### 3.1239 $$\int \frac{1}{(b d+2 c d x)^2 \sqrt{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=37 $\frac{2 \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}$

[Out]

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

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Rubi [A]  time = 0.0151481, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.038, Rules used = {682} $\frac{2 \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 682

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

Rubi steps

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

Mathematica [A]  time = 0.0148778, size = 36, normalized size = 0.97 $\frac{2 \sqrt{a+x (b+c x)}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 38, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{c{x}^{2}+bx+a}}{ \left ( 2\,cx+b \right ){d}^{2} \left ( 4\,ac-{b}^{2} \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)^(1/2),x)

[Out]

-2*(c*x^2+b*x+a)^(1/2)/(2*c*x+b)/d^2/(4*a*c-b^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(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.04856, size = 101, normalized size = 2.73 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x + a}}{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} - 4 \, a b 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)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{b^{2} \sqrt{a + b x + c x^{2}} + 4 b c x \sqrt{a + b x + c x^{2}} + 4 c^{2} x^{2} \sqrt{a + b x + c x^{2}}}\, dx}{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)**(1/2),x)

[Out]

Integral(1/(b**2*sqrt(a + b*x + c*x**2) + 4*b*c*x*sqrt(a + b*x + c*x**2) + 4*c**2*x**2*sqrt(a + b*x + c*x**2))
, x)/d**2

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Giac [B]  time = 1.17538, size = 194, normalized size = 5.24 \begin{align*} -\frac{\sqrt{c} \mathrm{sgn}\left (\frac{1}{2 \, c d x + b d}\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (d\right )}{b^{2} c d^{2} - 4 \, a c^{2} d^{2}} + \frac{\sqrt{-\frac{b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} c^{2}}{b^{2} c^{3} d^{2} \mathrm{sgn}\left (\frac{1}{2 \, c d x + b d}\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (d\right ) - 4 \, a c^{4} d^{2} \mathrm{sgn}\left (\frac{1}{2 \, c d x + b d}\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (d\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)^(1/2),x, algorithm="giac")

[Out]

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