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

Optimal. Leaf size=42 \[ \frac{\sqrt{c x^2} \log (x)}{a x}-\frac{\sqrt{c x^2} \log (a+b x)}{a x} \]

[Out]

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

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Rubi [A]  time = 0.007027, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 36, 29, 31} \[ \frac{\sqrt{c x^2} \log (x)}{a x}-\frac{\sqrt{c x^2} \log (a+b x)}{a x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x^2*(a + b*x)),x]

[Out]

(Sqrt[c*x^2]*Log[x])/(a*x) - (Sqrt[c*x^2]*Log[a + b*x])/(a*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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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

Mathematica [A]  time = 0.007199, size = 26, normalized size = 0.62 \[ \frac{c x (\log (x)-\log (a+b x))}{a \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 26, normalized size = 0.6 \begin{align*}{\frac{\ln \left ( x \right ) -\ln \left ( bx+a \right ) }{ax}\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07241, size = 32, normalized size = 0.76 \begin{align*} -\frac{\sqrt{c} \log \left (b x + a\right )}{a} + \frac{\sqrt{c} \log \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63121, size = 139, normalized size = 3.31 \begin{align*} \left [\frac{\sqrt{c x^{2}} \log \left (\frac{x}{b x + a}\right )}{a x}, \frac{2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )} \sqrt{-c}}{a c x}\right )}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(c*x^2)*log(x/(b*x + a))/(a*x), 2*sqrt(-c)*arctan(sqrt(c*x^2)*(2*b*x + a)*sqrt(-c)/(a*c*x))/a]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError