3.106 \(\int \frac{f^{a+b x^3}}{x^{13}} \, dx\)

Optimal. Leaf size=24 \[ -\frac{1}{3} b^4 f^a \log ^4(f) \text{Gamma}\left (-4,-b x^3 \log (f)\right ) \]

[Out]

-(b^4*f^a*Gamma[-4, -(b*x^3*Log[f])]*Log[f]^4)/3

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Rubi [A]  time = 0.0375154, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{3} b^4 f^a \log ^4(f) \text{Gamma}\left (-4,-b x^3 \log (f)\right ) \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^3)/x^13,x]

[Out]

-(b^4*f^a*Gamma[-4, -(b*x^3*Log[f])]*Log[f]^4)/3

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Rubi in Sympy [A]  time = 3.65257, size = 27, normalized size = 1.12 \[ - \frac{b^{4} f^{a} \Gamma{\left (-4,- b x^{3} \log{\left (f \right )} \right )} \log{\left (f \right )}^{4}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**3+a)/x**13,x)

[Out]

-b**4*f**a*Gamma(-4, -b*x**3*log(f))*log(f)**4/3

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Mathematica [B]  time = 0.0472257, size = 71, normalized size = 2.96 \[ \frac{f^a \left (b^4 x^{12} \log ^4(f) \text{ExpIntegralEi}\left (b x^3 \log (f)\right )-f^{b x^3} \left (b^3 x^9 \log ^3(f)+b^2 x^6 \log ^2(f)+2 b x^3 \log (f)+6\right )\right )}{72 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^3)/x^13,x]

[Out]

(f^a*(b^4*x^12*ExpIntegralEi[b*x^3*Log[f]]*Log[f]^4 - f^(b*x^3)*(6 + 2*b*x^3*Log
[f] + b^2*x^6*Log[f]^2 + b^3*x^9*Log[f]^3)))/(72*x^12)

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Maple [B]  time = 0.073, size = 213, normalized size = 8.9 \[{\frac{{f}^{a}{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{3} \left ( -{\frac{1}{4\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}}}-{\frac{1}{3\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}}}-{\frac{1}{4\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}}}-{\frac{1}{6\,b{x}^{3}\ln \left ( f \right ) }}-{\frac{25}{288}}+{\frac{\ln \left ( x \right ) }{8}}+{\frac{\ln \left ( -b \right ) }{24}}+{\frac{\ln \left ( \ln \left ( f \right ) \right ) }{24}}+{\frac{125\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}+240\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}+360\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+480\,b{x}^{3}\ln \left ( f \right ) +360}{1440\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}}}-{\frac{ \left ( 5\,{b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}+5\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+10\,b{x}^{3}\ln \left ( f \right ) +30 \right ){{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{120\,{b}^{4}{x}^{12} \left ( \ln \left ( f \right ) \right ) ^{4}}}-{\frac{\ln \left ( -b{x}^{3}\ln \left ( f \right ) \right ) }{24}}-{\frac{{\it Ei} \left ( 1,-b{x}^{3}\ln \left ( f \right ) \right ) }{24}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^3+a)/x^13,x)

[Out]

1/3*f^a*b^4*ln(f)^4*(-1/4/x^12/b^4/ln(f)^4-1/3/x^9/b^3/ln(f)^3-1/4/x^6/b^2/ln(f)
^2-1/6/x^3/b/ln(f)-25/288+1/8*ln(x)+1/24*ln(-b)+1/24*ln(ln(f))+1/1440/b^4/x^12/l
n(f)^4*(125*b^4*x^12*ln(f)^4+240*b^3*x^9*ln(f)^3+360*b^2*x^6*ln(f)^2+480*b*x^3*l
n(f)+360)-1/120/b^4/x^12/ln(f)^4*(5*b^3*x^9*ln(f)^3+5*b^2*x^6*ln(f)^2+10*b*x^3*l
n(f)+30)*exp(b*x^3*ln(f))-1/24*ln(-b*x^3*ln(f))-1/24*Ei(1,-b*x^3*ln(f)))

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Maxima [A]  time = 0.830885, size = 30, normalized size = 1.25 \[ -\frac{1}{3} \, b^{4} f^{a} \Gamma \left (-4, -b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)/x^13,x, algorithm="maxima")

[Out]

-1/3*b^4*f^a*gamma(-4, -b*x^3*log(f))*log(f)^4

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Fricas [A]  time = 0.265577, size = 96, normalized size = 4. \[ \frac{b^{4} f^{a} x^{12}{\rm Ei}\left (b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{4} -{\left (b^{3} x^{9} \log \left (f\right )^{3} + b^{2} x^{6} \log \left (f\right )^{2} + 2 \, b x^{3} \log \left (f\right ) + 6\right )} f^{b x^{3} + a}}{72 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)/x^13,x, algorithm="fricas")

[Out]

1/72*(b^4*f^a*x^12*Ei(b*x^3*log(f))*log(f)^4 - (b^3*x^9*log(f)^3 + b^2*x^6*log(f
)^2 + 2*b*x^3*log(f) + 6)*f^(b*x^3 + a))/x^12

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**3+a)/x**13,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{b x^{3} + a}}{x^{13}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)/x^13,x, algorithm="giac")

[Out]

integrate(f^(b*x^3 + a)/x^13, x)