### 3.1237 $$\int \frac{b d+2 c d x}{\sqrt{a+b x+c x^2}} \, dx$$

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

[Out]

2*d*Sqrt[a + b*x + c*x^2]

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Rubi [A]  time = 0.0060926, 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} $2 d \sqrt{a+b x+c x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)/Sqrt[a + b*x + c*x^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}{\sqrt{a+b x+c x^2}} \, dx &=2 d \sqrt{a+b x+c x^2}\\ \end{align*}

Mathematica [A]  time = 0.005941, size = 16, normalized size = 0.94 $2 d \sqrt{a+x (b+c x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

2*d*Sqrt[a + x*(b + c*x)]

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Maple [A]  time = 0.041, size = 16, normalized size = 0.9 \begin{align*} 2\,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)^(1/2),x)

[Out]

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

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Maxima [A]  time = 2.27746, size = 20, normalized size = 1.18 \begin{align*} 2 \, \sqrt{c x^{2} + b x + a} d \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)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(c*x^2 + b*x + a)*d

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Fricas [A]  time = 2.63262, size = 36, normalized size = 2.12 \begin{align*} 2 \, \sqrt{c x^{2} + b x + a} d \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)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(c*x^2 + b*x + a)*d

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Sympy [A]  time = 0.194262, size = 15, normalized size = 0.88 \begin{align*} 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)**(1/2),x)

[Out]

2*d*sqrt(a + b*x + c*x**2)

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Giac [A]  time = 1.18096, size = 20, normalized size = 1.18 \begin{align*} 2 \, \sqrt{c x^{2} + b x + a} d \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)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(c*x^2 + b*x + a)*d