3.1062 \(\int \frac{1}{(a+b x)^2 (a c-b c x)^3} \, dx\)

Optimal. Leaf size=83 \[ \frac{1}{4 a^3 b c^3 (a-b x)}-\frac{1}{8 a^3 b c^3 (a+b x)}+\frac{1}{8 a^2 b c^3 (a-b x)^2}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b c^3} \]

[Out]

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

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Rubi [A]  time = 0.0495295, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {44, 208} \[ \frac{1}{4 a^3 b c^3 (a-b x)}-\frac{1}{8 a^3 b c^3 (a+b x)}+\frac{1}{8 a^2 b c^3 (a-b x)^2}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 44

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] && L
tQ[m + n + 2, 0])

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^2 (a c-b c x)^3} \, dx &=\int \left (\frac{1}{4 a^2 c^3 (a-b x)^3}+\frac{1}{4 a^3 c^3 (a-b x)^2}+\frac{1}{8 a^3 c^3 (a+b x)^2}+\frac{3}{8 a^3 c^3 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{8 a^2 b c^3 (a-b x)^2}+\frac{1}{4 a^3 b c^3 (a-b x)}-\frac{1}{8 a^3 b c^3 (a+b x)}+\frac{3 \int \frac{1}{a^2-b^2 x^2} \, dx}{8 a^3 c^3}\\ &=\frac{1}{8 a^2 b c^3 (a-b x)^2}+\frac{1}{4 a^3 b c^3 (a-b x)}-\frac{1}{8 a^3 b c^3 (a+b x)}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b c^3}\\ \end{align*}

Mathematica [A]  time = 0.0355216, size = 68, normalized size = 0.82 \[ \frac{\frac{2 a \left (2 a^2+3 a b x-3 b^2 x^2\right )}{(a-b x)^2 (a+b x)}-3 \log (a-b x)+3 \log (a+b x)}{16 a^4 b c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*(2*a^2 + 3*a*b*x - 3*b^2*x^2))/((a - b*x)^2*(a + b*x)) - 3*Log[a - b*x] + 3*Log[a + b*x])/(16*a^4*b*c^3)

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Maple [A]  time = 0.01, size = 96, normalized size = 1.2 \begin{align*}{\frac{3\,\ln \left ( bx+a \right ) }{16\,{c}^{3}{a}^{4}b}}-{\frac{1}{8\,{a}^{3}b{c}^{3} \left ( bx+a \right ) }}-{\frac{3\,\ln \left ( bx-a \right ) }{16\,{c}^{3}{a}^{4}b}}-{\frac{1}{4\,{a}^{3}b{c}^{3} \left ( bx-a \right ) }}+{\frac{1}{8\,{c}^{3}{a}^{2}b \left ( bx-a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16/c^3/a^4/b*ln(b*x+a)-1/8/a^3/b/c^3/(b*x+a)-3/16/c^3/a^4/b*ln(b*x-a)-1/4/c^3/a^3/b/(b*x-a)+1/8/c^3/a^2/b/(b
*x-a)^2

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Maxima [A]  time = 1.07872, size = 146, normalized size = 1.76 \begin{align*} -\frac{3 \, b^{2} x^{2} - 3 \, a b x - 2 \, a^{2}}{8 \,{\left (a^{3} b^{4} c^{3} x^{3} - a^{4} b^{3} c^{3} x^{2} - a^{5} b^{2} c^{3} x + a^{6} b c^{3}\right )}} + \frac{3 \, \log \left (b x + a\right )}{16 \, a^{4} b c^{3}} - \frac{3 \, \log \left (b x - a\right )}{16 \, a^{4} b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*b^2*x^2 - 3*a*b*x - 2*a^2)/(a^3*b^4*c^3*x^3 - a^4*b^3*c^3*x^2 - a^5*b^2*c^3*x + a^6*b*c^3) + 3/16*log(
b*x + a)/(a^4*b*c^3) - 3/16*log(b*x - a)/(a^4*b*c^3)

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Fricas [A]  time = 1.66517, size = 290, normalized size = 3.49 \begin{align*} -\frac{6 \, a b^{2} x^{2} - 6 \, a^{2} b x - 4 \, a^{3} - 3 \,{\left (b^{3} x^{3} - a b^{2} x^{2} - a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{3} x^{3} - a b^{2} x^{2} - a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{16 \,{\left (a^{4} b^{4} c^{3} x^{3} - a^{5} b^{3} c^{3} x^{2} - a^{6} b^{2} c^{3} x + a^{7} b c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(6*a*b^2*x^2 - 6*a^2*b*x - 4*a^3 - 3*(b^3*x^3 - a*b^2*x^2 - a^2*b*x + a^3)*log(b*x + a) + 3*(b^3*x^3 - a
*b^2*x^2 - a^2*b*x + a^3)*log(b*x - a))/(a^4*b^4*c^3*x^3 - a^5*b^3*c^3*x^2 - a^6*b^2*c^3*x + a^7*b*c^3)

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Sympy [A]  time = 0.655645, size = 104, normalized size = 1.25 \begin{align*} - \frac{- 2 a^{2} - 3 a b x + 3 b^{2} x^{2}}{8 a^{6} b c^{3} - 8 a^{5} b^{2} c^{3} x - 8 a^{4} b^{3} c^{3} x^{2} + 8 a^{3} b^{4} c^{3} x^{3}} - \frac{\frac{3 \log{\left (- \frac{a}{b} + x \right )}}{16} - \frac{3 \log{\left (\frac{a}{b} + x \right )}}{16}}{a^{4} b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-2*a**2 - 3*a*b*x + 3*b**2*x**2)/(8*a**6*b*c**3 - 8*a**5*b**2*c**3*x - 8*a**4*b**3*c**3*x**2 + 8*a**3*b**4*c
**3*x**3) - (3*log(-a/b + x)/16 - 3*log(a/b + x)/16)/(a**4*b*c**3)

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Giac [A]  time = 1.06237, size = 109, normalized size = 1.31 \begin{align*} -\frac{3 \, \log \left ({\left | -\frac{2 \, a}{b x + a} + 1 \right |}\right )}{16 \, a^{4} b c^{3}} - \frac{1}{8 \,{\left (b x + a\right )} a^{3} b c^{3}} + \frac{\frac{12 \, a}{b x + a} - 5}{32 \, a^{4} b c^{3}{\left (\frac{2 \, a}{b x + a} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*log(abs(-2*a/(b*x + a) + 1))/(a^4*b*c^3) - 1/8/((b*x + a)*a^3*b*c^3) + 1/32*(12*a/(b*x + a) - 5)/(a^4*b*
c^3*(2*a/(b*x + a) - 1)^2)