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

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

[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 = 0.0681349, 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 \[ \frac{1}{3} b f^a \log (f) \text{ExpIntegralEi}\left (b x^3 \log (f)\right )-\frac{f^{a+b x^3}}{3 x^3} \]

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

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Rubi in Sympy [A]  time = 5.53877, size = 32, normalized size = 0.91 \[ \frac{b f^{a} \log{\left (f \right )} \operatorname{Ei}{\left (b x^{3} \log{\left (f \right )} \right )}}{3} - \frac{f^{a + b x^{3}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**3+a)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.016307, size = 32, normalized size = 0.91 \[ \frac{1}{3} f^a \left (b \log (f) \text{ExpIntegralEi}\left (b x^3 \log (f)\right )-\frac{f^{b x^3}}{x^3}\right ) \]

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 = 0.033, size = 97, normalized size = 2.8 \[ -{\frac{{f}^{a}b\ln \left ( f \right ) }{3} \left ({\frac{1}{b{x}^{3}\ln \left ( f \right ) }}+1-3\,\ln \left ( x \right ) -\ln \left ( -b \right ) -\ln \left ( \ln \left ( f \right ) \right ) -{\frac{2\,b{x}^{3}\ln \left ( f \right ) +2}{2\,b{x}^{3}\ln \left ( f \right ) }}+{\frac{{{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{b{x}^{3}\ln \left ( f \right ) }}+\ln \left ( -b{x}^{3}\ln \left ( f \right ) \right ) +{\it Ei} \left ( 1,-b{x}^{3}\ln \left ( f \right ) \right ) \right ) } \]

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*b*
x^3*ln(f)+2)+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 = 0.845398, size = 24, normalized size = 0.69 \[ \frac{1}{3} \, b f^{a} \Gamma \left (-1, -b x^{3} \log \left (f\right )\right ) \log \left (f\right ) \]

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 = 0.256073, size = 47, normalized size = 1.34 \[ \frac{b f^{a} x^{3}{\rm Ei}\left (b x^{3} \log \left (f\right )\right ) \log \left (f\right ) - f^{b x^{3} + a}}{3 \, x^{3}} \]

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. \[ \int \frac{f^{a + b x^{3}}}{x^{4}}\, dx \]

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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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{b x^{3} + a}}{x^{4}}\,{d x} \]

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)