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

Optimal. Leaf size=23 \[ -\frac{2 \sqrt{x}}{c \sqrt{b x+c x^2}} \]

[Out]

(-2*Sqrt[x])/(c*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.0281259, antiderivative size = 23, 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 \[ -\frac{2 \sqrt{x}}{c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x])/(c*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 3.55875, size = 20, normalized size = 0.87 \[ - \frac{2 \sqrt{x}}{c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(x)/(c*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.0124512, size = 21, normalized size = 0.91 \[ -\frac{2 \sqrt{x}}{c \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x])/(c*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.004, size = 25, normalized size = 1.1 \[ -2\,{\frac{ \left ( cx+b \right ){x}^{3/2}}{c \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.702482, size = 16, normalized size = 0.7 \[ -\frac{2}{\sqrt{c x + b} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sqrt(c*x + b)*c)

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Fricas [A]  time = 0.222603, size = 41, normalized size = 1.78 \[ -\frac{2 \, \sqrt{c x^{2} + b x} \sqrt{x}}{c^{2} x^{2} + b c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209788, size = 28, normalized size = 1.22 \[ -\frac{2}{\sqrt{c x + b} c} + \frac{2}{\sqrt{b} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sqrt(c*x + b)*c) + 2/(sqrt(b)*c)