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

Optimal. Leaf size=52 \[ \frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}} \]

[Out]

(-4*b*Sqrt[b*x + c*x^2])/(3*c^2*Sqrt[x]) + (2*Sqrt[x]*Sqrt[b*x + c*x^2])/(3*c)

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Rubi [A]  time = 0.0590356, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*b*Sqrt[b*x + c*x^2])/(3*c^2*Sqrt[x]) + (2*Sqrt[x]*Sqrt[b*x + c*x^2])/(3*c)

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Rubi in Sympy [A]  time = 6.18155, size = 46, normalized size = 0.88 \[ - \frac{4 b \sqrt{b x + c x^{2}}}{3 c^{2} \sqrt{x}} + \frac{2 \sqrt{x} \sqrt{b x + c x^{2}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(c*x**2+b*x)**(1/2),x)

[Out]

-4*b*sqrt(b*x + c*x**2)/(3*c**2*sqrt(x)) + 2*sqrt(x)*sqrt(b*x + c*x**2)/(3*c)

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Mathematica [A]  time = 0.0247225, size = 30, normalized size = 0.58 \[ \frac{2 (c x-2 b) \sqrt{x (b+c x)}}{3 c^2 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 33, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -cx+2\,b \right ) }{3\,{c}^{2}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(c*x^2+b*x)^(1/2),x)

[Out]

-2/3*(c*x+b)*(-c*x+2*b)*x^(1/2)/c^2/(c*x^2+b*x)^(1/2)

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Maxima [A]  time = 0.711942, size = 41, normalized size = 0.79 \[ \frac{2 \,{\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )}}{3 \, \sqrt{c x + b} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218899, size = 54, normalized size = 1.04 \[ \frac{2 \,{\left (c^{2} x^{3} - b c x^{2} - 2 \, b^{2} x\right )}}{3 \, \sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211046, size = 43, normalized size = 0.83 \[ \frac{4 \, b^{\frac{3}{2}}}{3 \, c^{2}} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x + b} b\right )}}{3 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*b^(3/2)/c^2 + 2/3*((c*x + b)^(3/2) - 3*sqrt(c*x + b)*b)/c^2