### 3.324 $$\int \frac{d+e x}{(b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=33 $-\frac{2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}$

[Out]

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

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Rubi [A]  time = 0.0103688, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {636} $-\frac{2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[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 (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0118774, size = 30, normalized size = 0.91 $\frac{2 (-b d+b e x-2 c d x)}{b^2 \sqrt{x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.047, size = 37, normalized size = 1.1 \begin{align*} -2\,{\frac{x \left ( cx+b \right ) \left ( -bxe+2\,cdx+bd \right ) }{{b}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*(c*x+b)*(-b*e*x+2*c*d*x+b*d)/b^2/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 1.05328, size = 74, normalized size = 2.24 \begin{align*} -\frac{4 \, c d x}{\sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, e x}{\sqrt{c x^{2} + b x} b} - \frac{2 \, d}{\sqrt{c x^{2} + b x} b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.96475, size = 89, normalized size = 2.7 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{b^{2} c x^{2} + b^{3} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.32997, size = 46, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (\frac{d}{b} + \frac{{\left (2 \, c d - b e\right )} x}{b^{2}}\right )}}{\sqrt{c x^{2} + b x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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