3.28 \(\int x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx\)

Optimal. Leaf size=167 \[ \frac{3 a b^2 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{3 a^2 b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{b^3 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{a^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]

[Out]

(a^3*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (3*a^2*b*x^8*Sqrt[a^
2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (3*a*b^2*x^11*Sqrt[a^2 + 2*a*b*x^3 +
 b^2*x^6])/(11*(a + b*x^3)) + (b^3*x^14*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*(a
+ b*x^3))

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Rubi [A]  time = 0.122196, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 a b^2 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{3 a^2 b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{b^3 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac{a^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]

[Out]

(a^3*x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*(a + b*x^3)) + (3*a^2*b*x^8*Sqrt[a^
2 + 2*a*b*x^3 + b^2*x^6])/(8*(a + b*x^3)) + (3*a*b^2*x^11*Sqrt[a^2 + 2*a*b*x^3 +
 b^2*x^6])/(11*(a + b*x^3)) + (b^3*x^14*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(14*(a
+ b*x^3))

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Rubi in Sympy [A]  time = 17.3258, size = 136, normalized size = 0.81 \[ \frac{81 a^{3} x^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3080 \left (a + b x^{3}\right )} + \frac{27 a^{2} x^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{616} + \frac{9 a x^{5} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{154} + \frac{x^{5} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

81*a**3*x**5*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(3080*(a + b*x**3)) + 27*a**2*x
**5*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/616 + 9*a*x**5*(a + b*x**3)*sqrt(a**2 +
2*a*b*x**3 + b**2*x**6)/154 + x**5*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/14

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Mathematica [A]  time = 0.0268024, size = 61, normalized size = 0.37 \[ \frac{x^5 \sqrt{\left (a+b x^3\right )^2} \left (616 a^3+1155 a^2 b x^3+840 a b^2 x^6+220 b^3 x^9\right )}{3080 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]

[Out]

(x^5*Sqrt[(a + b*x^3)^2]*(616*a^3 + 1155*a^2*b*x^3 + 840*a*b^2*x^6 + 220*b^3*x^9
))/(3080*(a + b*x^3))

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Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[{\frac{{x}^{5} \left ( 220\,{b}^{3}{x}^{9}+840\,a{b}^{2}{x}^{6}+1155\,{a}^{2}b{x}^{3}+616\,{a}^{3} \right ) }{3080\, \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

1/3080*x^5*(220*b^3*x^9+840*a*b^2*x^6+1155*a^2*b*x^3+616*a^3)*((b*x^3+a)^2)^(3/2
)/(b*x^3+a)^3

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Maxima [A]  time = 0.76952, size = 47, normalized size = 0.28 \[ \frac{1}{14} \, b^{3} x^{14} + \frac{3}{11} \, a b^{2} x^{11} + \frac{3}{8} \, a^{2} b x^{8} + \frac{1}{5} \, a^{3} x^{5} \]

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^4,x, algorithm="maxima")

[Out]

1/14*b^3*x^14 + 3/11*a*b^2*x^11 + 3/8*a^2*b*x^8 + 1/5*a^3*x^5

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Fricas [A]  time = 0.252331, size = 47, normalized size = 0.28 \[ \frac{1}{14} \, b^{3} x^{14} + \frac{3}{11} \, a b^{2} x^{11} + \frac{3}{8} \, a^{2} b x^{8} + \frac{1}{5} \, a^{3} x^{5} \]

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^4,x, algorithm="fricas")

[Out]

1/14*b^3*x^14 + 3/11*a*b^2*x^11 + 3/8*a^2*b*x^8 + 1/5*a^3*x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

Integral(x**4*((a + b*x**3)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.269335, size = 90, normalized size = 0.54 \[ \frac{1}{14} \, b^{3} x^{14}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{11} \, a b^{2} x^{11}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{8} \, a^{2} b x^{8}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{5} \, a^{3} x^{5}{\rm sign}\left (b x^{3} + a\right ) \]

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^4,x, algorithm="giac")

[Out]

1/14*b^3*x^14*sign(b*x^3 + a) + 3/11*a*b^2*x^11*sign(b*x^3 + a) + 3/8*a^2*b*x^8*
sign(b*x^3 + a) + 1/5*a^3*x^5*sign(b*x^3 + a)