\(\int \frac {1}{(a+b x) (a c-b c x)^3} \, dx\) [1055]

Optimal result

Integrand size = 19, antiderivative size = 63 \[ \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx=\frac {1}{4 a b c^3 (a-b x)^2}+\frac {1}{4 a^2 b c^3 (a-b x)}+\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{4 a^3 b c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {46, 214} \[ \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx=\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{4 a^3 b c^3}+\frac {1}{4 a^2 b c^3 (a-b x)}+\frac {1}{4 a b c^3 (a-b x)^2} \]

[In]

Int[1/((a + b*x)*(a*c - b*c*x)^3),x]

[Out]

1/(4*a*b*c^3*(a - b*x)^2) + 1/(4*a^2*b*c^3*(a - b*x)) + ArcTanh[(b*x)/a]/(4*a^3*b*c^3)

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])

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 \left (\frac {1}{2 a c^3 (a-b x)^3}+\frac {1}{4 a^2 c^3 (a-b x)^2}+\frac {1}{4 a^2 c^3 \left (a^2-b^2 x^2\right )}\right ) \, dx \\ & = \frac {1}{4 a b c^3 (a-b x)^2}+\frac {1}{4 a^2 b c^3 (a-b x)}+\frac {\int \frac {1}{a^2-b^2 x^2} \, dx}{4 a^2 c^3} \\ & = \frac {1}{4 a b c^3 (a-b x)^2}+\frac {1}{4 a^2 b c^3 (a-b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^3 b c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx=\frac {2 a (2 a-b x)-(a-b x)^2 \log (a-b x)+(a-b x)^2 \log (a+b x)}{8 a^3 b c^3 (a-b x)^2} \]

[In]

Integrate[1/((a + b*x)*(a*c - b*c*x)^3),x]

[Out]

(2*a*(2*a - b*x) - (a - b*x)^2*Log[a - b*x] + (a - b*x)^2*Log[a + b*x])/(8*a^3*b*c^3*(a - b*x)^2)

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02

method	result	size
risch	\(\frac {-\frac {x}{4 a^{2}}+\frac {1}{2 b a}}{c^{3} \left (-b x +a \right )^{2}}-\frac {\ln \left (-b x +a \right )}{8 a^{3} c^{3} b}+\frac {\ln \left (b x +a \right )}{8 a^{3} c^{3} b}\)	\(64\)
default	\(\frac {\frac {\ln \left (b x +a \right )}{8 a^{3} b}-\frac {\ln \left (-b x +a \right )}{8 a^{3} b}+\frac {1}{4 a^{2} b \left (-b x +a \right )}+\frac {1}{4 b a \left (-b x +a \right )^{2}}}{c^{3}}\)	\(67\)
norman	\(\frac {\frac {3 x}{4 a^{2} c}-\frac {b \,x^{2}}{2 a^{3} c}}{c^{2} \left (-b x +a \right )^{2}}-\frac {\ln \left (-b x +a \right )}{8 a^{3} c^{3} b}+\frac {\ln \left (b x +a \right )}{8 a^{3} c^{3} b}\)	\(71\)
parallelrisch	\(\frac {-\ln \left (b x -a \right ) x^{2} b^{2}+b^{2} \ln \left (b x +a \right ) x^{2}+2 \ln \left (b x -a \right ) x a b -2 \ln \left (b x +a \right ) x a b -4 b^{2} x^{2}-a^{2} \ln \left (b x -a \right )+a^{2} \ln \left (b x +a \right )+6 a b x}{8 a^{3} c^{3} \left (b x -a \right )^{2} b}\)	\(111\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx=-\frac {2 \, a b x - 4 \, a^{2} - {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{8 \, {\left (a^{3} b^{3} c^{3} x^{2} - 2 \, a^{4} b^{2} c^{3} x + a^{5} b c^{3}\right )}} \]

[In]

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

[Out]

-1/8*(2*a*b*x - 4*a^2 - (b^2*x^2 - 2*a*b*x + a^2)*log(b*x + a) + (b^2*x^2 - 2*a*b*x + a^2)*log(b*x - a))/(a^3*
b^3*c^3*x^2 - 2*a^4*b^2*c^3*x + a^5*b*c^3)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx=- \frac {- 2 a + b x}{4 a^{4} b c^{3} - 8 a^{3} b^{2} c^{3} x + 4 a^{2} b^{3} c^{3} x^{2}} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{8} - \frac {\log {\left (\frac {a}{b} + x \right )}}{8}}{a^{3} b c^{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx=-\frac {b x - 2 \, a}{4 \, {\left (a^{2} b^{3} c^{3} x^{2} - 2 \, a^{3} b^{2} c^{3} x + a^{4} b c^{3}\right )}} + \frac {\log \left (b x + a\right )}{8 \, a^{3} b c^{3}} - \frac {\log \left (b x - a\right )}{8 \, a^{3} b c^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{8 \, a^{3} b c^{3}} - \frac {\log \left ({\left | b x - a \right |}\right )}{8 \, a^{3} b c^{3}} - \frac {a b x - 2 \, a^{2}}{4 \, {\left (b x - a\right )}^{2} a^{3} b c^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx=\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{4\,a^3\,b\,c^3}-\frac {\frac {x}{4\,a^2}-\frac {1}{2\,a\,b}}{a^2\,c^3-2\,a\,b\,c^3\,x+b^2\,c^3\,x^2} \]

[In]

int(1/((a*c - b*c*x)^3*(a + b*x)),x)

[Out]

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