\(\int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx\) [882]

Optimal result

Integrand size = 20, antiderivative size = 54 \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx=-\frac {1}{a \sqrt {c x^2}}-\frac {b x \log (x)}{a^2 \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 46} \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx=-\frac {b x \log (x)}{a^2 \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 \sqrt {c x^2}}-\frac {1}{a \sqrt {c x^2}} \]

[In]

Int[1/(x*Sqrt[c*x^2]*(a + b*x)),x]

[Out]

-(1/(a*Sqrt[c*x^2])) - (b*x*Log[x])/(a^2*Sqrt[c*x^2]) + (b*x*Log[a + b*x])/(a^2*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 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx=\frac {c x^2 (-a-b x \log (x)+b x \log (a+b x))}{a^2 \left (c x^2\right )^{3/2}} \]

[In]

Integrate[1/(x*Sqrt[c*x^2]*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.56

method	result	size
default	\(-\frac {b \ln \left (x \right ) x -b \ln \left (b x +a \right ) x +a}{\sqrt {c \,x^{2}}\, a^{2}}\)	\(30\)
risch	\(-\frac {1}{a \sqrt {c \,x^{2}}}+\frac {x b \ln \left (-b x -a \right )}{\sqrt {c \,x^{2}}\, a^{2}}-\frac {b x \ln \left (x \right )}{a^{2} \sqrt {c \,x^{2}}}\)	\(52\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx=\frac {\sqrt {c x^{2}} {\left (b x \log \left (\frac {b x + a}{x}\right ) - a\right )}}{a^{2} c x^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx=\int \frac {1}{x \sqrt {c x^{2}} \left (a + b x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx=\frac {b \log \left (b x + a\right )}{a^{2} \sqrt {c}} - \frac {b \log \left (x\right )}{a^{2} \sqrt {c}} - \frac {1}{a \sqrt {c} x} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx=\int \frac {1}{x\,\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \]

[In]

int(1/(x*(c*x^2)^(1/2)*(a + b*x)),x)

[Out]

int(1/(x*(c*x^2)^(1/2)*(a + b*x)), x)