\(\int \frac {\sqrt {c x^2}}{a+b x} \, dx\) [855]

Optimal result

Integrand size = 17, antiderivative size = 38 \[ \int \frac {\sqrt {c x^2}}{a+b x} \, dx=\frac {\sqrt {c x^2}}{b}-\frac {a \sqrt {c x^2} \log (a+b x)}{b^2 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 45} \[ \int \frac {\sqrt {c x^2}}{a+b x} \, dx=\frac {\sqrt {c x^2}}{b}-\frac {a \sqrt {c x^2} \log (a+b x)}{b^2 x} \]

[In]

Int[Sqrt[c*x^2]/(a + b*x),x]

[Out]

Sqrt[c*x^2]/b - (a*Sqrt[c*x^2]*Log[a + b*x])/(b^2*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 45

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {c x^2}}{a+b x} \, dx=\sqrt {c x^2} \left (\frac {1}{b}-\frac {a \log (a+b x)}{b^2 x}\right ) \]

[In]

Integrate[Sqrt[c*x^2]/(a + b*x),x]

[Out]

Sqrt[c*x^2]*(b^(-1) - (a*Log[a + b*x])/(b^2*x))

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {\sqrt {c \,x^{2}}\, \left (a \ln \left (b x +a \right )-b x \right )}{b^{2} x}\)	\(29\)
risch	\(\frac {\sqrt {c \,x^{2}}}{b}-\frac {a \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{2} x}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {c x^2}}{a+b x} \, dx=\frac {\sqrt {c x^{2}} {\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} x} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {c x^2}}{a+b x} \, dx=\int \frac {\sqrt {c x^{2}}}{a + b x}\, dx \]

[In]

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

[Out]

Integral(sqrt(c*x**2)/(a + b*x), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (34) = 68\).

Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {c x^2}}{a+b x} \, dx=-\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} + \frac {\sqrt {c x^{2}}}{b} \]

[In]

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

[Out]

-(-1)^(2*c*x/b)*a*sqrt(c)*log(2*c*x/b)/b^2 - (-1)^(2*a*c*x/b)*a*sqrt(c)*log(-2*a*c*x/(b*abs(b*x + a)))/b^2 + s
qrt(c*x^2)/b

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {c x^2}}{a+b x} \, dx=\sqrt {c} {\left (\frac {x \mathrm {sgn}\left (x\right )}{b} - \frac {a \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{2}} + \frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{2}}\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c x^2}}{a+b x} \, dx=\int \frac {\sqrt {c\,x^2}}{a+b\,x} \,d x \]

[In]

int((c*x^2)^(1/2)/(a + b*x),x)

[Out]

int((c*x^2)^(1/2)/(a + b*x), x)