### 3.1808 $$\int \frac{1}{(a+b x)^3 (a c+(b c+a d) x+b d x^2)} \, dx$$

Optimal. Leaf size=107 $-\frac{d^2}{(a+b x) (b c-a d)^3}-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (b c-a d)}$

[Out]

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

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Rubi [A]  time = 0.0712196, antiderivative size = 107, 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} $-\frac{d^2}{(a+b x) (b c-a d)^3}-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (b c-a d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.042251, size = 107, normalized size = 1. $-\frac{d^2}{(a+b x) (b c-a d)^3}-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}+\frac{1}{3 (a+b x)^3 (a d-b c)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.051, size = 103, normalized size = 1. \begin{align*}{\frac{{d}^{3}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}}+{\frac{1}{ \left ( 3\,ad-3\,bc \right ) \left ( bx+a \right ) ^{3}}}+{\frac{d}{2\, \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{d}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}}+{\frac{{d}^{2}}{ \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)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x)

[Out]

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

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Maxima [B]  time = 1.24775, size = 487, normalized size = 4.55 \begin{align*} -\frac{d^{3} \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{d^{3} \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} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{6 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^3*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) + d^3*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) - 1/6*(6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*
b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3 + (b^
6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2
- a^4*b^2*d^3)*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*x)

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Fricas [B]  time = 1.60064, size = 856, normalized size = 8. \begin{align*} -\frac{2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, 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 - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x + 6 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (d x + c\right )}{6 \,{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4} +{\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} x^{3} + 3 \,{\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 11*a^3*d^3 + 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 3*(b^3*c^2*d -
6*a*b^2*c*d^2 + 5*a^2*b*d^3)*x + 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6
*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c))/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^
5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4 + (b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4
*b^3*d^4)*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*x^2 + 3*(a
^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*x)

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Sympy [B]  time = 2.87301, size = 570, normalized size = 5.33 \begin{align*} \frac{d^{3} \log{\left (x + \frac{- \frac{a^{5} d^{8}}{\left (a d - b c\right )^{4}} + \frac{5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} - \frac{10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac{10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} - \frac{5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} + \frac{b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} - \frac{d^{3} \log{\left (x + \frac{\frac{a^{5} d^{8}}{\left (a d - b c\right )^{4}} - \frac{5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} + \frac{10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac{10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} + \frac{5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} - \frac{b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} + \frac{11 a^{2} d^{2} - 7 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (15 a b d^{2} - 3 b^{2} c d\right )}{6 a^{6} d^{3} - 18 a^{5} b c d^{2} + 18 a^{4} b^{2} c^{2} d - 6 a^{3} b^{3} c^{3} + x^{3} \left (6 a^{3} b^{3} d^{3} - 18 a^{2} b^{4} c d^{2} + 18 a b^{5} c^{2} d - 6 b^{6} c^{3}\right ) + x^{2} \left (18 a^{4} b^{2} d^{3} - 54 a^{3} b^{3} c d^{2} + 54 a^{2} b^{4} c^{2} d - 18 a b^{5} c^{3}\right ) + x \left (18 a^{5} b d^{3} - 54 a^{4} b^{2} c d^{2} + 54 a^{3} b^{3} c^{2} d - 18 a^{2} b^{4} c^{3}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**3*log(x + (-a**5*d**8/(a*d - b*c)**4 + 5*a**4*b*c*d**7/(a*d - b*c)**4 - 10*a**3*b**2*c**2*d**6/(a*d - b*c)*
*4 + 10*a**2*b**3*c**3*d**5/(a*d - b*c)**4 - 5*a*b**4*c**4*d**4/(a*d - b*c)**4 + a*d**4 + b**5*c**5*d**3/(a*d
- b*c)**4 + b*c*d**3)/(2*b*d**4))/(a*d - b*c)**4 - d**3*log(x + (a**5*d**8/(a*d - b*c)**4 - 5*a**4*b*c*d**7/(a
*d - b*c)**4 + 10*a**3*b**2*c**2*d**6/(a*d - b*c)**4 - 10*a**2*b**3*c**3*d**5/(a*d - b*c)**4 + 5*a*b**4*c**4*d
**4/(a*d - b*c)**4 + a*d**4 - b**5*c**5*d**3/(a*d - b*c)**4 + b*c*d**3)/(2*b*d**4))/(a*d - b*c)**4 + (11*a**2*
d**2 - 7*a*b*c*d + 2*b**2*c**2 + 6*b**2*d**2*x**2 + x*(15*a*b*d**2 - 3*b**2*c*d))/(6*a**6*d**3 - 18*a**5*b*c*d
**2 + 18*a**4*b**2*c**2*d - 6*a**3*b**3*c**3 + x**3*(6*a**3*b**3*d**3 - 18*a**2*b**4*c*d**2 + 18*a*b**5*c**2*d
- 6*b**6*c**3) + x**2*(18*a**4*b**2*d**3 - 54*a**3*b**3*c*d**2 + 54*a**2*b**4*c**2*d - 18*a*b**5*c**3) + x*(1
8*a**5*b*d**3 - 54*a**4*b**2*c*d**2 + 54*a**3*b**3*c**2*d - 18*a**2*b**4*c**3))

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Giac [B]  time = 1.2146, size = 328, normalized size = 3.07 \begin{align*} -\frac{b d^{3} \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{d^{4} \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{2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, 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 - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x}{6 \,{\left (b c - a d\right )}^{4}{\left (b x + a\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*d^3*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) + d^4*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) - 1/6*(2*b^3*c^3 -
9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 11*a^3*d^3 + 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 3*(b^3*c^2*d - 6*a*b^2*c*d^2 + 5
*a^2*b*d^3)*x)/((b*c - a*d)^4*(b*x + a)^3)