\(\int \frac {\sqrt {c x^2}}{x^3 (a+b x)^2} \, dx\) [899]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)^2} \, dx=-\frac {\sqrt {c x^2}}{a^2 x^2}-\frac {b \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {2 b \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b \sqrt {c x^2} \log (a+b x)}{a^3 x} \]

[Out]

-(c*x^2)^(1/2)/a^2/x^2-b*(c*x^2)^(1/2)/a^2/x/(b*x+a)-2*b*ln(x)*(c*x^2)^(1/2)/a^3/x+2*b*ln(b*x+a)*(c*x^2)^(1/2)
/a^3/x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, 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} \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)^2} \, dx=-\frac {2 b \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b \sqrt {c x^2} \log (a+b x)}{a^3 x}-\frac {b \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {\sqrt {c x^2}}{a^2 x^2} \]

[In]

Int[Sqrt[c*x^2]/(x^3*(a + b*x)^2),x]

[Out]

-(Sqrt[c*x^2]/(a^2*x^2)) - (b*Sqrt[c*x^2])/(a^2*x*(a + b*x)) - (2*b*Sqrt[c*x^2]*Log[x])/(a^3*x) + (2*b*Sqrt[c*
x^2]*Log[a + b*x])/(a^3*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)^2} \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx}{x} \\ & = -\frac {\sqrt {c x^2}}{a^2 x^2}-\frac {b \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {2 b \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b \sqrt {c x^2} \log (a+b x)}{a^3 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)^2} \, dx=\sqrt {c x^2} \left (\frac {-a-2 b x}{a^2 x^2 (a+b x)}-\frac {2 b \log (x)}{a^3 x}+\frac {2 b \log (a+b x)}{a^3 x}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b \ln \left (x \right ) x -2 \ln \left (b x +a \right ) x a b +2 a b x +a^{2}\right )}{x^{2} a^{3} \left (b x +a \right )}\)	\(74\)
risch	\(\frac {\sqrt {c \,x^{2}}\, \left (-\frac {2 b x}{a^{2}}-\frac {1}{a}\right )}{x^{2} \left (b x +a \right )}-\frac {2 b \ln \left (x \right ) \sqrt {c \,x^{2}}}{a^{3} x}+\frac {2 \sqrt {c \,x^{2}}\, b \ln \left (-b x -a \right )}{x \,a^{3}}\)	\(76\)

[In]

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

[Out]

-(c*x^2)^(1/2)*(2*b^2*ln(x)*x^2-2*b^2*ln(b*x+a)*x^2+2*a*b*ln(x)*x-2*ln(b*x+a)*x*a*b+2*a*b*x+a^2)/x^2/a^3/(b*x+
a)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)^2} \, dx=-\frac {{\left (2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b x^{3} + a^{4} x^{2}} \]

[In]

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

[Out]

-(2*a*b*x + a^2 - 2*(b^2*x^2 + a*b*x)*log((b*x + a)/x))*sqrt(c*x^2)/(a^3*b*x^3 + a^4*x^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c x^2}}{x^3 (a+b x)^2} \, dx=-\frac {2 \, b \sqrt {c} x + a \sqrt {c}}{a^{2} b x^{2} + a^{3} x} + \frac {2 \, b \sqrt {c} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b \sqrt {c} \log \left (x\right )}{a^{3}} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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