3.233 \(\int x \sqrt{a x^2+b x^3} \, dx\)

Optimal. Leaf size=80 \[ \frac{16 a^2 \left (a x^2+b x^3\right )^{3/2}}{105 b^3 x^3}-\frac{8 a \left (a x^2+b x^3\right )^{3/2}}{35 b^2 x^2}+\frac{2 \left (a x^2+b x^3\right )^{3/2}}{7 b x} \]

[Out]

(16*a^2*(a*x^2 + b*x^3)^(3/2))/(105*b^3*x^3) - (8*a*(a*x^2 + b*x^3)^(3/2))/(35*b
^2*x^2) + (2*(a*x^2 + b*x^3)^(3/2))/(7*b*x)

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Rubi [A]  time = 0.131132, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{16 a^2 \left (a x^2+b x^3\right )^{3/2}}{105 b^3 x^3}-\frac{8 a \left (a x^2+b x^3\right )^{3/2}}{35 b^2 x^2}+\frac{2 \left (a x^2+b x^3\right )^{3/2}}{7 b x} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[a*x^2 + b*x^3],x]

[Out]

(16*a^2*(a*x^2 + b*x^3)^(3/2))/(105*b^3*x^3) - (8*a*(a*x^2 + b*x^3)^(3/2))/(35*b
^2*x^2) + (2*(a*x^2 + b*x^3)^(3/2))/(7*b*x)

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Rubi in Sympy [A]  time = 13.9309, size = 71, normalized size = 0.89 \[ \frac{16 a^{2} \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{105 b^{3} x^{3}} - \frac{8 a \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{35 b^{2} x^{2}} + \frac{2 \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{7 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*a**2*(a*x**2 + b*x**3)**(3/2)/(105*b**3*x**3) - 8*a*(a*x**2 + b*x**3)**(3/2)/
(35*b**2*x**2) + 2*(a*x**2 + b*x**3)**(3/2)/(7*b*x)

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Mathematica [A]  time = 0.020636, size = 53, normalized size = 0.66 \[ \frac{2 \sqrt{x^2 (a+b x)} \left (8 a^3-4 a^2 b x+3 a b^2 x^2+15 b^3 x^3\right )}{105 b^3 x} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[x^2*(a + b*x)]*(8*a^3 - 4*a^2*b*x + 3*a*b^2*x^2 + 15*b^3*x^3))/(105*b^3*
x)

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Maple [A]  time = 0.007, size = 46, normalized size = 0.6 \[{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 15\,{b}^{2}{x}^{2}-12\,abx+8\,{a}^{2} \right ) }{105\,{b}^{3}x}\sqrt{b{x}^{3}+a{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a*x^2)^(1/2),x)

[Out]

2/105*(b*x+a)*(15*b^2*x^2-12*a*b*x+8*a^2)*(b*x^3+a*x^2)^(1/2)/b^3/x

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Maxima [A]  time = 1.39356, size = 57, normalized size = 0.71 \[ \frac{2 \,{\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{105 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*b^3*x^3 + 3*a*b^2*x^2 - 4*a^2*b*x + 8*a^3)*sqrt(b*x + a)/b^3

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Fricas [A]  time = 0.226787, size = 69, normalized size = 0.86 \[ \frac{2 \,{\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{105 \, b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*b^3*x^3 + 3*a*b^2*x^2 - 4*a^2*b*x + 8*a^3)*sqrt(b*x^3 + a*x^2)/(b^3*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(x**2*(a + b*x)), x)

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GIAC/XCAS [A]  time = 0.219522, size = 80, normalized size = 1. \[ -\frac{16 \, a^{\frac{7}{2}}{\rm sign}\left (x\right )}{105 \, b^{3}} + \frac{2 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )}{\rm sign}\left (x\right )}{105 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/105*a^(7/2)*sign(x)/b^3 + 2/105*(15*(b*x + a)^(7/2)*b^12 - 42*(b*x + a)^(5/2
)*a*b^12 + 35*(b*x + a)^(3/2)*a^2*b^12)*sign(x)/b^15