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

Optimal. Leaf size=105 \[ -\frac{32 a^3 \left (a x^2+b x^3\right )^{3/2}}{315 b^4 x^3}+\frac{16 a^2 \left (a x^2+b x^3\right )^{3/2}}{105 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{3/2}}{21 b^2 x}+\frac{2 \left (a x^2+b x^3\right )^{3/2}}{9 b} \]

[Out]

(2*(a*x^2 + b*x^3)^(3/2))/(9*b) - (32*a^3*(a*x^2 + b*x^3)^(3/2))/(315*b^4*x^3) +
(16*a^2*(a*x^2 + b*x^3)^(3/2))/(105*b^3*x^2) - (4*a*(a*x^2 + b*x^3)^(3/2))/(21*
b^2*x)

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Rubi [A]  time = 0.212287, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{32 a^3 \left (a x^2+b x^3\right )^{3/2}}{315 b^4 x^3}+\frac{16 a^2 \left (a x^2+b x^3\right )^{3/2}}{105 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{3/2}}{21 b^2 x}+\frac{2 \left (a x^2+b x^3\right )^{3/2}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*(a*x^2 + b*x^3)^(3/2))/(9*b) - (32*a^3*(a*x^2 + b*x^3)^(3/2))/(315*b^4*x^3) +
(16*a^2*(a*x^2 + b*x^3)^(3/2))/(105*b^3*x^2) - (4*a*(a*x^2 + b*x^3)^(3/2))/(21*
b^2*x)

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Rubi in Sympy [A]  time = 21.6329, size = 95, normalized size = 0.9 \[ - \frac{32 a^{3} \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{315 b^{4} x^{3}} + \frac{16 a^{2} \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{105 b^{3} x^{2}} - \frac{4 a \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{21 b^{2} x} + \frac{2 \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*a**3*(a*x**2 + b*x**3)**(3/2)/(315*b**4*x**3) + 16*a**2*(a*x**2 + b*x**3)**(
3/2)/(105*b**3*x**2) - 4*a*(a*x**2 + b*x**3)**(3/2)/(21*b**2*x) + 2*(a*x**2 + b*
x**3)**(3/2)/(9*b)

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Mathematica [A]  time = 0.0275515, size = 64, normalized size = 0.61 \[ \frac{2 \sqrt{x^2 (a+b x)} \left (-16 a^4+8 a^3 b x-6 a^2 b^2 x^2+5 a b^3 x^3+35 b^4 x^4\right )}{315 b^4 x} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x^2*(a + b*x)]*(-16*a^4 + 8*a^3*b*x - 6*a^2*b^2*x^2 + 5*a*b^3*x^3 + 35*b
^4*x^4))/(315*b^4*x)

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Maple [A]  time = 0.008, size = 57, normalized size = 0.5 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -35\,{x}^{3}{b}^{3}+30\,a{b}^{2}{x}^{2}-24\,{a}^{2}xb+16\,{a}^{3} \right ) }{315\,{b}^{4}x}\sqrt{b{x}^{3}+a{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^3+a*x^2)^(1/2),x)

[Out]

-2/315*(b*x+a)*(-35*b^3*x^3+30*a*b^2*x^2-24*a^2*b*x+16*a^3)*(b*x^3+a*x^2)^(1/2)/
b^4/x

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Maxima [A]  time = 1.43845, size = 72, normalized size = 0.69 \[ \frac{2 \,{\left (35 \, b^{4} x^{4} + 5 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 8 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt{b x + a}}{315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*b^4*x^4 + 5*a*b^3*x^3 - 6*a^2*b^2*x^2 + 8*a^3*b*x - 16*a^4)*sqrt(b*x +
a)/b^4

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Fricas [A]  time = 0.217481, size = 84, normalized size = 0.8 \[ \frac{2 \,{\left (35 \, b^{4} x^{4} + 5 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 8 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt{b x^{3} + a x^{2}}}{315 \, b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*b^4*x^4 + 5*a*b^3*x^3 - 6*a^2*b^2*x^2 + 8*a^3*b*x - 16*a^4)*sqrt(b*x^3
+ a*x^2)/(b^4*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*sqrt(x**2*(a + b*x)), x)

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GIAC/XCAS [A]  time = 0.221052, size = 100, normalized size = 0.95 \[ \frac{32 \, a^{\frac{9}{2}}{\rm sign}\left (x\right )}{315 \, b^{4}} + \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )}{\rm sign}\left (x\right )}{315 \, b^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/315*a^(9/2)*sign(x)/b^4 + 2/315*(35*(b*x + a)^(9/2)*b^24 - 135*(b*x + a)^(7/2
)*a*b^24 + 189*(b*x + a)^(5/2)*a^2*b^24 - 105*(b*x + a)^(3/2)*a^3*b^24)*sign(x)/
b^28