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

Optimal. Leaf size=25 \[ \frac{2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5} \]

[Out]

(2*(a*x^2 + b*x^3)^(5/2))/(5*b*x^5)

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Rubi [A]  time = 0.0671833, 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 \[ \frac{2 \left (a x^2+b x^3\right )^{5/2}}{5 b x^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*x^2 + b*x^3)^(5/2))/(5*b*x^5)

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Rubi in Sympy [A]  time = 7.45821, size = 20, normalized size = 0.8 \[ \frac{2 \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{5 b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*x**2 + b*x**3)**(5/2)/(5*b*x**5)

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Mathematica [A]  time = 0.0215035, size = 23, normalized size = 0.92 \[ \frac{2 \left (x^2 (a+b x)\right )^{5/2}}{5 b x^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x^2*(a + b*x))^(5/2))/(5*b*x^5)

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Maple [A]  time = 0.003, size = 27, normalized size = 1.1 \[{\frac{2\,bx+2\,a}{5\,b{x}^{3}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^(3/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.51756, size = 38, normalized size = 1.52 \[ \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)/b

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Fricas [A]  time = 0.22818, size = 50, normalized size = 2. \[ \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x^{3} + a x^{2}}}{5 \, b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221377, size = 70, normalized size = 2.8 \[ -\frac{2 \, a^{\frac{5}{2}}{\rm sign}\left (x\right )}{5 \, b} + \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{3}{2}} a{\rm sign}\left (x\right ) +{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )}{\rm sign}\left (x\right )\right )}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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