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

Optimal. Leaf size=47 \[ \frac{4 \sqrt{a x^2+b x^3}}{b^2 x}-\frac{2 x^2}{b \sqrt{a x^2+b x^3}} \]

[Out]

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

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Rubi [A]  time = 0.0567734, antiderivative size = 47, 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 = {2015, 1588} \[ \frac{4 \sqrt{a x^2+b x^3}}{b^2 x}-\frac{2 x^2}{b \sqrt{a x^2+b x^3}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a*x^2 + b*x^3)^(3/2),x]

[Out]

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

Rule 2015

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] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n, j
] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || 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^4}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=-\frac{2 x^2}{b \sqrt{a x^2+b x^3}}+\frac{2 \int \frac{x}{\sqrt{a x^2+b x^3}} \, dx}{b}\\ &=-\frac{2 x^2}{b \sqrt{a x^2+b x^3}}+\frac{4 \sqrt{a x^2+b x^3}}{b^2 x}\\ \end{align*}

Mathematica [A]  time = 0.0124749, size = 26, normalized size = 0.55 \[ \frac{2 x (2 a+b x)}{b^2 \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 34, normalized size = 0.7 \begin{align*} 2\,{\frac{ \left ( bx+a \right ) \left ( bx+2\,a \right ){x}^{3}}{{b}^{2} \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^4/(b*x^3+a*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.11266, size = 26, normalized size = 0.55 \begin{align*} \frac{2 \,{\left (b x + 2 \, a\right )}}{\sqrt{b x + a} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x + 2*a)/(sqrt(b*x + a)*b^2)

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Fricas [A]  time = 0.82862, size = 74, normalized size = 1.57 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )}}{b^{3} x^{2} + a b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\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**4/(b*x**3+a*x**2)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.13415, size = 38, normalized size = 0.81 \begin{align*} \frac{2 \,{\left (\frac{1}{b} + \frac{2 \, a}{b^{2} x}\right )}}{\sqrt{\frac{b}{x} + \frac{a}{x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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