### 3.1826 $$\int \frac{1}{(a c+(b c+a d) x+b d x^2)^3} \, dx$$

Optimal. Leaf size=143 $\frac{6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x)}{(b c-a d)^5}+\frac{3 b d (a d+b c+2 b d x)}{(b c-a d)^4 \left (x (a d+b c)+a c+b d x^2\right )}-\frac{a d+b c+2 b d x}{2 (b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )^2}$

[Out]

-(b*c + a*d + 2*b*d*x)/(2*(b*c - a*d)^2*(a*c + (b*c + a*d)*x + b*d*x^2)^2) + (3*b*d*(b*c + a*d + 2*b*d*x))/((b
*c - a*d)^4*(a*c + (b*c + a*d)*x + b*d*x^2)) + (6*b^2*d^2*Log[a + b*x])/(b*c - a*d)^5 - (6*b^2*d^2*Log[c + d*x
])/(b*c - a*d)^5

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Rubi [A]  time = 0.0486094, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {614, 616, 31} $\frac{6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x)}{(b c-a d)^5}+\frac{3 b d (a d+b c+2 b d x)}{(b c-a d)^4 \left (x (a d+b c)+a c+b d x^2\right )}-\frac{a d+b c+2 b d x}{2 (b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b*c + a*d + 2*b*d*x)/(2*(b*c - a*d)^2*(a*c + (b*c + a*d)*x + b*d*x^2)^2) + (3*b*d*(b*c + a*d + 2*b*d*x))/((b
*c - a*d)^4*(a*c + (b*c + a*d)*x + b*d*x^2)) + (6*b^2*d^2*Log[a + b*x])/(b*c - a*d)^5 - (6*b^2*d^2*Log[c + d*x
])/(b*c - a*d)^5

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=-\frac{b c+a d+2 b d x}{2 (b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )^2}-\frac{(3 b d) \int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx}{(b c-a d)^2}\\ &=-\frac{b c+a d+2 b d x}{2 (b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )^2}+\frac{3 b d (b c+a d+2 b d x)}{(b c-a d)^4 \left (a c+(b c+a d) x+b d x^2\right )}+\frac{\left (6 b^2 d^2\right ) \int \frac{1}{a c+(b c+a d) x+b d x^2} \, dx}{(b c-a d)^4}\\ &=-\frac{b c+a d+2 b d x}{2 (b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )^2}+\frac{3 b d (b c+a d+2 b d x)}{(b c-a d)^4 \left (a c+(b c+a d) x+b d x^2\right )}-\frac{\left (6 b^3 d^3\right ) \int \frac{1}{b c+b d x} \, dx}{(b c-a d)^5}+\frac{\left (6 b^3 d^3\right ) \int \frac{1}{a d+b d x} \, dx}{(b c-a d)^5}\\ &=-\frac{b c+a d+2 b d x}{2 (b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )^2}+\frac{3 b d (b c+a d+2 b d x)}{(b c-a d)^4 \left (a c+(b c+a d) x+b d x^2\right )}+\frac{6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x)}{(b c-a d)^5}\\ \end{align*}

Mathematica [A]  time = 0.109183, size = 128, normalized size = 0.9 $\frac{\frac{6 b^2 d (b c-a d)}{a+b x}-\frac{b^2 (b c-a d)^2}{(a+b x)^2}+12 b^2 d^2 \log (a+b x)+\frac{6 b d^2 (b c-a d)}{c+d x}+\frac{d^2 (b c-a d)^2}{(c+d x)^2}-12 b^2 d^2 \log (c+d x)}{2 (b c-a d)^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-((b^2*(b*c - a*d)^2)/(a + b*x)^2) + (6*b^2*d*(b*c - a*d))/(a + b*x) + (d^2*(b*c - a*d)^2)/(c + d*x)^2 + (6*b
*d^2*(b*c - a*d))/(c + d*x) + 12*b^2*d^2*Log[a + b*x] - 12*b^2*d^2*Log[c + d*x])/(2*(b*c - a*d)^5)

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Maple [A]  time = 0.053, size = 140, normalized size = 1. \begin{align*} -{\frac{{d}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}+6\,{\frac{{b}^{2}{d}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{5}}}+3\,{\frac{b{d}^{2}}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+{\frac{{b}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{b}^{2}{d}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{5}}}+3\,{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.15268, size = 802, normalized size = 5.61 \begin{align*} \frac{6 \, b^{2} d^{2} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{6 \, b^{2} d^{2} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac{12 \, b^{3} d^{3} x^{3} - b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} - a^{3} d^{3} + 18 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d + 7 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*b^2*d^2*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^
5*d^5) - 6*b^2*d^2*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c
*d^4 - a^5*d^5) + 1/2*(12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*
b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b*d^3)*x)/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2
- 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*
b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*
b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c
^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)

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Fricas [B]  time = 1.75451, size = 1488, normalized size = 10.41 \begin{align*} -\frac{b^{4} c^{4} - 8 \, a b^{3} c^{3} d + 8 \, a^{3} b c d^{3} - a^{4} d^{4} - 12 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} - 18 \,{\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (b^{4} c^{3} d + 6 \, a b^{3} c^{2} d^{2} - 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x - 12 \,{\left (b^{4} d^{4} x^{4} + a^{2} b^{2} c^{2} d^{2} + 2 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} +{\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} d^{4} x^{4} + a^{2} b^{2} c^{2} d^{2} + 2 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} +{\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} +{\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \,{\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} +{\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} x^{2} + 2 \,{\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^4*c^4 - 8*a*b^3*c^3*d + 8*a^3*b*c*d^3 - a^4*d^4 - 12*(b^4*c*d^3 - a*b^3*d^4)*x^3 - 18*(b^4*c^2*d^2 - a
^2*b^2*d^4)*x^2 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 - 6*a^2*b^2*c*d^3 - a^3*b*d^4)*x - 12*(b^4*d^4*x^4 + a^2*b^2*
c^2*d^2 + 2*(b^4*c*d^3 + a*b^3*d^4)*x^3 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 2*(a*b^3*c^2*d^2 +
a^2*b^2*c*d^3)*x)*log(b*x + a) + 12*(b^4*d^4*x^4 + a^2*b^2*c^2*d^2 + 2*(b^4*c*d^3 + a*b^3*d^4)*x^3 + (b^4*c^2
*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 2*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*x)*log(d*x + c))/(a^2*b^5*c^7 - 5*
a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5 + (b^7*c^5*d^2 - 5*a*b
^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^4 + 2*(b^7*c^6*d - 4*a
*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 - a^6*b*d^7)*x^3 + (b^7*c^7 - a*b^6*c^6
*d - 9*a^2*b^5*c^5*d^2 + 25*a^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^7*d^7)*
x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*x)

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Sympy [B]  time = 3.79196, size = 881, normalized size = 6.16 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*b**2*d**2*log(x + (-6*a**6*b**2*d**8/(a*d - b*c)**5 + 36*a**5*b**3*c*d**7/(a*d - b*c)**5 - 90*a**4*b**4*c**2
*d**6/(a*d - b*c)**5 + 120*a**3*b**5*c**3*d**5/(a*d - b*c)**5 - 90*a**2*b**6*c**4*d**4/(a*d - b*c)**5 + 36*a*b
**7*c**5*d**3/(a*d - b*c)**5 + 6*a*b**2*d**3 - 6*b**8*c**6*d**2/(a*d - b*c)**5 + 6*b**3*c*d**2)/(12*b**3*d**3)
)/(a*d - b*c)**5 - 6*b**2*d**2*log(x + (6*a**6*b**2*d**8/(a*d - b*c)**5 - 36*a**5*b**3*c*d**7/(a*d - b*c)**5 +
90*a**4*b**4*c**2*d**6/(a*d - b*c)**5 - 120*a**3*b**5*c**3*d**5/(a*d - b*c)**5 + 90*a**2*b**6*c**4*d**4/(a*d
- b*c)**5 - 36*a*b**7*c**5*d**3/(a*d - b*c)**5 + 6*a*b**2*d**3 + 6*b**8*c**6*d**2/(a*d - b*c)**5 + 6*b**3*c*d*
*2)/(12*b**3*d**3))/(a*d - b*c)**5 + (-a**3*d**3 + 7*a**2*b*c*d**2 + 7*a*b**2*c**2*d - b**3*c**3 + 12*b**3*d**
3*x**3 + x**2*(18*a*b**2*d**3 + 18*b**3*c*d**2) + x*(4*a**2*b*d**3 + 28*a*b**2*c*d**2 + 4*b**3*c**2*d))/(2*a**
6*c**2*d**4 - 8*a**5*b*c**3*d**3 + 12*a**4*b**2*c**4*d**2 - 8*a**3*b**3*c**5*d + 2*a**2*b**4*c**6 + x**4*(2*a*
*4*b**2*d**6 - 8*a**3*b**3*c*d**5 + 12*a**2*b**4*c**2*d**4 - 8*a*b**5*c**3*d**3 + 2*b**6*c**4*d**2) + x**3*(4*
a**5*b*d**6 - 12*a**4*b**2*c*d**5 + 8*a**3*b**3*c**2*d**4 + 8*a**2*b**4*c**3*d**3 - 12*a*b**5*c**4*d**2 + 4*b*
*6*c**5*d) + x**2*(2*a**6*d**6 - 18*a**4*b**2*c**2*d**4 + 32*a**3*b**3*c**3*d**3 - 18*a**2*b**4*c**4*d**2 + 2*
b**6*c**6) + x*(4*a**6*c*d**5 - 12*a**5*b*c**2*d**4 + 8*a**4*b**2*c**3*d**3 + 8*a**3*b**3*c**4*d**2 - 12*a**2*
b**4*c**5*d + 4*a*b**5*c**6))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError