### 3.1814 $$\int \frac{(a+b x)^2}{(a c+(b c+a d) x+b d x^2)^2} \, dx$$

Optimal. Leaf size=12 $-\frac{1}{d (c+d x)}$

[Out]

-(1/(d*(c + d*x)))

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Rubi [A]  time = 0.0081565, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 32} $-\frac{1}{d (c+d x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(1/(d*(c + d*x)))

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0029981, size = 12, normalized size = 1. $-\frac{1}{d (c+d x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(1/(d*(c + d*x)))

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Maple [A]  time = 0.037, size = 13, normalized size = 1.1 \begin{align*} -{\frac{1}{d \left ( dx+c \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/(d*x+c)

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Maxima [A]  time = 1.02133, size = 18, normalized size = 1.5 \begin{align*} -\frac{1}{d^{2} x + c d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(d^2*x + c*d)

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Fricas [A]  time = 1.51142, size = 24, normalized size = 2. \begin{align*} -\frac{1}{d^{2} x + c d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(d^2*x + c*d)

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Sympy [A]  time = 0.325532, size = 10, normalized size = 0.83 \begin{align*} - \frac{1}{c d + d^{2} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(c*d + d**2*x)

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Giac [A]  time = 1.17501, size = 16, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (d x + c\right )} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((d*x + c)*d)