\(\int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx\) [1061]

Optimal result

Integrand size = 19, antiderivative size = 46 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {x}{2 a^2 c^2 \left (a^2-b^2 x^2\right )}+\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{2 a^3 b c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {41, 205, 214} \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{2 a^3 b c^2}+\frac {x}{2 a^2 c^2 \left (a^2-b^2 x^2\right )} \]

[In]

Int[1/((a + b*x)^2*(a*c - b*c*x)^2),x]

[Out]

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a^2 c-b^2 c x^2\right )^2} \, dx \\ & = \frac {x}{2 a^2 c^2 \left (a^2-b^2 x^2\right )}+\frac {\int \frac {1}{a^2 c-b^2 c x^2} \, dx}{2 a^2 c} \\ & = \frac {x}{2 a^2 c^2 \left (a^2-b^2 x^2\right )}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{2 a^3 b c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {2 a b x+\left (-a^2+b^2 x^2\right ) \log (a-b x)+\left (a^2-b^2 x^2\right ) \log (a+b x)}{4 a^3 b c^2 (a-b x) (a+b x)} \]

[In]

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

[Out]

(2*a*b*x + (-a^2 + b^2*x^2)*Log[a - b*x] + (a^2 - b^2*x^2)*Log[a + b*x])/(4*a^3*b*c^2*(a - b*x)*(a + b*x))

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.33

method	result	size
norman	\(\frac {x}{2 a^{2} c^{2} \left (b x +a \right ) \left (-b x +a \right )}-\frac {\ln \left (-b x +a \right )}{4 a^{3} b \,c^{2}}+\frac {\ln \left (b x +a \right )}{4 a^{3} b \,c^{2}}\)	\(61\)
risch	\(\frac {x}{2 a^{2} c^{2} \left (b x +a \right ) \left (-b x +a \right )}-\frac {\ln \left (-b x +a \right )}{4 a^{3} b \,c^{2}}+\frac {\ln \left (b x +a \right )}{4 a^{3} b \,c^{2}}\)	\(61\)
default	\(\frac {\frac {\ln \left (b x +a \right )}{4 a^{3} b}-\frac {1}{4 a^{2} b \left (b x +a \right )}-\frac {\ln \left (-b x +a \right )}{4 a^{3} b}+\frac {1}{4 a^{2} b \left (-b x +a \right )}}{c^{2}}\)	\(66\)
parallelrisch	\(\frac {-\ln \left (b x -a \right ) x^{2} b^{3}+b^{3} \ln \left (b x +a \right ) x^{2}+\ln \left (b x -a \right ) a^{2} b -\ln \left (b x +a \right ) a^{2} b -2 a \,b^{2} x}{4 a^{3} b^{2} c^{2} \left (b x +a \right ) \left (b x -a \right )}\)	\(90\)

[In]

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

[Out]

1/2/a^2/c^2*x/(b*x+a)/(-b*x+a)-1/4/a^3/b/c^2*ln(-b*x+a)+1/4/a^3/b/c^2*ln(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx=-\frac {2 \, a b x - {\left (b^{2} x^{2} - a^{2}\right )} \log \left (b x + a\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \log \left (b x - a\right )}{4 \, {\left (a^{3} b^{3} c^{2} x^{2} - a^{5} b c^{2}\right )}} \]

[In]

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

[Out]

-1/4*(2*a*b*x - (b^2*x^2 - a^2)*log(b*x + a) + (b^2*x^2 - a^2)*log(b*x - a))/(a^3*b^3*c^2*x^2 - a^5*b*c^2)

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx=- \frac {x}{- 2 a^{4} c^{2} + 2 a^{2} b^{2} c^{2} x^{2}} + \frac {- \frac {\log {\left (- \frac {a}{b} + x \right )}}{4} + \frac {\log {\left (\frac {a}{b} + x \right )}}{4}}{a^{3} b c^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx=-\frac {x}{2 \, {\left (a^{2} b^{2} c^{2} x^{2} - a^{4} c^{2}\right )}} + \frac {\log \left (b x + a\right )}{4 \, a^{3} b c^{2}} - \frac {\log \left (b x - a\right )}{4 \, a^{3} b c^{2}} \]

[In]

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

[Out]

-1/2*x/(a^2*b^2*c^2*x^2 - a^4*c^2) + 1/4*log(b*x + a)/(a^3*b*c^2) - 1/4*log(b*x - a)/(a^3*b*c^2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.80 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx=-\frac {1}{4 \, {\left (b c x - a c\right )} a^{2} b c} + \frac {\log \left ({\left | -\frac {2 \, a c}{b c x - a c} - 1 \right |}\right )}{4 \, a^{3} b c^{2}} + \frac {1}{8 \, a^{3} b {\left (\frac {2 \, a c}{b c x - a c} + 1\right )} c^{2}} \]

[In]

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

[Out]

-1/4/((b*c*x - a*c)*a^2*b*c) + 1/4*log(abs(-2*a*c/(b*c*x - a*c) - 1))/(a^3*b*c^2) + 1/8/(a^3*b*(2*a*c/(b*c*x -
a*c) + 1)*c^2)

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {x}{2\,a^2\,\left (a^2\,c^2-b^2\,c^2\,x^2\right )}+\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{2\,a^3\,b\,c^2} \]

[In]

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

[Out]

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