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]

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

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Rubi [A]  time = 0.0420116, antiderivative size = 35, normalized size of antiderivative = 1., 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 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^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*}

Mathematica [A]  time = 0.0092715, 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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Maple [B]  time = 0.027, size = 97, normalized size = 2.8 \begin{align*} -{\frac{{f}^{a}b\ln \left ( f \right ) }{3} \left ({\frac{1}{b{x}^{3}\ln \left ( f \right ) }}+1-3\,\ln \left ( x \right ) -\ln \left ( -b \right ) -\ln \left ( \ln \left ( f \right ) \right ) -{\frac{2+2\,b{x}^{3}\ln \left ( f \right ) }{2\,b{x}^{3}\ln \left ( f \right ) }}+{\frac{{{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{b{x}^{3}\ln \left ( f \right ) }}+\ln \left ( -b{x}^{3}\ln \left ( f \right ) \right ) +{\it Ei} \left ( 1,-b{x}^{3}\ln \left ( f \right ) \right ) \right ) } \end{align*}

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

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Maxima [A]  time = 1.23272, size = 24, normalized size = 0.69 \begin{align*} \frac{1}{3} \, b f^{a} \Gamma \left (-1, -b x^{3} \log \left (f\right )\right ) \log \left (f\right ) \end{align*}

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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Fricas [A]  time = 1.79514, size = 82, normalized size = 2.34 \begin{align*} \frac{b f^{a} x^{3}{\rm Ei}\left (b x^{3} \log \left (f\right )\right ) \log \left (f\right ) - f^{b x^{3} + a}}{3 \, x^{3}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + b x^{3}}}{x^{4}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{3} + a}}{x^{4}}\,{d x} \end{align*}

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)