3.16 \(\int \frac {(b x^2)^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac {1}{2} b x \sqrt {b x^2} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ \frac {1}{2} b x \sqrt {b x^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(3/2)/x^2,x]

[Out]

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

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 \frac {\left (b x^2\right )^{3/2}}{x^2} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int x \, dx}{x}\\ &=\frac {1}{2} b x \sqrt {b x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {1}{2} b x \sqrt {b x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(3/2)/x^2,x]

[Out]

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

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fricas [A]  time = 0.63, size = 11, normalized size = 0.73 \[ \frac {1}{2} \, \sqrt {b x^{2}} b x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2)/x^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 10, normalized size = 0.67 \[ \frac {1}{2} \, b^{\frac {3}{2}} x^{2} \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2)/x^2,x, algorithm="giac")

[Out]

1/2*b^(3/2)*x^2*sgn(x)

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maple [A]  time = 0.00, size = 13, normalized size = 0.87 \[ \frac {\left (b \,x^{2}\right )^{\frac {3}{2}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(3/2)/x^2,x)

[Out]

1/2*(b*x^2)^(3/2)/x

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.97, size = 8, normalized size = 0.53 \[ \frac {b^{3/2}\,x\,\relax |x|}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(3/2)/x^2,x)

[Out]

(b^(3/2)*x*abs(x))/2

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sympy [A]  time = 0.42, size = 14, normalized size = 0.93 \[ \frac {b^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(3/2)/x**2,x)

[Out]

b**(3/2)*(x**2)**(3/2)/(2*x)

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