3.35 \(\int \frac {x^3}{\sqrt {b x^2}} \, dx\)

Optimal. Leaf size=16 \[ \frac {x^4}{3 \sqrt {b x^2}} \]

[Out]

1/3*x^4/(b*x^2)^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 16, 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} \[ \frac {x^4}{3 \sqrt {b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[b*x^2],x]

[Out]

x^4/(3*Sqrt[b*x^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 {x^3}{\sqrt {b x^2}} \, dx &=\frac {x \int x^2 \, dx}{\sqrt {b x^2}}\\ &=\frac {x^4}{3 \sqrt {b x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \frac {x^4}{3 \sqrt {b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[b*x^2],x]

[Out]

x^4/(3*Sqrt[b*x^2])

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fricas [A]  time = 0.51, size = 15, normalized size = 0.94 \[ \frac {\sqrt {b x^{2}} x^{2}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 15, normalized size = 0.94 \[ \frac {\sqrt {b x^{2}} x^{2}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \[ \frac {x^{4}}{3 \sqrt {b \,x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^4/(b*x^2)^(1/2)

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maxima [A]  time = 1.35, size = 15, normalized size = 0.94 \[ \frac {\sqrt {b x^{2}} x^{2}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.97, size = 10, normalized size = 0.62 \[ \frac {\sqrt {x^6}}{3\,\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^6)^(1/2)/(3*b^(1/2))

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sympy [A]  time = 0.49, size = 15, normalized size = 0.94 \[ \frac {x^{4}}{3 \sqrt {b} \sqrt {x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**2)**(1/2),x)

[Out]

x**4/(3*sqrt(b)*sqrt(x**2))

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