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

Optimal. Leaf size=16 \[ \frac{x \left (b x^2\right )^p}{2 p+1} \]

[Out]

(x*(b*x^2)^p)/(1 + 2*p)

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Rubi [A]  time = 0.012752, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x \left (b x^2\right )^p}{2 p+1} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2)^p,x]

[Out]

(x*(b*x^2)^p)/(1 + 2*p)

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Rubi in Sympy [A]  time = 2.0107, size = 22, normalized size = 1.38 \[ \frac{x^{- 2 p} x^{2 p + 1} \left (b x^{2}\right )^{p}}{2 p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0026293, size = 16, normalized size = 1. \[ \frac{x \left (b x^2\right )^p}{2 p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2)^p,x]

[Out]

(x*(b*x^2)^p)/(1 + 2*p)

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Maple [A]  time = 0.003, size = 17, normalized size = 1.1 \[{\frac{x \left ( b{x}^{2} \right ) ^{p}}{1+2\,p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2)^p,x)

[Out]

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

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Maxima [A]  time = 1.44597, size = 23, normalized size = 1.44 \[ \frac{b^{p} x x^{2 \, p}}{2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^p*x*x^(2*p)/(2*p + 1)

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Fricas [A]  time = 0.253608, size = 22, normalized size = 1.38 \[ \frac{\left (b x^{2}\right )^{p} x}{2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x^2)^p*x/(2*p + 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.223256, size = 24, normalized size = 1.5 \[ \frac{x e^{\left (p{\rm ln}\left (b x^{2}\right )\right )}}{2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^(p*ln(b*x^2))/(2*p + 1)