3.103 \(\int \frac {f^{a+b x^3}}{x^4} \, dx\)

Optimal. Leaf size=35 \[ \frac {1}{3} b f^a \log (f) \text {Ei}\left (b x^3 \log (f)\right )-\frac {f^{a+b x^3}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2214, 2210} \[ \frac {1}{3} b f^a \log (f) \text {Ei}\left (b x^3 \log (f)\right )-\frac {f^{a+b x^3}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)/x^4,x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 32, normalized size = 0.91 \[ \frac {1}{3} f^a \left (b \log (f) \text {Ei}\left (b x^3 \log (f)\right )-\frac {f^{b x^3}}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)/x^4,x]

[Out]

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

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fricas [A]  time = 0.41, size = 35, normalized size = 1.00 \[ \frac {b f^{a} x^{3} {\rm Ei}\left (b x^{3} \log \relax (f)\right ) \log \relax (f) - f^{b x^{3} + a}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^4,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{b x^{3} + a}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^4,x, algorithm="giac")

[Out]

integrate(f^(b*x^3 + a)/x^4, x)

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maple [B]  time = 0.05, size = 97, normalized size = 2.77 \[ -\frac {\left (\Ei \left (1, -b \,x^{3} \ln \relax (f )\right )-3 \ln \relax (x )-\ln \left (-b \right )+\ln \left (-b \,x^{3} \ln \relax (f )\right )-\ln \left (\ln \relax (f )\right )+\frac {{\mathrm e}^{b \,x^{3} \ln \relax (f )}}{b \,x^{3} \ln \relax (f )}-\frac {2 b \,x^{3} \ln \relax (f )+2}{2 b \,x^{3} \ln \relax (f )}+\frac {1}{b \,x^{3} \ln \relax (f )}+1\right ) b \,f^{a} \ln \relax (f )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^3+a)/x^4,x)

[Out]

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

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maxima [A]  time = 1.36, size = 18, normalized size = 0.51 \[ \frac {1}{3} \, b f^{a} \Gamma \left (-1, -b x^{3} \log \relax (f)\right ) \log \relax (f) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.53, size = 32, normalized size = 0.91 \[ -\frac {f^a\,\left (f^{b\,x^3}+b\,x^3\,\ln \relax (f)\,\mathrm {expint}\left (-b\,x^3\,\ln \relax (f)\right )\right )}{3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^3)/x^4,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x^{3}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**3+a)/x**4,x)

[Out]

Integral(f**(a + b*x**3)/x**4, x)

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