\(\int \frac {f^{a+b x^3}}{x^7} \, dx\) [104]

Optimal result

Integrand size = 13, antiderivative size = 58 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=-\frac {f^{a+b x^3}}{6 x^6}-\frac {b f^{a+b x^3} \log (f)}{6 x^3}+\frac {1}{6} b^2 f^a \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right ) \log ^2(f) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2245, 2241} \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\frac {1}{6} b^2 f^a \log ^2(f) \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right )-\frac {b \log (f) f^{a+b x^3}}{6 x^3}-\frac {f^{a+b x^3}}{6 x^6} \]

[In]

Int[f^(a + b*x^3)/x^7,x]

[Out]

-1/6*f^(a + b*x^3)/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 2241

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 2245

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*} \text {integral}& = -\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] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\frac {f^a \left (b^2 x^6 \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right ) \log ^2(f)-f^{b x^3} \left (1+b x^3 \log (f)\right )\right )}{6 x^6} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs. \(2(52)=104\).

Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.43

method	result	size
meijerg	\(\frac {f^{a} b^{2} \ln \left (f \right )^{2} \left (-\frac {1}{2 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {1}{b \,x^{3} \ln \left (f \right )}-\frac {3}{4}+\frac {3 \ln \left (x \right )}{2}+\frac {\ln \left (-b \right )}{2}+\frac {\ln \left (\ln \left (f \right )\right )}{2}+\frac {9 b^{2} x^{6} \ln \left (f \right )^{2}+12 b \,x^{3} \ln \left (f \right )+6}{12 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {\left (3+3 b \,x^{3} \ln \left (f \right )\right ) {\mathrm e}^{b \,x^{3} \ln \left (f \right )}}{6 b^{2} x^{6} \ln \left (f \right )^{2}}-\frac {\ln \left (-b \,x^{3} \ln \left (f \right )\right )}{2}-\frac {\operatorname {Ei}_{1}\left (-b \,x^{3} \ln \left (f \right )\right )}{2}\right )}{3}\)	\(141\)

[In]

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

[Out]

1/3*f^a*b^2*ln(f)^2*(-1/2/b^2/x^6/ln(f)^2-1/b/x^3/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)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\frac {b^{2} f^{a} x^{6} {\rm Ei}\left (b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{2} - {\left (b x^{3} \log \left (f\right ) + 1\right )} f^{b x^{3} + a}}{6 \, x^{6}} \]

[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

Sympy [F]

\[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\int \frac {f^{a + b x^{3}}}{x^{7}}\, dx \]

[In]

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

[Out]

Integral(f**(a + b*x**3)/x**7, x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.38 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=-\frac {1}{3} \, b^{2} f^{a} \Gamma \left (-2, -b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{2} \]

[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

Giac [F]

\[ \int \frac {f^{a+b x^3}}{x^7} \, dx=\int { \frac {f^{b x^{3} + a}}{x^{7}} \,d x } \]

[In]

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

[Out]

integrate(f^(b*x^3 + a)/x^7, x)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \frac {f^{a+b x^3}}{x^7} \, dx=-\frac {b^2\,f^a\,{\ln \left (f\right )}^2\,\left (f^{b\,x^3}\,\left (\frac {1}{2\,b\,x^3\,\ln \left (f\right )}+\frac {1}{2\,b^2\,x^6\,{\ln \left (f\right )}^2}\right )+\frac {\mathrm {expint}\left (-b\,x^3\,\ln \left (f\right )\right )}{2}\right )}{3} \]

[In]

int(f^(a + b*x^3)/x^7,x)

[Out]

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