\(\int \frac {\sqrt {c x^2}}{x (a+b x)} \, dx\) [856]

Optimal result

Integrand size = 20, antiderivative size = 22 \[ \int \frac {\sqrt {c x^2}}{x (a+b x)} \, dx=\frac {\sqrt {c x^2} \log (a+b x)}{b x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 31} \[ \int \frac {\sqrt {c x^2}}{x (a+b x)} \, dx=\frac {\sqrt {c x^2} \log (a+b x)}{b x} \]

[In]

Int[Sqrt[c*x^2]/(x*(a + b*x)),x]

[Out]

(Sqrt[c*x^2]*Log[a + b*x])/(b*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 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*} \text {integral}& = \frac {\sqrt {c x^2} \int \frac {1}{a+b x} \, dx}{x} \\ & = \frac {\sqrt {c x^2} \log (a+b x)}{b x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c x^2}}{x (a+b x)} \, dx=\frac {c x \log (a+b x)}{b \sqrt {c x^2}} \]

[In]

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

[Out]

(c*x*Log[a + b*x])/(b*Sqrt[c*x^2])

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b x}\)	\(21\)
risch	\(\frac {\ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b x}\)	\(21\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {c x^2}}{x (a+b x)} \, dx=\frac {\sqrt {c x^{2}} \log \left (b x + a\right )}{b x} \]

[In]

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

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c x^2}}{x (a+b x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {c x^2}}{x (a+b x)} \, dx=\sqrt {c} {\left (\frac {\log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b} - \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b}\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((c*x^2)^(1/2)/(x*(a + b*x)),x)

[Out]

int((c*x^2)^(1/2)/(x*(a + b*x)), x)