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

Optimal. Leaf size=22 \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]

[Out]

-(x/(b*Sqrt[c*x^2]*(a + b*x)))

________________________________________________________________________________________

Rubi [A]  time = 0.0033827, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 32} \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

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

Mathematica [A]  time = 0.0028723, size = 22, normalized size = 1. \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/(b*Sqrt[c*x^2]*(a + b*x)))

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 21, normalized size = 1. \begin{align*} -{\frac{x}{b \left ( bx+a \right ) }{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.33952, size = 49, normalized size = 2.23 \begin{align*} -\frac{\sqrt{c x^{2}}}{b^{2} c x^{2} + a b c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^2)/(b^2*c*x^2 + a*b*c*x)

________________________________________________________________________________________

Sympy [A]  time = 1.11411, size = 85, normalized size = 3.86 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{c} \sqrt{x^{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{b^{2} \sqrt{c} \sqrt{x^{2}}} & \text{for}\: a = 0 \\\frac{\tilde{\infty } x^{2}}{\sqrt{c} \sqrt{x^{2}}} & \text{for}\: b = - \frac{a}{x} \\\frac{x^{2}}{a^{2} \sqrt{c} \sqrt{x^{2}} + a b \sqrt{c} x \sqrt{x^{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/(sqrt(c)*sqrt(x**2)), Eq(a, 0) & Eq(b, 0)), (-1/(b**2*sqrt(c)*sqrt(x**2)), Eq(a, 0)), (zoo*x**2
/(sqrt(c)*sqrt(x**2)), Eq(b, -a/x)), (x**2/(a**2*sqrt(c)*sqrt(x**2) + a*b*sqrt(c)*x*sqrt(x**2)), True))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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