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

Optimal. Leaf size=25 \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]

[Out]

(2*(b*x + c*x^2)^(3/2))/(3*c*x^(3/2))

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Rubi [A]  time = 0.0263756, antiderivative size = 25, 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 \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*x + c*x^2)^(3/2))/(3*c*x^(3/2))

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Rubi in Sympy [A]  time = 3.61731, size = 20, normalized size = 0.8 \[ \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 c x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0191625, size = 23, normalized size = 0.92 \[ \frac{2 (x (b+c x))^{3/2}}{3 c x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(3/2))/(3*c*x^(3/2))

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Maple [A]  time = 0.006, size = 25, normalized size = 1. \[{\frac{2\,cx+2\,b}{3\,c}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.717153, size = 16, normalized size = 0.64 \[ \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(c*x + b)^(3/2)/c

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Fricas [A]  time = 0.222882, size = 53, normalized size = 2.12 \[ \frac{2 \,{\left (c^{2} x^{3} + 2 \, b c x^{2} + b^{2} x\right )}}{3 \, \sqrt{c x^{2} + b x} c \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208015, size = 28, normalized size = 1.12 \[ \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}}}{3 \, c} - \frac{2 \, b^{\frac{3}{2}}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(c*x + b)^(3/2)/c - 2/3*b^(3/2)/c