3.69 \(\int x^3 \left (b x^2\right )^p \, dx\)

Optimal. Leaf size=19 \[ \frac{x^4 \left (b x^2\right )^p}{2 (p+2)} \]

[Out]

(x^4*(b*x^2)^p)/(2*(2 + p))

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Rubi [A]  time = 0.0143023, antiderivative size = 19, 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 \[ \frac{x^4 \left (b x^2\right )^p}{2 (p+2)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(b*x^2)^p,x]

[Out]

(x^4*(b*x^2)^p)/(2*(2 + p))

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Rubi in Sympy [A]  time = 3.56362, size = 15, normalized size = 0.79 \[ \frac{\left (b x^{2}\right )^{p + 2}}{2 b^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2)**p,x)

[Out]

(b*x**2)**(p + 2)/(2*b**2*(p + 2))

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Mathematica [A]  time = 0.00425513, size = 18, normalized size = 0.95 \[ \frac{x^4 \left (b x^2\right )^p}{2 p+4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(b*x^2)^p,x]

[Out]

(x^4*(b*x^2)^p)/(4 + 2*p)

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Maple [A]  time = 0.001, size = 18, normalized size = 1. \[{\frac{{x}^{4} \left ( b{x}^{2} \right ) ^{p}}{4+2\,p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2)^p,x)

[Out]

1/2*x^4*(b*x^2)^p/(2+p)

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Maxima [A]  time = 1.44136, size = 24, normalized size = 1.26 \[ \frac{b^{p}{\left (x^{2}\right )}^{p} x^{4}}{2 \,{\left (p + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^p*x^3,x, algorithm="maxima")

[Out]

1/2*b^p*(x^2)^p*x^4/(p + 2)

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Fricas [A]  time = 0.249202, size = 23, normalized size = 1.21 \[ \frac{\left (b x^{2}\right )^{p} x^{4}}{2 \,{\left (p + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^p*x^3,x, algorithm="fricas")

[Out]

1/2*(b*x^2)^p*x^4/(p + 2)

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Sympy [A]  time = 1.19385, size = 24, normalized size = 1.26 \[ \begin{cases} \frac{b^{p} x^{4} \left (x^{2}\right )^{p}}{2 p + 4} & \text{for}\: p \neq -2 \\\frac{\log{\left (x \right )}}{b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2)**p,x)

[Out]

Piecewise((b**p*x**4*(x**2)**p/(2*p + 4), Ne(p, -2)), (log(x)/b**2, True))

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GIAC/XCAS [A]  time = 0.231338, size = 26, normalized size = 1.37 \[ \frac{x^{4} e^{\left (p{\rm ln}\left (b x^{2}\right )\right )}}{2 \,{\left (p + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^p*x^3,x, algorithm="giac")

[Out]

1/2*x^4*e^(p*ln(b*x^2))/(p + 2)