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

Optimal. Leaf size=21 \[ -\frac{2 x}{b \sqrt{a x^2+b x^3}} \]

[Out]

(-2*x)/(b*Sqrt[a*x^2 + b*x^3])

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Rubi [A]  time = 0.0175054, antiderivative size = 21, 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 = {1588} \[ -\frac{2 x}{b \sqrt{a x^2+b x^3}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a*x^2 + b*x^3)^(3/2),x]

[Out]

(-2*x)/(b*Sqrt[a*x^2 + b*x^3])

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

Mathematica [A]  time = 0.0059717, size = 19, normalized size = 0.9 \[ -\frac{2 x}{b \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/(b*Sqrt[x^2*(a + b*x)])

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Maple [A]  time = 0.002, size = 27, normalized size = 1.3 \begin{align*} -2\,{\frac{ \left ( bx+a \right ){x}^{3}}{b \left ( 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/(b*x^3+a*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.07839, size = 16, normalized size = 0.76 \begin{align*} -\frac{2}{\sqrt{b x + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(b*x + a)*b)

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Fricas [A]  time = 0.83839, size = 57, normalized size = 2.71 \begin{align*} -\frac{2 \, \sqrt{b x^{3} + a x^{2}}}{b^{2} x^{2} + a b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18193, size = 50, normalized size = 2.38 \begin{align*} \frac{2}{{\left (\sqrt{a}{\left (\sqrt{\frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}\right )} - b\right )} \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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