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

Optimal. Leaf size=49 \[ \frac{2 \sqrt{a x^2+b x^3}}{3 b}-\frac{4 a \sqrt{a x^2+b x^3}}{3 b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.0545702, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 1588} \[ \frac{2 \sqrt{a x^2+b x^3}}{3 b}-\frac{4 a \sqrt{a x^2+b x^3}}{3 b^2 x} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a*x^2 + b*x^3],x]

[Out]

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

Rule 2016

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*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{a x^2+b x^3}} \, dx &=\frac{2 \sqrt{a x^2+b x^3}}{3 b}-\frac{(2 a) \int \frac{x}{\sqrt{a x^2+b x^3}} \, dx}{3 b}\\ &=\frac{2 \sqrt{a x^2+b x^3}}{3 b}-\frac{4 a \sqrt{a x^2+b x^3}}{3 b^2 x}\\ \end{align*}

Mathematica [A]  time = 0.0148682, size = 30, normalized size = 0.61 \[ \frac{2 (b x-2 a) \sqrt{x^2 (a+b x)}}{3 b^2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*(-2*a + b*x)*Sqrt[x^2*(a + b*x)])/(3*b^2*x)

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Maple [A]  time = 0.003, size = 33, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -bx+2\,a \right ) x}{3\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03395, size = 41, normalized size = 0.84 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} - a b x - 2 \, a^{2}\right )}}{3 \, \sqrt{b x + a} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.858497, size = 61, normalized size = 1.24 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a x^{2}}{\left (b x - 2 \, a\right )}}{3 \, b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/sqrt(x**2*(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{b x^{3} + a x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/sqrt(b*x^3 + a*x^2), x)