### 3.2385 $$\int \frac{d+e x}{(a+b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=45 $-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}$

[Out]

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

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Rubi [A]  time = 0.0110652, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {636} $-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 636

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

Rubi steps

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

Mathematica [A]  time = 0.198597, size = 43, normalized size = 0.96 $\frac{4 a e-2 b d+2 b e x-4 c d x}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 45, normalized size = 1. \begin{align*} -2\,{\frac{bxe-2\,cdx+2\,ae-bd}{\sqrt{c{x}^{2}+bx+a} \left ( 4\,ac-{b}^{2} \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(c*x^2+b*x+a)^(1/2)*(b*e*x-2*c*d*x+2*a*e-b*d)/(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((e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.44906, size = 162, normalized size = 3.6 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((d + e*x)/(a + b*x + c*x**2)**(3/2), x)

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Giac [A]  time = 1.11703, size = 77, normalized size = 1.71 \begin{align*} -\frac{2 \,{\left (\frac{{\left (2 \, c d - b e\right )} x}{b^{2} - 4 \, a c} + \frac{b d - 2 \, a e}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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