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

Optimal. Leaf size=29 \[ \frac{a x \log (x)}{\sqrt{c x^2}}+\frac{b x^2}{\sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0051277, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 43} \[ \frac{a x \log (x)}{\sqrt{c x^2}}+\frac{b x^2}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/Sqrt[c*x^2],x]

[Out]

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

Mathematica [A]  time = 0.0016435, size = 19, normalized size = 0.66 \[ \frac{x (a \log (x)+b x)}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/Sqrt[c*x^2],x]

[Out]

(x*(b*x + a*Log[x]))/Sqrt[c*x^2]

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Maple [A]  time = 0.002, size = 18, normalized size = 0.6 \begin{align*}{x \left ( bx+a\ln \left ( x \right ) \right ){\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04629, size = 27, normalized size = 0.93 \begin{align*} \frac{a \log \left (x\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x)/sqrt(c) + sqrt(c*x^2)*b/c

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Fricas [A]  time = 1.79276, size = 49, normalized size = 1.69 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/(c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*x)/sqrt(c*x**2), x)

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Giac [A]  time = 1.07634, size = 47, normalized size = 1.62 \begin{align*} -\frac{a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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