[f]^3)/12

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Rubi [A]  time = 0.09055, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2214, 2210} \[ \frac{1}{12} b^3 f^a \log ^3(f) \text{Ei}\left (b x^2 \log (f)\right )-\frac{b^2 \log ^2(f) f^{a+b x^2}}{12 x^2}-\frac{f^{a+b x^2}}{6 x^6}-\frac{b \log (f) f^{a+b x^2}}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^7,x]

[Out]

-f^(a + b*x^2)/(6*x^6) - (b*f^(a + b*x^2)*Log[f])/(12*x^4) - (b^2*f^(a + b*x^2)*Log[f]^2)/(12*x^2) + (b^3*f^a*
ExpIntegralEi[b*x^2*Log[f]]*Log[f]^3)/12

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]))

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]

Rubi steps

\begin{align*} \int \frac{f^{a+b x^2}}{x^7} \, dx &=-\frac{f^{a+b x^2}}{6 x^6}+\frac{1}{3} (b \log (f)) \int \frac{f^{a+b x^2}}{x^5} \, dx\\ &=-\frac{f^{a+b x^2}}{6 x^6}-\frac{b f^{a+b x^2} \log (f)}{12 x^4}+\frac{1}{6} \left (b^2 \log ^2(f)\right ) \int \frac{f^{a+b x^2}}{x^3} \, dx\\ &=-\frac{f^{a+b x^2}}{6 x^6}-\frac{b f^{a+b x^2} \log (f)}{12 x^4}-\frac{b^2 f^{a+b x^2} \log ^2(f)}{12 x^2}+\frac{1}{6} \left (b^3 \log ^3(f)\right ) \int \frac{f^{a+b x^2}}{x} \, dx\\ &=-\frac{f^{a+b x^2}}{6 x^6}-\frac{b f^{a+b x^2} \log (f)}{12 x^4}-\frac{b^2 f^{a+b x^2} \log ^2(f)}{12 x^2}+\frac{1}{12} b^3 f^a \text{Ei}\left (b x^2 \log (f)\right ) \log ^3(f)\\ \end{align*}

Mathematica [A]  time = 0.0237001, size = 59, normalized size = 0.73 \[ \frac{f^a \left (b^3 x^6 \log ^3(f) \text{Ei}\left (b x^2 \log (f)\right )-f^{b x^2} \left (b^2 x^4 \log ^2(f)+b x^2 \log (f)+2\right )\right )}{12 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^7,x]

[Out]

(f^a*(b^3*x^6*ExpIntegralEi[b*x^2*Log[f]]*Log[f]^3 - f^(b*x^2)*(2 + b*x^2*Log[f] + b^2*x^4*Log[f]^2)))/(12*x^6
)

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Maple [A]  time = 0.033, size = 79, normalized size = 1. \begin{align*} -{\frac{{f}^{a}{f}^{b{x}^{2}}}{6\,{x}^{6}}}-{\frac{{f}^{a}\ln \left ( f \right ) b{f}^{b{x}^{2}}}{12\,{x}^{4}}}-{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{f}^{b{x}^{2}}}{12\,{x}^{2}}}-{\frac{{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{\it Ei} \left ( 1,-b{x}^{2}\ln \left ( f \right ) \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)/x^7,x)

[Out]

-1/6*f^a/x^6*f^(b*x^2)-1/12*f^a*ln(f)*b/x^4*f^(b*x^2)-1/12*f^a*ln(f)^2*b^2/x^2*f^(b*x^2)-1/12*f^a*ln(f)^3*b^3*
Ei(1,-b*x^2*ln(f))

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Maxima [A]  time = 1.24856, size = 30, normalized size = 0.37 \begin{align*} \frac{1}{2} \, b^{3} f^{a} \Gamma \left (-3, -b x^{2} \log \left (f\right )\right ) \log \left (f\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^7,x, algorithm="maxima")

[Out]

1/2*b^3*f^a*gamma(-3, -b*x^2*log(f))*log(f)^3

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Fricas [A]  time = 1.85051, size = 140, normalized size = 1.73 \begin{align*} \frac{b^{3} f^{a} x^{6}{\rm Ei}\left (b x^{2} \log \left (f\right )\right ) \log \left (f\right )^{3} -{\left (b^{2} x^{4} \log \left (f\right )^{2} + b x^{2} \log \left (f\right ) + 2\right )} f^{b x^{2} + a}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^7,x, algorithm="fricas")

[Out]

1/12*(b^3*f^a*x^6*Ei(b*x^2*log(f))*log(f)^3 - (b^2*x^4*log(f)^2 + b*x^2*log(f) + 2)*f^(b*x^2 + a))/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + b x^{2}}}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)/x**7,x)

[Out]

Integral(f**(a + b*x**2)/x**7, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{2} + a}}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^7,x, algorithm="giac")

[Out]

integrate(f^(b*x^2 + a)/x^7, x)