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

Optimal. Leaf size=25 \[ \frac{2 \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}} \]

[Out]

(2*(b*x + c*x^2)^(5/2))/(5*c*x^(5/2))

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Rubi [A]  time = 0.0287649, 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 \[ \frac{2 \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*x + c*x^2)^(5/2))/(5*c*x^(5/2))

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Rubi in Sympy [A]  time = 3.57468, size = 20, normalized size = 0.8 \[ \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 c x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b*x + c*x**2)**(5/2)/(5*c*x**(5/2))

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Mathematica [A]  time = 0.0235751, size = 23, normalized size = 0.92 \[ \frac{2 (x (b+c x))^{5/2}}{5 c x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(5/2))/(5*c*x^(5/2))

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Maple [A]  time = 0.004, size = 25, normalized size = 1. \[{\frac{2\,cx+2\,b}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(c*x+b)*(c*x^2+b*x)^(3/2)/c/x^(3/2)

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Maxima [A]  time = 0.708193, size = 66, normalized size = 2.64 \[ \frac{2 \,{\left (5 \, b c x^{2} + 5 \, b^{2} x +{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} x\right )} \sqrt{c x + b}}{15 \, c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(5*b*c*x^2 + 5*b^2*x + (3*c^2*x^2 + b*c*x - 2*b^2)*x)*sqrt(c*x + b)/(c*x)

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Fricas [A]  time = 0.226789, size = 68, normalized size = 2.72 \[ \frac{2 \,{\left (c^{3} x^{4} + 3 \, b c^{2} x^{3} + 3 \, b^{2} c x^{2} + b^{3} x\right )}}{5 \, \sqrt{c x^{2} + b x} c \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(c^3*x^4 + 3*b*c^2*x^3 + 3*b^2*c*x^2 + b^3*x)/(sqrt(c*x^2 + b*x)*c*sqrt(x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212616, size = 81, normalized size = 3.24 \[ \frac{2}{15} \, c{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} + \frac{2}{3} \, b{\left (\frac{{\left (c x + b\right )}^{\frac{3}{2}}}{c} - \frac{b^{\frac{3}{2}}}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*c*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x + b)^(3/2)*b)/c^2) + 2/3*b*(
(c*x + b)^(3/2)/c - b^(3/2)/c)