3.31 \(\int x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx\)

Optimal. Leaf size=167 \[ \frac{3 a b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

[Out]

(a^3*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (3*a^2*b*x^5*Sqrt[a^
2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (3*a*b^2*x^8*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])/(8*(a + b*x^3)) + (b^3*x^11*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*(a +
b*x^3))

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Rubi [A]  time = 0.105138, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{3 a b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]

[Out]

(a^3*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*(a + b*x^3)) + (3*a^2*b*x^5*Sqrt[a^
2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (3*a*b^2*x^8*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])/(8*(a + b*x^3)) + (b^3*x^11*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*(a +
b*x^3))

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Rubi in Sympy [A]  time = 15.5806, size = 136, normalized size = 0.81 \[ \frac{81 a^{3} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{440 \left (a + b x^{3}\right )} + \frac{27 a^{2} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{220} + \frac{9 a x^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88} + \frac{x^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

81*a**3*x**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(440*(a + b*x**3)) + 27*a**2*x*
*2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/220 + 9*a*x**2*(a + b*x**3)*sqrt(a**2 + 2
*a*b*x**3 + b**2*x**6)/88 + x**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/11

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Mathematica [A]  time = 0.0288461, size = 61, normalized size = 0.37 \[ \frac{x^2 \sqrt{\left (a+b x^3\right )^2} \left (220 a^3+264 a^2 b x^3+165 a b^2 x^6+40 b^3 x^9\right )}{440 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]

[Out]

(x^2*Sqrt[(a + b*x^3)^2]*(220*a^3 + 264*a^2*b*x^3 + 165*a*b^2*x^6 + 40*b^3*x^9))
/(440*(a + b*x^3))

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Maple [A]  time = 0.007, size = 58, normalized size = 0.4 \[{\frac{{x}^{2} \left ( 40\,{b}^{3}{x}^{9}+165\,a{b}^{2}{x}^{6}+264\,{a}^{2}b{x}^{3}+220\,{a}^{3} \right ) }{440\, \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

1/440*x^2*(40*b^3*x^9+165*a*b^2*x^6+264*a^2*b*x^3+220*a^3)*((b*x^3+a)^2)^(3/2)/(
b*x^3+a)^3

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Maxima [A]  time = 0.797611, size = 47, normalized size = 0.28 \[ \frac{1}{11} \, b^{3} x^{11} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x,x, algorithm="maxima")

[Out]

1/11*b^3*x^11 + 3/8*a*b^2*x^8 + 3/5*a^2*b*x^5 + 1/2*a^3*x^2

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Fricas [A]  time = 0.249562, size = 47, normalized size = 0.28 \[ \frac{1}{11} \, b^{3} x^{11} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x,x, algorithm="fricas")

[Out]

1/11*b^3*x^11 + 3/8*a*b^2*x^8 + 3/5*a^2*b*x^5 + 1/2*a^3*x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.296884, size = 90, normalized size = 0.54 \[ \frac{1}{11} \, b^{3} x^{11}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{8} \, a b^{2} x^{8}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{5} \, a^{2} b x^{5}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{2} \, a^{3} x^{2}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x,x, algorithm="giac")

[Out]

1/11*b^3*x^11*sign(b*x^3 + a) + 3/8*a*b^2*x^8*sign(b*x^3 + a) + 3/5*a^2*b*x^5*si
gn(b*x^3 + a) + 1/2*a^3*x^2*sign(b*x^3 + a)