3.73 \(\int f^{a+b x^2} x^5 \, dx\)

Optimal. Leaf size=62 \[ \frac{f^{a+b x^2}}{b^3 \log ^3(f)}-\frac{x^2 f^{a+b x^2}}{b^2 \log ^2(f)}+\frac{x^4 f^{a+b x^2}}{2 b \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.103509, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{f^{a+b x^2}}{b^3 \log ^3(f)}-\frac{x^2 f^{a+b x^2}}{b^2 \log ^2(f)}+\frac{x^4 f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2)*x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.92569, size = 54, normalized size = 0.87 \[ \frac{f^{a + b x^{2}} x^{4}}{2 b \log{\left (f \right )}} - \frac{f^{a + b x^{2}} x^{2}}{b^{2} \log{\left (f \right )}^{2}} + \frac{f^{a + b x^{2}}}{b^{3} \log{\left (f \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**2+a)*x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0140354, size = 41, normalized size = 0.66 \[ \frac{f^{a+b x^2} \left (b^2 x^4 \log ^2(f)-2 b x^2 \log (f)+2\right )}{2 b^3 \log ^3(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2)*x^5,x]

[Out]

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

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Maple [A]  time = 0.01, size = 40, normalized size = 0.7 \[{\frac{ \left ({b}^{2}{x}^{4} \left ( \ln \left ( f \right ) \right ) ^{2}-2\,b{x}^{2}\ln \left ( f \right ) +2 \right ){f}^{b{x}^{2}+a}}{2\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)*x^5,x)

[Out]

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

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Maxima [A]  time = 0.774554, size = 63, normalized size = 1.02 \[ \frac{{\left (b^{2} f^{a} x^{4} \log \left (f\right )^{2} - 2 \, b f^{a} x^{2} \log \left (f\right ) + 2 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{3} \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.244174, size = 53, normalized size = 0.85 \[ \frac{{\left (b^{2} x^{4} \log \left (f\right )^{2} - 2 \, b x^{2} \log \left (f\right ) + 2\right )} f^{b x^{2} + a}}{2 \, b^{3} \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.239402, size = 54, normalized size = 0.87 \[ \begin{cases} \frac{f^{a + b x^{2}} \left (b^{2} x^{4} \log{\left (f \right )}^{2} - 2 b x^{2} \log{\left (f \right )} + 2\right )}{2 b^{3} \log{\left (f \right )}^{3}} & \text{for}\: 2 b^{3} \log{\left (f \right )}^{3} \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**2+a)*x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.248299, size = 58, normalized size = 0.94 \[ \frac{{\left (b^{2} x^{4}{\rm ln}\left (f\right )^{2} - 2 \, b x^{2}{\rm ln}\left (f\right ) + 2\right )} e^{\left (b x^{2}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{2 \, b^{3}{\rm ln}\left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^2 + a)*x^5,x, algorithm="giac")

[Out]

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