### 3.55 $$\int \frac{x}{(b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=19 $\frac{2 x}{b \sqrt{b x+c x^2}}$

[Out]

(2*x)/(b*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.0053672, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {636} $\frac{2 x}{b \sqrt{b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/(b*x + c*x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.006217, size = 17, normalized size = 0.89 $\frac{2 x}{b \sqrt{x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/(b*x + c*x^2)^(3/2),x]

[Out]

(2*x)/(b*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.047, size = 25, normalized size = 1.3 \begin{align*} 2\,{\frac{{x}^{2} \left ( cx+b \right ) }{b \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^2*(c*x+b)/b/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 1.09251, size = 23, normalized size = 1.21 \begin{align*} \frac{2 \, x}{\sqrt{c x^{2} + b x} b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/(sqrt(c*x^2 + b*x)*b)

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Fricas [A]  time = 1.96491, size = 47, normalized size = 2.47 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}}{b c x + b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(c*x^2 + b*x)/(b*c*x + b^2)

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

[Out]

Integral(x/(x*(b + c*x))**(3/2), x)

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Giac [A]  time = 1.20963, size = 43, normalized size = 2.26 \begin{align*} \frac{2}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b\right )} \sqrt{c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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