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

Optimal. Leaf size=24 \[ -\frac{\sqrt{c x^2}}{b x (a+b x)} \]

[Out]

-(Sqrt[c*x^2]/(b*x*(a + b*x)))

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Rubi [A]  time = 0.0036378, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ -\frac{\sqrt{c x^2}}{b x (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x*(a + b*x)^2),x]

[Out]

-(Sqrt[c*x^2]/(b*x*(a + b*x)))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0061281, size = 23, normalized size = 0.96 \[ -\frac{c x}{b \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*x)/(b*Sqrt[c*x^2]*(a + b*x)))

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Maple [A]  time = 0.003, size = 23, normalized size = 1. \begin{align*} -{\frac{1}{bx \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03162, size = 22, normalized size = 0.92 \begin{align*} -\frac{\sqrt{c}}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c)/(b^2*x + a*b)

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Fricas [A]  time = 1.26492, size = 43, normalized size = 1.79 \begin{align*} -\frac{\sqrt{c x^{2}}}{b^{2} x^{2} + a b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^2)/(b^2*x^2 + a*b*x)

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Sympy [A]  time = 0.801927, size = 39, normalized size = 1.62 \begin{align*} \begin{cases} - \frac{\sqrt{c} \sqrt{x^{2}}}{a b x + b^{2} x^{2}} & \text{for}\: b \neq 0 \\\frac{\sqrt{c} \sqrt{x^{2}}}{a^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(c)*sqrt(x**2)/(a*b*x + b**2*x**2), Ne(b, 0)), (sqrt(c)*sqrt(x**2)/a**2, True))

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Giac [A]  time = 1.06232, size = 39, normalized size = 1.62 \begin{align*} -\sqrt{c}{\left (\frac{\mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b} - \frac{\mathrm{sgn}\left (x\right )}{a b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c)*(sgn(x)/((b*x + a)*b) - sgn(x)/(a*b))