### 3.1245 $$\int \frac{b d+2 c d x}{(a+b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=17 $-\frac{2 d}{\sqrt{a+b x+c x^2}}$

[Out]

(-2*d)/Sqrt[a + b*x + c*x^2]

________________________________________________________________________________________

Rubi [A]  time = 0.0061819, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {629} $-\frac{2 d}{\sqrt{a+b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*d)/Sqrt[a + b*x + c*x^2]

Rule 629

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

Rubi steps

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

Mathematica [A]  time = 0.0069606, size = 16, normalized size = 0.94 $-\frac{2 d}{\sqrt{a+x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*d)/Sqrt[a + x*(b + c*x)]

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 16, normalized size = 0.9 \begin{align*} -2\,{\frac{d}{\sqrt{c{x}^{2}+bx+a}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.1367, size = 20, normalized size = 1.18 \begin{align*} -\frac{2 \, d}{\sqrt{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)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-2*d/sqrt(c*x^2 + b*x + a)

________________________________________________________________________________________

Fricas [A]  time = 3.19535, size = 38, normalized size = 2.24 \begin{align*} -\frac{2 \, d}{\sqrt{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)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2*d/sqrt(c*x^2 + b*x + a)

________________________________________________________________________________________

Sympy [A]  time = 1.49275, size = 17, normalized size = 1. \begin{align*} - \frac{2 d}{\sqrt{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)/(c*x**2+b*x+a)**(3/2),x)

[Out]

-2*d/sqrt(a + b*x + c*x**2)

________________________________________________________________________________________

Giac [B]  time = 1.16329, size = 47, normalized size = 2.76 \begin{align*} -\frac{2 \,{\left (b^{2} d - 4 \, a c d\right )}}{\sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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