3.104 \(\int \frac{f^{a+b x^3}}{x^7} \, dx\)

Optimal. Leaf size=58 \[ \frac{1}{6} b^2 f^a \log ^2(f) \text{Ei}\left (b x^3 \log (f)\right )-\frac{f^{a+b x^3}}{6 x^6}-\frac{b \log (f) f^{a+b x^3}}{6 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0658597, antiderivative size = 58, normalized size of antiderivative = 1., 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} \[ \frac{1}{6} b^2 f^a \log ^2(f) \text{Ei}\left (b x^3 \log (f)\right )-\frac{f^{a+b x^3}}{6 x^6}-\frac{b \log (f) f^{a+b x^3}}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)/x^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0185699, size = 48, normalized size = 0.83 \[ \frac{f^a \left (b^2 x^6 \log ^2(f) \text{Ei}\left (b x^3 \log (f)\right )-f^{b x^3} \left (b x^3 \log (f)+1\right )\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)/x^7,x]

[Out]

(f^a*(b^2*x^6*ExpIntegralEi[b*x^3*Log[f]]*Log[f]^2 - f^(b*x^3)*(1 + b*x^3*Log[f])))/(6*x^6)

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Maple [B]  time = 0.035, size = 141, normalized size = 2.4 \begin{align*}{\frac{{f}^{a}{b}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}}{3} \left ( -{\frac{1}{2\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}}}-{\frac{1}{b{x}^{3}\ln \left ( f \right ) }}-{\frac{3}{4}}+{\frac{3\,\ln \left ( x \right ) }{2}}+{\frac{\ln \left ( -b \right ) }{2}}+{\frac{\ln \left ( \ln \left ( f \right ) \right ) }{2}}+{\frac{9\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+12\,b{x}^{3}\ln \left ( f \right ) +6}{12\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}}}-{\frac{ \left ( 3+3\,b{x}^{3}\ln \left ( f \right ) \right ){{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{6\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}}}-{\frac{\ln \left ( -b{x}^{3}\ln \left ( f \right ) \right ) }{2}}-{\frac{{\it Ei} \left ( 1,-b{x}^{3}\ln \left ( f \right ) \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^3+a)/x^7,x)

[Out]

1/3*f^a*b^2*ln(f)^2*(-1/2/x^6/b^2/ln(f)^2-1/x^3/b/ln(f)-3/4+3/2*ln(x)+1/2*ln(-b)+1/2*ln(ln(f))+1/12/b^2/x^6/ln
(f)^2*(9*b^2*x^6*ln(f)^2+12*b*x^3*ln(f)+6)-1/6/b^2/x^6/ln(f)^2*(3+3*b*x^3*ln(f))*exp(b*x^3*ln(f))-1/2*ln(-b*x^
3*ln(f))-1/2*Ei(1,-b*x^3*ln(f)))

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Maxima [A]  time = 1.17171, size = 30, normalized size = 0.52 \begin{align*} -\frac{1}{3} \, b^{2} f^{a} \Gamma \left (-2, -b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76726, size = 113, normalized size = 1.95 \begin{align*} \frac{b^{2} f^{a} x^{6}{\rm Ei}\left (b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{2} -{\left (b x^{3} \log \left (f\right ) + 1\right )} f^{b x^{3} + a}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + b x^{3}}}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**3+a)/x**7,x)

[Out]

Integral(f**(a + b*x**3)/x**7, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{3} + a}}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^7,x, algorithm="giac")

[Out]

integrate(f^(b*x^3 + a)/x^7, x)