3.24 \(\int (b x^2)^{5/2} \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{6} b^2 x^5 \sqrt{b x^2} \]

[Out]

(b^2*x^5*Sqrt[b*x^2])/6

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Rubi [A]  time = 0.0017534, 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, Rules used = {15, 30} \[ \frac{1}{6} b^2 x^5 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(5/2),x]

[Out]

(b^2*x^5*Sqrt[b*x^2])/6

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (b x^2\right )^{5/2} \, dx &=\frac{\left (b^2 \sqrt{b x^2}\right ) \int x^5 \, dx}{x}\\ &=\frac{1}{6} b^2 x^5 \sqrt{b x^2}\\ \end{align*}

Mathematica [A]  time = 0.0006825, size = 14, normalized size = 0.74 \[ \frac{1}{6} x \left (b x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(5/2),x]

[Out]

(x*(b*x^2)^(5/2))/6

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Maple [A]  time = 0.003, size = 11, normalized size = 0.6 \begin{align*}{\frac{x}{6} \left ( b{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2),x)

[Out]

1/6*x*(b*x^2)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.1466, size = 34, normalized size = 1.79 \begin{align*} \frac{1}{6} \, \sqrt{b x^{2}} b^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2),x, algorithm="fricas")

[Out]

1/6*sqrt(b*x^2)*b^2*x^5

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Sympy [A]  time = 1.43159, size = 14, normalized size = 0.74 \begin{align*} \frac{b^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(5/2),x)

[Out]

b**(5/2)*x*(x**2)**(5/2)/6

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Giac [A]  time = 1.17278, size = 14, normalized size = 0.74 \begin{align*} \frac{1}{6} \, b^{\frac{5}{2}} x^{6} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2),x, algorithm="giac")

[Out]

1/6*b^(5/2)*x^6*sgn(x)