\(\int \frac {\sqrt {c x^2}}{x^2 (a+b x)^2} \, dx\) [898]

Optimal result

Integrand size = 20, antiderivative size = 65 \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)^2} \, dx=\frac {\sqrt {c x^2}}{a x (a+b x)}+\frac {\sqrt {c x^2} \log (x)}{a^2 x}-\frac {\sqrt {c x^2} \log (a+b x)}{a^2 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, 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} \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)^2} \, dx=-\frac {\sqrt {c x^2} \log (a+b x)}{a^2 x}+\frac {\sqrt {c x^2} \log (x)}{a^2 x}+\frac {\sqrt {c x^2}}{a x (a+b x)} \]

[In]

Int[Sqrt[c*x^2]/(x^2*(a + b*x)^2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)^2} \, dx=\frac {c x (a+(a+b x) \log (x)-(a+b x) \log (a+b x))}{a^2 \sqrt {c x^2} (a+b x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {\sqrt {c \,x^{2}}\, \left (b \ln \left (x \right ) x -b \ln \left (b x +a \right ) x +a \ln \left (x \right )-a \ln \left (b x +a \right )+a \right )}{x \,a^{2} \left (b x +a \right )}\)	\(52\)
risch	\(\frac {\sqrt {c \,x^{2}}}{a x \left (b x +a \right )}+\frac {\sqrt {c \,x^{2}}\, \ln \left (-x \right )}{x \,a^{2}}-\frac {\ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a^{2} x}\)	\(62\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)^2} \, dx=\frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (\frac {x}{b x + a}\right ) + a\right )}}{a^{2} b x^{2} + a^{3} x} \]

[In]

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

[Out]

sqrt(c*x^2)*((b*x + a)*log(x/(b*x + a)) + a)/(a^2*b*x^2 + a^3*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)^2} \, dx=\frac {\sqrt {c}}{a b x + a^{2}} - \frac {\sqrt {c} \log \left (b x + a\right )}{a^{2}} + \frac {\sqrt {c} \log \left (x\right )}{a^{2}} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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