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

Optimal. Leaf size=28 \[ \frac{a \sqrt{c x^2} \log (x)}{x}+b \sqrt{c x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0049333, antiderivative size = 28, 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} \[ \frac{a \sqrt{c x^2} \log (x)}{x}+b \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x))/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0042371, size = 20, normalized size = 0.71 \[ \frac{c x (a \log (x)+b x)}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(b*x + a*Log[x]))/Sqrt[c*x^2]

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 20, normalized size = 0.7 \begin{align*}{\frac{bx+a\ln \left ( x \right ) }{x}\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^2,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.52884, size = 43, normalized size = 1.54 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/x

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.06821, size = 23, normalized size = 0.82 \begin{align*}{\left (b x \mathrm{sgn}\left (x\right ) + a \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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