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

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

[Out]

(Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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Rubi [A]  time = 0.0032198, antiderivative size = 22, 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, 31} \[ \frac{\sqrt{c x^2} \log (a+b x)}{b x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x*(a + b*x)),x]

[Out]

(Sqrt[c*x^2]*Log[a + b*x])/(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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0036721, size = 21, normalized size = 0.95 \[ \frac{c x \log (a+b x)}{b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*Log[a + b*x])/(b*Sqrt[c*x^2])

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Maple [A]  time = 0.002, size = 21, normalized size = 1. \begin{align*}{\frac{\ln \left ( bx+a \right ) }{bx}\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),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.46506, size = 43, normalized size = 1.95 \begin{align*} \frac{\sqrt{c x^{2}} \log \left (b x + a\right )}{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),x, algorithm="fricas")

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05504, size = 38, normalized size = 1.73 \begin{align*} \sqrt{c}{\left (\frac{\log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b} - \frac{\log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{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),x, algorithm="giac")

[Out]

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