3.98 \(\int f^{a+b x^3} x^{11} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 f^{a+b x^3}}{b^4 \log ^4(f)}+\frac{2 x^3 f^{a+b x^3}}{b^3 \log ^3(f)}-\frac{x^6 f^{a+b x^3}}{b^2 \log ^2(f)}+\frac{x^9 f^{a+b x^3}}{3 b \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.154276, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 f^{a+b x^3}}{b^4 \log ^4(f)}+\frac{2 x^3 f^{a+b x^3}}{b^3 \log ^3(f)}-\frac{x^6 f^{a+b x^3}}{b^2 \log ^2(f)}+\frac{x^9 f^{a+b x^3}}{3 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^3)*x^11,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.0034, size = 78, normalized size = 0.93 \[ \frac{f^{a + b x^{3}} x^{9}}{3 b \log{\left (f \right )}} - \frac{f^{a + b x^{3}} x^{6}}{b^{2} \log{\left (f \right )}^{2}} + \frac{2 f^{a + b x^{3}} x^{3}}{b^{3} \log{\left (f \right )}^{3}} - \frac{2 f^{a + b x^{3}}}{b^{4} \log{\left (f \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**3+a)*x**11,x)

[Out]

f**(a + b*x**3)*x**9/(3*b*log(f)) - f**(a + b*x**3)*x**6/(b**2*log(f)**2) + 2*f*
*(a + b*x**3)*x**3/(b**3*log(f)**3) - 2*f**(a + b*x**3)/(b**4*log(f)**4)

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Mathematica [A]  time = 0.0157771, size = 53, normalized size = 0.63 \[ \frac{f^{a+b x^3} \left (b^3 x^9 \log ^3(f)-3 b^2 x^6 \log ^2(f)+6 b x^3 \log (f)-6\right )}{3 b^4 \log ^4(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^3)*x^11,x]

[Out]

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

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Maple [A]  time = 0.014, size = 52, normalized size = 0.6 \[{\frac{ \left ({b}^{3}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{3}-3\,{b}^{2}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{2}+6\,b{x}^{3}\ln \left ( f \right ) -6 \right ){f}^{b{x}^{3}+a}}{3\, \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^3+a)*x^11,x)

[Out]

1/3*(b^3*x^9*ln(f)^3-3*b^2*x^6*ln(f)^2+6*b*x^3*ln(f)-6)*f^(b*x^3+a)/ln(f)^4/b^4

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Maxima [A]  time = 0.972793, size = 84, normalized size = 1. \[ \frac{{\left (b^{3} f^{a} x^{9} \log \left (f\right )^{3} - 3 \, b^{2} f^{a} x^{6} \log \left (f\right )^{2} + 6 \, b f^{a} x^{3} \log \left (f\right ) - 6 \, f^{a}\right )} f^{b x^{3}}}{3 \, b^{4} \log \left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^11,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.271136, size = 69, normalized size = 0.82 \[ \frac{{\left (b^{3} x^{9} \log \left (f\right )^{3} - 3 \, b^{2} x^{6} \log \left (f\right )^{2} + 6 \, b x^{3} \log \left (f\right ) - 6\right )} f^{b x^{3} + a}}{3 \, b^{4} \log \left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^11,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.285147, size = 68, normalized size = 0.81 \[ \begin{cases} \frac{f^{a + b x^{3}} \left (b^{3} x^{9} \log{\left (f \right )}^{3} - 3 b^{2} x^{6} \log{\left (f \right )}^{2} + 6 b x^{3} \log{\left (f \right )} - 6\right )}{3 b^{4} \log{\left (f \right )}^{4}} & \text{for}\: 3 b^{4} \log{\left (f \right )}^{4} \neq 0 \\\frac{x^{12}}{12} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**3+a)*x**11,x)

[Out]

Piecewise((f**(a + b*x**3)*(b**3*x**9*log(f)**3 - 3*b**2*x**6*log(f)**2 + 6*b*x*
*3*log(f) - 6)/(3*b**4*log(f)**4), Ne(3*b**4*log(f)**4, 0)), (x**12/12, True))

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GIAC/XCAS [A]  time = 0.232021, size = 128, normalized size = 1.52 \[ \frac{b^{3} x^{9} e^{\left (b x^{3}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}{\rm ln}\left (f\right )^{3} - 3 \, b^{2} x^{6} e^{\left (b x^{3}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}{\rm ln}\left (f\right )^{2} + 6 \, b x^{3} e^{\left (b x^{3}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}{\rm ln}\left (f\right ) - 6 \, e^{\left (b x^{3}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{3 \, b^{4}{\rm ln}\left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^11,x, algorithm="giac")

[Out]

1/3*(b^3*x^9*e^(b*x^3*ln(f) + a*ln(f))*ln(f)^3 - 3*b^2*x^6*e^(b*x^3*ln(f) + a*ln
(f))*ln(f)^2 + 6*b*x^3*e^(b*x^3*ln(f) + a*ln(f))*ln(f) - 6*e^(b*x^3*ln(f) + a*ln
(f)))/(b^4*ln(f)^4)