3.71 \(\int f^{a+b x^2} x^9 \, dx\)

Optimal. Leaf size=24 \[ \frac{f^a \text{Gamma}\left (5,-b x^2 \log (f)\right )}{2 b^5 \log ^5(f)} \]

[Out]

(f^a*Gamma[5, -(b*x^2*Log[f])])/(2*b^5*Log[f]^5)

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Rubi [A]  time = 0.0415684, 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 \[ \frac{f^a \text{Gamma}\left (5,-b x^2 \log (f)\right )}{2 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2)*x^9,x]

[Out]

(f^a*Gamma[5, -(b*x^2*Log[f])])/(2*b^5*Log[f]^5)

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Rubi in Sympy [A]  time = 3.68162, size = 24, normalized size = 1. \[ \frac{f^{a} \Gamma{\left (5,- b x^{2} \log{\left (f \right )} \right )}}{2 b^{5} \log{\left (f \right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**2+a)*x**9,x)

[Out]

f**a*Gamma(5, -b*x**2*log(f))/(2*b**5*log(f)**5)

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Mathematica [B]  time = 0.017864, size = 65, normalized size = 2.71 \[ \frac{f^{a+b x^2} \left (b^4 x^8 \log ^4(f)-4 b^3 x^6 \log ^3(f)+12 b^2 x^4 \log ^2(f)-24 b x^2 \log (f)+24\right )}{2 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2)*x^9,x]

[Out]

(f^(a + b*x^2)*(24 - 24*b*x^2*Log[f] + 12*b^2*x^4*Log[f]^2 - 4*b^3*x^6*Log[f]^3
+ b^4*x^8*Log[f]^4))/(2*b^5*Log[f]^5)

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Maple [A]  time = 0.012, size = 64, normalized size = 2.7 \[{\frac{ \left ({b}^{4}{x}^{8} \left ( \ln \left ( f \right ) \right ) ^{4}-4\,{b}^{3}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{3}+12\,{b}^{2}{x}^{4} \left ( \ln \left ( f \right ) \right ) ^{2}-24\,b{x}^{2}\ln \left ( f \right ) +24 \right ){f}^{b{x}^{2}+a}}{2\, \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)*x^9,x)

[Out]

1/2*(b^4*x^8*ln(f)^4-4*b^3*x^6*ln(f)^3+12*b^2*x^4*ln(f)^2-24*b*x^2*ln(f)+24)*f^(
b*x^2+a)/ln(f)^5/b^5

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Maxima [A]  time = 0.832096, size = 104, normalized size = 4.33 \[ \frac{{\left (b^{4} f^{a} x^{8} \log \left (f\right )^{4} - 4 \, b^{3} f^{a} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} f^{a} x^{4} \log \left (f\right )^{2} - 24 \, b f^{a} x^{2} \log \left (f\right ) + 24 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{5} \log \left (f\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^9,x, algorithm="maxima")

[Out]

1/2*(b^4*f^a*x^8*log(f)^4 - 4*b^3*f^a*x^6*log(f)^3 + 12*b^2*f^a*x^4*log(f)^2 - 2
4*b*f^a*x^2*log(f) + 24*f^a)*f^(b*x^2)/(b^5*log(f)^5)

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Fricas [A]  time = 0.263398, size = 85, normalized size = 3.54 \[ \frac{{\left (b^{4} x^{8} \log \left (f\right )^{4} - 4 \, b^{3} x^{6} \log \left (f\right )^{3} + 12 \, b^{2} x^{4} \log \left (f\right )^{2} - 24 \, b x^{2} \log \left (f\right ) + 24\right )} f^{b x^{2} + a}}{2 \, b^{5} \log \left (f\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^9,x, algorithm="fricas")

[Out]

1/2*(b^4*x^8*log(f)^4 - 4*b^3*x^6*log(f)^3 + 12*b^2*x^4*log(f)^2 - 24*b*x^2*log(
f) + 24)*f^(b*x^2 + a)/(b^5*log(f)^5)

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Sympy [A]  time = 0.299704, size = 82, normalized size = 3.42 \[ \begin{cases} \frac{f^{a + b x^{2}} \left (b^{4} x^{8} \log{\left (f \right )}^{4} - 4 b^{3} x^{6} \log{\left (f \right )}^{3} + 12 b^{2} x^{4} \log{\left (f \right )}^{2} - 24 b x^{2} \log{\left (f \right )} + 24\right )}{2 b^{5} \log{\left (f \right )}^{5}} & \text{for}\: 2 b^{5} \log{\left (f \right )}^{5} \neq 0 \\\frac{x^{10}}{10} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**2+a)*x**9,x)

[Out]

Piecewise((f**(a + b*x**2)*(b**4*x**8*log(f)**4 - 4*b**3*x**6*log(f)**3 + 12*b**
2*x**4*log(f)**2 - 24*b*x**2*log(f) + 24)/(2*b**5*log(f)**5), Ne(2*b**5*log(f)**
5, 0)), (x**10/10, True))

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GIAC/XCAS [A]  time = 0.254026, size = 90, normalized size = 3.75 \[ \frac{{\left (b^{4} x^{8}{\rm ln}\left (f\right )^{4} - 4 \, b^{3} x^{6}{\rm ln}\left (f\right )^{3} + 12 \, b^{2} x^{4}{\rm ln}\left (f\right )^{2} - 24 \, b x^{2}{\rm ln}\left (f\right ) + 24\right )} e^{\left (b x^{2}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{2 \, b^{5}{\rm ln}\left (f\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^9,x, algorithm="giac")

[Out]

1/2*(b^4*x^8*ln(f)^4 - 4*b^3*x^6*ln(f)^3 + 12*b^2*x^4*ln(f)^2 - 24*b*x^2*ln(f) +
24)*e^(b*x^2*ln(f) + a*ln(f))/(b^5*ln(f)^5)