3.71 \(\int x \left (b x^2\right )^p \, dx\)

Optimal. Leaf size=19 \[ \frac{x^2 \left (b x^2\right )^p}{2 (p+1)} \]

[Out]

(x^2*(b*x^2)^p)/(2*(1 + p))

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Rubi [A]  time = 0.0143378, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x^2 \left (b x^2\right )^p}{2 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x*(b*x^2)^p,x]

[Out]

(x^2*(b*x^2)^p)/(2*(1 + p))

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Rubi in Sympy [A]  time = 2.98443, size = 14, normalized size = 0.74 \[ \frac{\left (b x^{2}\right )^{p + 1}}{2 b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x**2)**p,x)

[Out]

(b*x**2)**(p + 1)/(2*b*(p + 1))

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Mathematica [A]  time = 0.00399403, size = 18, normalized size = 0.95 \[ \frac{x^2 \left (b x^2\right )^p}{2 p+2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(b*x^2)^p,x]

[Out]

(x^2*(b*x^2)^p)/(2 + 2*p)

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Maple [A]  time = 0.002, size = 18, normalized size = 1. \[{\frac{{x}^{2} \left ( b{x}^{2} \right ) ^{p}}{2+2\,p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2)^p,x)

[Out]

1/2*x^2*(b*x^2)^p/(1+p)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235345, size = 23, normalized size = 1.21 \[ \frac{\left (b x^{2}\right )^{p} x^{2}}{2 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2)^p*x^2/(p + 1)

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Sympy [A]  time = 0.582597, size = 22, normalized size = 1.16 \[ \begin{cases} \frac{b^{p} x^{2} \left (x^{2}\right )^{p}}{2 p + 2} & \text{for}\: p \neq -1 \\\frac{\log{\left (x \right )}}{b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**2)**p,x)

[Out]

Piecewise((b**p*x**2*(x**2)**p/(2*p + 2), Ne(p, -1)), (log(x)/b, True))

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GIAC/XCAS [A]  time = 0.222965, size = 26, normalized size = 1.37 \[ \frac{\left (b x^{2}\right )^{p + 1}}{2 \, b{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2)^(p + 1)/(b*(p + 1))