3.59 \(\int \frac{x^3}{(a x^2+b x^3+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{2 x (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0402122, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1915} \[ -\frac{2 x (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]

[Out]

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

Rule 1915

Int[(x_)^(m_.)/((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(3/2), x_Symbol] :> Simp[(-2*x^((n - 1
)/2)*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a*x^(n - 1) + b*x^n + c*x^(n + 1)]), x] /; FreeQ[{a, b, c, n}, x] && EqQ
[m, (3*(n - 1))/2] && EqQ[q, n - 1] && EqQ[r, n + 1] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

Mathematica [A]  time = 0.0244011, size = 36, normalized size = 0.92 \[ -\frac{2 x (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 52, normalized size = 1.3 \begin{align*} 2\,{\frac{ \left ( c{x}^{2}+bx+a \right ) \left ( 2\,cx+b \right ){x}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^4+b*x^3+a*x^2)^(3/2),x)

[Out]

2*(c*x^2+b*x+a)*(2*c*x+b)*x^3/(4*a*c-b^2)/(c*x^4+b*x^3+a*x^2)^(3/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^3/(c*x^4 + b*x^3 + a*x^2)^(3/2), x)

________________________________________________________________________________________

Fricas [A]  time = 1.94596, size = 151, normalized size = 3.87 \begin{align*} -\frac{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} +{\left (b^{3} - 4 \, a b c\right )} x^{2} +{\left (a b^{2} - 4 \, a^{2} c\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)/((b^2*c - 4*a*c^2)*x^3 + (b^3 - 4*a*b*c)*x^2 + (a*b^2 - 4*a^2*c)*x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(c*x**4+b*x**3+a*x**2)**(3/2),x)

[Out]

Integral(x**3/(x**2*(a + b*x + c*x**2))**(3/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.16545, size = 61, normalized size = 1.56 \begin{align*} -\frac{2 \,{\left (\frac{2 \, c}{b^{2} - 4 \, a c} + \frac{b}{{\left (b^{2} - 4 \, a c\right )} x}\right )}}{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="giac")

[Out]

-2*(2*c/(b^2 - 4*a*c) + b/((b^2 - 4*a*c)*x))/sqrt(c + b/x + a/x^2)