### 3.1163 $$\int \frac{1}{(b d+2 c d x)^2 (a+b x+c x^2)} \, dx$$

Optimal. Leaf size=61 $\frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.040675, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {693, 618, 206} $\frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 693

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

Rule 618

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0507395, size = 63, normalized size = 1.03 $\frac{\frac{2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}}{d^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.152, size = 64, normalized size = 1.1 \begin{align*} -2\,{\frac{1}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{1}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.11649, size = 564, normalized size = 9.25 \begin{align*} \left [-\frac{\sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 2 \, b^{2} + 8 \, a c}{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2} x +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2}}, -\frac{2 \,{\left (\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - b^{2} + 4 \, a c\right )}}{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2} x +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(sqrt(b^2 - 4*a*c)*(2*c*x + b)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^
2 + b*x + a)) - 2*b^2 + 8*a*c)/(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2*x + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*
d^2), -2*(sqrt(-b^2 + 4*a*c)*(2*c*x + b)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - b^2 + 4*a*c)/
(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2*x + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2)]

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Sympy [B]  time = 1.21653, size = 240, normalized size = 3.93 \begin{align*} - \frac{2}{4 a b c d^{2} - b^{3} d^{2} + x \left (8 a c^{2} d^{2} - 2 b^{2} c d^{2}\right )} + \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b}{2 c} \right )}}{d^{2}} - \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b}{2 c} \right )}}{d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(4*a*b*c*d**2 - b**3*d**2 + x*(8*a*c**2*d**2 - 2*b**2*c*d**2)) + sqrt(-1/(4*a*c - b**2)**3)*log(x + (-16*a*
*2*c**2*sqrt(-1/(4*a*c - b**2)**3) + 8*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3) - b**4*sqrt(-1/(4*a*c - b**2)**3) +
b)/(2*c))/d**2 - sqrt(-1/(4*a*c - b**2)**3)*log(x + (16*a**2*c**2*sqrt(-1/(4*a*c - b**2)**3) - 8*a*b**2*c*sqr
t(-1/(4*a*c - b**2)**3) + b**4*sqrt(-1/(4*a*c - b**2)**3) + b)/(2*c))/d**2

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Giac [B]  time = 1.17961, size = 158, normalized size = 2.59 \begin{align*} \frac{2 \, c^{2} d^{3}}{{\left (b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4}\right )}{\left (2 \, c d x + b d\right )}} - \frac{2 \, \arctan \left (-\frac{\frac{b^{2} d}{2 \, c d x + b d} - \frac{4 \, a c d}{2 \, c d x + b d}}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c^2*d^3/((b^2*c^2*d^4 - 4*a*c^3*d^4)*(2*c*d*x + b*d)) - 2*arctan(-(b^2*d/(2*c*d*x + b*d) - 4*a*c*d/(2*c*d*x
+ b*d))/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(-b^2 + 4*a*c)*d^2)