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

Optimal. Leaf size=27 \[ \frac{b x \log (x)}{\sqrt{c x^2}}-\frac{a}{\sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0060676, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{b x \log (x)}{\sqrt{c x^2}}-\frac{a}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x*Sqrt[c*x^2]),x]

[Out]

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

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0060521, size = 23, normalized size = 0.85 \[ \frac{c x^2 (b x \log (x)-a)}{\left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^2*(-a + b*x*Log[x]))/(c*x^2)^(3/2)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.7 \begin{align*}{(b\ln \left ( x \right ) x-a){\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07141, size = 23, normalized size = 0.85 \begin{align*} \frac{b \log \left (x\right )}{\sqrt{c}} - \frac{a}{\sqrt{c} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(x)/sqrt(c) - a/(sqrt(c)*x)

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Fricas [A]  time = 1.82753, size = 51, normalized size = 1.89 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x \log \left (x\right ) - a\right )}}{c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*(b*x*log(x) - a)/(c*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.07331, size = 63, normalized size = 2.33 \begin{align*} -\frac{b \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right ) - \frac{2 \, a \sqrt{c}}{\sqrt{c} x - \sqrt{c x^{2}}}}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*log(abs(-sqrt(c)*x + sqrt(c*x^2))) - 2*a*sqrt(c)/(sqrt(c)*x - sqrt(c*x^2)))/sqrt(c)