\(\int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx\) [858]

Optimal result

Integrand size = 20, antiderivative size = 61 \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx=-\frac {\sqrt {c x^2}}{a x^2}-\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, 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} \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx=-\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {\sqrt {c x^2}}{a x^2} \]

[In]

Int[Sqrt[c*x^2]/(x^3*(a + b*x)),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx=-\frac {c (a+b x \log (x)-b x \log (a+b x))}{a^2 \sqrt {c x^2}} \]

[In]

Integrate[Sqrt[c*x^2]/(x^3*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.54

method	result	size
default	\(-\frac {\sqrt {c \,x^{2}}\, \left (b \ln \left (x \right ) x -b \ln \left (b x +a \right ) x +a \right )}{a^{2} x^{2}}\)	\(33\)
risch	\(-\frac {\sqrt {c \,x^{2}}}{a \,x^{2}}+\frac {\sqrt {c \,x^{2}}\, b \ln \left (-b x -a \right )}{x \,a^{2}}-\frac {b \ln \left (x \right ) \sqrt {c \,x^{2}}}{a^{2} x}\)	\(59\)

[In]

int((c*x^2)^(1/2)/x^3/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx=\frac {\sqrt {c x^{2}} {\left (b x \log \left (\frac {b x + a}{x}\right ) - a\right )}}{a^{2} x^{2}} \]

[In]

integrate((c*x^2)^(1/2)/x^3/(b*x+a),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx=\int \frac {\sqrt {c x^{2}}}{x^{3} \left (a + b x\right )}\, dx \]

[In]

integrate((c*x**2)**(1/2)/x**3/(b*x+a),x)

[Out]

Integral(sqrt(c*x**2)/(x**3*(a + b*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx=\frac {b \sqrt {c} \log \left (b x + a\right )}{a^{2}} - \frac {b \sqrt {c} \log \left (x\right )}{a^{2}} - \frac {\sqrt {c}}{a x} \]

[In]

integrate((c*x^2)^(1/2)/x^3/(b*x+a),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c*x^2)^(1/2)/x^3/(b*x+a),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. \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx=\int \frac {\sqrt {c\,x^2}}{x^3\,\left (a+b\,x\right )} \,d x \]

[In]

int((c*x^2)^(1/2)/(x^3*(a + b*x)),x)

[Out]

int((c*x^2)^(1/2)/(x^3*(a + b*x)), x)