3.315 \(\int \frac{1}{x \sqrt{a x^3+b x^4}} \, dx\)

Optimal. Leaf size=52 \[ \frac{4 b \sqrt{a x^3+b x^4}}{3 a^2 x^2}-\frac{2 \sqrt{a x^3+b x^4}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0450667, antiderivative size = 52, 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, 2000} \[ \frac{4 b \sqrt{a x^3+b x^4}}{3 a^2 x^2}-\frac{2 \sqrt{a x^3+b x^4}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a*x^3 + b*x^4]),x]

[Out]

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

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 2000

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

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{a x^3+b x^4}} \, dx &=-\frac{2 \sqrt{a x^3+b x^4}}{3 a x^3}-\frac{(2 b) \int \frac{1}{\sqrt{a x^3+b x^4}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a x^3+b x^4}}{3 a x^3}+\frac{4 b \sqrt{a x^3+b x^4}}{3 a^2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0110932, size = 29, normalized size = 0.56 \[ -\frac{2 (a-2 b x) \sqrt{x^3 (a+b x)}}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a*x^3 + b*x^4]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 30, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -2\,bx+a \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^4+a*x^3)^(1/2),x)

[Out]

-2/3*(b*x+a)*(-2*b*x+a)/a^2/(b*x^4+a*x^3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{4} + a x^{3}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^4+a*x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^4 + a*x^3)*x), x)

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Fricas [A]  time = 0.859516, size = 63, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{b x^{4} + a x^{3}}{\left (2 \, b x - a\right )}}{3 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^4+a*x^3)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{x^{3} \left (a + b x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**4+a*x**3)**(1/2),x)

[Out]

Integral(1/(x*sqrt(x**3*(a + b*x))), x)

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Giac [A]  time = 1.20618, size = 36, normalized size = 0.69 \begin{align*} -\frac{2 \,{\left ({\left (b + \frac{a}{x}\right )}^{\frac{3}{2}} - 3 \, \sqrt{b + \frac{a}{x}} b\right )}}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^4+a*x^3)^(1/2),x, algorithm="giac")

[Out]

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