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

Optimal. Leaf size=20 \[ \frac{f^{a+b x^2}}{2 b \log (f)} \]

[Out]

f^(a + b*x^2)/(2*b*Log[f])

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Rubi [A]  time = 0.023673, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2)*x,x]

[Out]

f^(a + b*x^2)/(2*b*Log[f])

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Rubi in Sympy [A]  time = 2.55136, size = 14, normalized size = 0.7 \[ \frac{f^{a + b x^{2}}}{2 b \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f**(a + b*x**2)/(2*b*log(f))

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Mathematica [A]  time = 0.00388811, size = 20, normalized size = 1. \[ \frac{f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2)*x,x]

[Out]

f^(a + b*x^2)/(2*b*Log[f])

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Maple [A]  time = 0.003, size = 19, normalized size = 1. \[{\frac{{f}^{b{x}^{2}+a}}{2\,b\ln \left ( f \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)*x,x)

[Out]

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

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Maxima [A]  time = 0.762497, size = 24, normalized size = 1.2 \[ \frac{f^{b x^{2} + a}}{2 \, b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*f^(b*x^2 + a)/(b*log(f))

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Fricas [A]  time = 0.236463, size = 24, normalized size = 1.2 \[ \frac{f^{b x^{2} + a}}{2 \, b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*f^(b*x^2 + a)/(b*log(f))

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Sympy [A]  time = 0.18018, size = 24, normalized size = 1.2 \[ \begin{cases} \frac{f^{a + b x^{2}}}{2 b \log{\left (f \right )}} & \text{for}\: 2 b \log{\left (f \right )} \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**2+a)*x,x)

[Out]

Piecewise((f**(a + b*x**2)/(2*b*log(f)), Ne(2*b*log(f), 0)), (x**2/2, True))

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GIAC/XCAS [A]  time = 0.229338, size = 24, normalized size = 1.2 \[ \frac{f^{b x^{2} + a}}{2 \, b{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*f^(b*x^2 + a)/(b*ln(f))