3.1172 \(\int \frac{(b d+2 c d x)^2}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=62 \[ -\frac{4 c d^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{d^2 (b+2 c x)}{a+b x+c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0359027, antiderivative size = 62, 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 = {686, 618, 206} \[ -\frac{4 c d^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{d^2 (b+2 c x)}{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 686

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

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

Mathematica [A]  time = 0.0362394, size = 65, normalized size = 1.05 \[ d^2 \left (\frac{4 c \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{-b-2 c x}{a+b x+c x^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.152, size = 77, normalized size = 1.2 \begin{align*} -2\,{\frac{c{d}^{2}x}{c{x}^{2}+bx+a}}-{\frac{{d}^{2}b}{c{x}^{2}+bx+a}}+4\,{\frac{c{d}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.07454, size = 667, normalized size = 10.76 \begin{align*} \left [-\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} - 4 \, a b c\right )} d^{2} - 2 \,{\left (c^{2} d^{2} x^{2} + b c d^{2} x + a c d^{2}\right )} \sqrt{b^{2} - 4 \, a c} \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 )}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x}, -\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} - 4 \, a b c\right )} d^{2} + 4 \,{\left (c^{2} d^{2} x^{2} + b c d^{2} x + a c d^{2}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(2*(b^2*c - 4*a*c^2)*d^2*x + (b^3 - 4*a*b*c)*d^2 - 2*(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2)*sqrt(b^2 - 4*a*c)*l
og((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)))/(a*b^2 - 4*a^2*c +
(b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x), -(2*(b^2*c - 4*a*c^2)*d^2*x + (b^3 - 4*a*b*c)*d^2 + 4*(c^2*d^2*x^2
+ b*c*d^2*x + a*c*d^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)))/(a*b^2 - 4*a
^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)]

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Sympy [B]  time = 1.0763, size = 209, normalized size = 3.37 \begin{align*} - 2 c d^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{- 8 a c^{2} d^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + 2 b^{2} c d^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + 2 b c d^{2}}{4 c^{2} d^{2}} \right )} + 2 c d^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{8 a c^{2} d^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} - 2 b^{2} c d^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + 2 b c d^{2}}{4 c^{2} d^{2}} \right )} - \frac{b d^{2} + 2 c d^{2} x}{a + b x + c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17002, size = 89, normalized size = 1.44 \begin{align*} \frac{4 \, c d^{2} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, c d^{2} x + b d^{2}}{c x^{2} + b x + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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