### 3.1817 $$\int \frac{1}{(a+b x) (a c+(b c+a d) x+b d x^2)^2} \, dx$$

Optimal. Leaf size=109 $\frac{d^2}{(c+d x) (b c-a d)^3}+\frac{3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac{3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac{2 b d}{(a+b x) (b c-a d)^3}-\frac{b}{2 (a+b x)^2 (b c-a d)^2}$

[Out]

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

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Rubi [A]  time = 0.0778673, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 44} $\frac{d^2}{(c+d x) (b c-a d)^3}+\frac{3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac{3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac{2 b d}{(a+b x) (b c-a d)^3}-\frac{b}{2 (a+b x)^2 (b c-a d)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

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])

Rubi steps

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

Mathematica [A]  time = 0.0723044, size = 98, normalized size = 0.9 $\frac{\frac{2 d^2 (b c-a d)}{c+d x}+\frac{4 b d (b c-a d)}{a+b x}-\frac{b (b c-a d)^2}{(a+b x)^2}+6 b d^2 \log (a+b x)-6 b d^2 \log (c+d x)}{2 (b c-a d)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-((b*(b*c - a*d)^2)/(a + b*x)^2) + (4*b*d*(b*c - a*d))/(a + b*x) + (2*d^2*(b*c - a*d))/(c + d*x) + 6*b*d^2*Lo
g[a + b*x] - 6*b*d^2*Log[c + d*x])/(2*(b*c - a*d)^4)

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Maple [A]  time = 0.052, size = 109, normalized size = 1. \begin{align*} -{\frac{{d}^{2}}{ \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-3\,{\frac{{d}^{2}b\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}}-{\frac{b}{2\, \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}+3\,{\frac{{d}^{2}b\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}}-2\,{\frac{bd}{ \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.1036, size = 521, normalized size = 4.78 \begin{align*} \frac{3 \, b d^{2} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{3 \, b d^{2} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac{6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \,{\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{2 \,{\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} +{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} +{\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} +{\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.61282, size = 991, normalized size = 9.09 \begin{align*} -\frac{b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \,{\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} +{\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} +{\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} +{\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} +{\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4} +{\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} x^{3} +{\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} x^{2} +{\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*c^3 - 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 2*a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 3*(b^3*c^2*d + 2*a
*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 +
a^2*b*d^3)*x)*log(b*x + a) + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a
^2*b*d^3)*x)*log(d*x + c))/(a^2*b^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^2*c^3*d^2 - 4*a^5*b*c^2*d^3 + a^6*c*d^4 +
(b^6*c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3 - 4*a^3*b^3*c*d^4 + a^4*b^2*d^5)*x^3 + (b^6*c^5 - 2*a*b^5*c^4
*d - 2*a^2*b^4*c^3*d^2 + 8*a^3*b^3*c^2*d^3 - 7*a^4*b^2*c*d^4 + 2*a^5*b*d^5)*x^2 + (2*a*b^5*c^5 - 7*a^2*b^4*c^4
*d + 8*a^3*b^3*c^3*d^2 - 2*a^4*b^2*c^2*d^3 - 2*a^5*b*c*d^4 + a^6*d^5)*x)

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Sympy [B]  time = 3.87319, size = 632, normalized size = 5.8 \begin{align*} - \frac{3 b d^{2} \log{\left (x + \frac{- \frac{3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} + \frac{15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} - \frac{30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} + \frac{30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} - \frac{15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} + \frac{3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} + \frac{3 b d^{2} \log{\left (x + \frac{\frac{3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} - \frac{15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} + \frac{30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} - \frac{30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} + \frac{15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} - \frac{3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} - \frac{2 a^{2} d^{2} + 5 a b c d - b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (9 a b d^{2} + 3 b^{2} c d\right )}{2 a^{5} c d^{3} - 6 a^{4} b c^{2} d^{2} + 6 a^{3} b^{2} c^{3} d - 2 a^{2} b^{3} c^{4} + x^{3} \left (2 a^{3} b^{2} d^{4} - 6 a^{2} b^{3} c d^{3} + 6 a b^{4} c^{2} d^{2} - 2 b^{5} c^{3} d\right ) + x^{2} \left (4 a^{4} b d^{4} - 10 a^{3} b^{2} c d^{3} + 6 a^{2} b^{3} c^{2} d^{2} + 2 a b^{4} c^{3} d - 2 b^{5} c^{4}\right ) + x \left (2 a^{5} d^{4} - 2 a^{4} b c d^{3} - 6 a^{3} b^{2} c^{2} d^{2} + 10 a^{2} b^{3} c^{3} d - 4 a b^{4} c^{4}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b*d**2*log(x + (-3*a**5*b*d**7/(a*d - b*c)**4 + 15*a**4*b**2*c*d**6/(a*d - b*c)**4 - 30*a**3*b**3*c**2*d**5
/(a*d - b*c)**4 + 30*a**2*b**4*c**3*d**4/(a*d - b*c)**4 - 15*a*b**5*c**4*d**3/(a*d - b*c)**4 + 3*a*b*d**3 + 3*
b**6*c**5*d**2/(a*d - b*c)**4 + 3*b**2*c*d**2)/(6*b**2*d**3))/(a*d - b*c)**4 + 3*b*d**2*log(x + (3*a**5*b*d**7
/(a*d - b*c)**4 - 15*a**4*b**2*c*d**6/(a*d - b*c)**4 + 30*a**3*b**3*c**2*d**5/(a*d - b*c)**4 - 30*a**2*b**4*c*
*3*d**4/(a*d - b*c)**4 + 15*a*b**5*c**4*d**3/(a*d - b*c)**4 + 3*a*b*d**3 - 3*b**6*c**5*d**2/(a*d - b*c)**4 + 3
*b**2*c*d**2)/(6*b**2*d**3))/(a*d - b*c)**4 - (2*a**2*d**2 + 5*a*b*c*d - b**2*c**2 + 6*b**2*d**2*x**2 + x*(9*a
*b*d**2 + 3*b**2*c*d))/(2*a**5*c*d**3 - 6*a**4*b*c**2*d**2 + 6*a**3*b**2*c**3*d - 2*a**2*b**3*c**4 + x**3*(2*a
**3*b**2*d**4 - 6*a**2*b**3*c*d**3 + 6*a*b**4*c**2*d**2 - 2*b**5*c**3*d) + x**2*(4*a**4*b*d**4 - 10*a**3*b**2*
c*d**3 + 6*a**2*b**3*c**2*d**2 + 2*a*b**4*c**3*d - 2*b**5*c**4) + x*(2*a**5*d**4 - 2*a**4*b*c*d**3 - 6*a**3*b*
*2*c**2*d**2 + 10*a**2*b**3*c**3*d - 4*a*b**4*c**4))

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Giac [B]  time = 1.21765, size = 342, normalized size = 3.14 \begin{align*} \frac{3 \, b^{2} d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac{3 \, b d^{3} \log \left ({\left | d x + c \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} - \frac{b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x}{2 \,{\left (b c - a d\right )}^{4}{\left (b x + a\right )}^{2}{\left (d x + c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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