\(\int \frac {1}{x \sqrt {c x^2} (a+b x)^2} \, dx\) [915]

Optimal result

Integrand size = 20, antiderivative size = 78 \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)^2} \, dx=-\frac {1}{a^2 \sqrt {c x^2}}-\frac {b x}{a^2 \sqrt {c x^2} (a+b x)}-\frac {2 b x \log (x)}{a^3 \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 78, 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} \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)^2} \, dx=-\frac {2 b x \log (x)}{a^3 \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 \sqrt {c x^2}}-\frac {b x}{a^2 \sqrt {c x^2} (a+b x)}-\frac {1}{a^2 \sqrt {c x^2}} \]

[In]

Int[1/(x*Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)^2} \, dx=\frac {c x^2 \left (-\frac {a (a+2 b x)}{a+b x}-2 b x \log (x)+2 b x \log (a+b x)\right )}{a^3 \left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {-\frac {2 b x}{a^{2}}-\frac {1}{a}}{\sqrt {c \,x^{2}}\, \left (b x +a \right )}-\frac {2 b x \ln \left (x \right )}{a^{3} \sqrt {c \,x^{2}}}+\frac {2 x b \ln \left (-b x -a \right )}{\sqrt {c \,x^{2}}\, a^{3}}\)	\(69\)
default	\(-\frac {2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b \ln \left (x \right ) x -2 \ln \left (b x +a \right ) x a b +2 a b x +a^{2}}{\sqrt {c \,x^{2}}\, a^{3} \left (b x +a \right )}\)	\(71\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)^2} \, dx=-\frac {{\left (2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b c x^{3} + a^{4} c x^{2}} \]

[In]

integrate(1/x/(b*x+a)^2/(c*x^2)^(1/2),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*c*x^3 + a^4*c*x^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x \sqrt {c x^2} (a+b x)^2} \, dx=-\frac {2 \, b x + a}{a^{2} b \sqrt {c} x^{2} + a^{3} \sqrt {c} x} + \frac {2 \, b \log \left (b x + a\right )}{a^{3} \sqrt {c}} - \frac {2 \, b \log \left (x\right )}{a^{3} \sqrt {c}} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate(1/x/(b*x+a)^2/(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:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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