3.235 \(\int \frac{\sqrt{a x^2+b x^3}}{x} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \left (a x^2+b x^3\right )^{3/2}}{3 b x^3} \]

[Out]

(2*(a*x^2 + b*x^3)^(3/2))/(3*b*x^3)

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Rubi [A]  time = 0.0358451, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2014} \[ \frac{2 \left (a x^2+b x^3\right )^{3/2}}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^2 + b*x^3]/x,x]

[Out]

(2*(a*x^2 + b*x^3)^(3/2))/(3*b*x^3)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

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

Mathematica [A]  time = 0.0090109, size = 23, normalized size = 0.92 \[ \frac{2 \left (x^2 (a+b x)\right )^{3/2}}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^2 + b*x^3]/x,x]

[Out]

(2*(x^2*(a + b*x))^(3/2))/(3*b*x^3)

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Maple [A]  time = 0.001, size = 27, normalized size = 1.1 \begin{align*}{\frac{2\,bx+2\,a}{3\,bx}\sqrt{b{x}^{3}+a{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00775, size = 16, normalized size = 0.64 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x + a)^(3/2)/b

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Fricas [A]  time = 0.823993, size = 55, normalized size = 2.2 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a x^{2}}{\left (b x + a\right )}}{3 \, b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x^3 + a*x^2)*(b*x + a)/(b*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (a + b x\right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**2*(a + b*x))/x, x)

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Giac [A]  time = 1.17417, size = 34, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, b} - \frac{2 \, a^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x + a)^(3/2)*sgn(x)/b - 2/3*a^(3/2)*sgn(x)/b