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

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

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Rubi [A]  time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \begin {gather*} -\frac {\sqrt {c x^2}}{b x (a+b x)} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.96 \begin {gather*} -\frac {c x}{b \sqrt {c x^2} (a+b x)} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c x^2}}{b x (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[c*x^2]/(b*x*(a + b*x)))

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fricas [A]  time = 1.25, size = 23, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {c x^{2}}}{b^{2} x^{2} + a b x} \end {gather*}

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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giac [A]  time = 1.02, size = 29, normalized size = 1.21 \begin {gather*} -\sqrt {c} {\left (\frac {\mathrm {sgn}\relax (x)}{{\left (b x + a\right )} b} - \frac {\mathrm {sgn}\relax (x)}{a b}\right )} \end {gather*}

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))

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maple [A]  time = 0.00, size = 23, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {c \,x^{2}}}{\left (b x +a \right ) b x} \end {gather*}

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 = 1.36, size = 16, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {c}}{b^{2} x + a b} \end {gather*}

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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mupad [B]  time = 0.16, size = 22, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {c\,x^2}}{b\,x\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(1/2)/(x*(a + b*x)^2),x)

[Out]

-(c*x^2)^(1/2)/(b*x*(a + b*x))

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sympy [A]  time = 0.82, size = 39, normalized size = 1.62 \begin {gather*} \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 {gather*}

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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