### 3.325 $$\int \frac{1}{(b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=24 $-\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0036398, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {613} $-\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^(-3/2),x]

[Out]

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

Rule 613

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

Rubi steps

\begin{align*} \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.006469, size = 22, normalized size = 0.92 $-\frac{2 (b+2 c x)}{b^2 \sqrt{x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^(-3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 29, normalized size = 1.2 \begin{align*} -2\,{\frac{x \left ( cx+b \right ) \left ( 2\,cx+b \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(1/(c*x^2+b*x)^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.12876, size = 47, normalized size = 1.96 \begin{align*} -\frac{4 \, c x}{\sqrt{c x^{2} + b x} b^{2}} - \frac{2}{\sqrt{c x^{2} + b x} b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.81908, size = 73, normalized size = 3.04 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x + b\right )}}{b^{2} c x^{2} + b^{3} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+b*x)**(3/2),x)

[Out]

Integral((b*x + c*x**2)**(-3/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.44096, size = 32, normalized size = 1.33 \begin{align*} -\frac{2 \,{\left (\frac{2 \, c x}{b^{2}} + \frac{1}{b}\right )}}{\sqrt{c x^{2} + b x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

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