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

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 = 0.0248089, 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 \[ \frac{2 x}{b \sqrt{b x+c x^2}} \]

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])

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Rubi in Sympy [A]  time = 3.24221, size = 15, normalized size = 0.79 \[ \frac{2 x}{b \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(c*x**2+b*x)**(3/2),x)

[Out]

2*x/(b*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.0153441, 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 = 0.006, size = 25, normalized size = 1.3 \[ 2\,{\frac{{x}^{2} \left ( cx+b \right ) }{b \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]

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 = 0.707546, size = 23, normalized size = 1.21 \[ \frac{2 \, x}{\sqrt{c x^{2} + b x} b} \]

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 = 0.217289, size = 23, normalized size = 1.21 \[ \frac{2 \, x}{\sqrt{c x^{2} + b x} b} \]

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*x/(sqrt(c*x^2 + b*x)*b)

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

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/XCAS [A]  time = 0.220794, size = 43, normalized size = 2.26 \[ \frac{2}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b\right )} \sqrt{c}} \]

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))