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

Optimal. Leaf size=18 \[ \frac{x^3 \left (b x^2\right )^p}{2 p+3} \]

[Out]

(x^3*(b*x^2)^p)/(3 + 2*p)

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Rubi [A]  time = 0.013493, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x^3 \left (b x^2\right )^p}{2 p+3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(b*x^2)^p,x]

[Out]

(x^3*(b*x^2)^p)/(3 + 2*p)

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Rubi in Sympy [A]  time = 3.18907, size = 22, normalized size = 1.22 \[ \frac{x^{- 2 p} x^{2 p + 3} \left (b x^{2}\right )^{p}}{2 p + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2)**p,x)

[Out]

x**(-2*p)*x**(2*p + 3)*(b*x**2)**p/(2*p + 3)

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Mathematica [A]  time = 0.00362381, size = 18, normalized size = 1. \[ \frac{x^3 \left (b x^2\right )^p}{2 p+3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(b*x^2)^p,x]

[Out]

(x^3*(b*x^2)^p)/(3 + 2*p)

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Maple [A]  time = 0.002, size = 19, normalized size = 1.1 \[{\frac{{x}^{3} \left ( b{x}^{2} \right ) ^{p}}{3+2\,p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2)^p,x)

[Out]

x^3*(b*x^2)^p/(3+2*p)

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Maxima [A]  time = 1.44536, size = 26, normalized size = 1.44 \[ \frac{b^{p} x^{3} x^{2 \, p}}{2 \, p + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^p*x^3*x^(2*p)/(2*p + 3)

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Fricas [A]  time = 0.231987, size = 24, normalized size = 1.33 \[ \frac{\left (b x^{2}\right )^{p} x^{3}}{2 \, p + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x^2)^p*x^3/(2*p + 3)

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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(x**2*(b*x**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.218028, size = 27, normalized size = 1.5 \[ \frac{x^{3} e^{\left (p{\rm ln}\left (b x^{2}\right )\right )}}{2 \, p + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^3*e^(p*ln(b*x^2))/(2*p + 3)