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

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 x}{b \sqrt {b x+c x^2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 x}{b \sqrt {x (b+c x)}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.26, size = 25, normalized size = 1.32 \begin {gather*} \frac {2 \sqrt {b x+c x^2}}{b (b+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/(b*x + c*x^2)^(3/2),x]

[Out]

(2*Sqrt[b*x + c*x^2])/(b*(b + c*x))

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fricas [A]  time = 0.40, size = 23, normalized size = 1.21 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x}}{b c x + b^{2}} \end {gather*}

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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giac [A]  time = 0.23, size = 32, normalized size = 1.68 \begin {gather*} \frac {2}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )} \sqrt {c}} \end {gather*}

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))

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maple [A]  time = 0.05, size = 25, normalized size = 1.32 \begin {gather*} \frac {2 \left (c x +b \right ) x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 \, x}{\sqrt {c x^{2} + b x} b} \end {gather*}

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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mupad [B]  time = 0.03, size = 15, normalized size = 0.79 \begin {gather*} \frac {2\,x}{b\,\sqrt {x\,\left (b+c\,x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x + c*x^2)^(3/2),x)

[Out]

(2*x)/(b*(x*(b + c*x))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

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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