### 3.1807 $$\int \frac{1}{(a+b x)^2 (a c+(b c+a d) x+b d x^2)} \, dx$$

Optimal. Leaf size=82 $\frac{d^2 \log (a+b x)}{(b c-a d)^3}-\frac{d^2 \log (c+d x)}{(b c-a d)^3}+\frac{d}{(a+b x) (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (b c-a d)}$

[Out]

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

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Rubi [A]  time = 0.0563078, antiderivative size = 82, 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} $\frac{d^2 \log (a+b x)}{(b c-a d)^3}-\frac{d^2 \log (c+d x)}{(b c-a d)^3}+\frac{d}{(a+b x) (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (b c-a d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0628019, size = 67, normalized size = 0.82 $\frac{\frac{(b c-a d) (3 a d-b c+2 b d x)}{(a+b x)^2}+2 d^2 \log (a+b x)-2 d^2 \log (c+d x)}{2 (b c-a d)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.051, size = 81, normalized size = 1. \begin{align*}{\frac{{d}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}+{\frac{1}{ \left ( 2\,ad-2\,bc \right ) \left ( bx+a \right ) ^{2}}}+{\frac{d}{ \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-{\frac{{d}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04429, size = 273, normalized size = 3.33 \begin{align*} \frac{d^{2} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{d^{2} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, b d x - b c + 3 \, a d}{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.65853, size = 491, normalized size = 5.99 \begin{align*} -\frac{b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2} - 2 \,{\left (b^{2} c d - a b d^{2}\right )} x - 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3} +{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.92071, size = 381, normalized size = 4.65 \begin{align*} \frac{d^{2} \log{\left (x + \frac{- \frac{a^{4} d^{6}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b c d^{5}}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + a d^{3} - \frac{b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + b c d^{2}}{2 b d^{3}} \right )}}{\left (a d - b c\right )^{3}} - \frac{d^{2} \log{\left (x + \frac{\frac{a^{4} d^{6}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b c d^{5}}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + a d^{3} + \frac{b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + b c d^{2}}{2 b d^{3}} \right )}}{\left (a d - b c\right )^{3}} + \frac{3 a d - b c + 2 b d x}{2 a^{4} d^{2} - 4 a^{3} b c d + 2 a^{2} b^{2} c^{2} + x^{2} \left (2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}\right ) + x \left (4 a^{3} b d^{2} - 8 a^{2} b^{2} c d + 4 a b^{3} c^{2}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19967, size = 196, normalized size = 2.39 \begin{align*} -\frac{b d^{2} \log \left ({\left | -\frac{b c}{b x + a} + \frac{a d}{b x + a} - d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{\frac{b^{3} c}{{\left (b x + a\right )}^{2}} - \frac{2 \, b^{2} d}{b x + a} - \frac{a b^{2} d}{{\left (b x + a\right )}^{2}}}{2 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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