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

Optimal. Leaf size=167 \[ \frac{3 a b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac{b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac{a^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \]

[Out]

(a^3*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (3*a^2*b*x^10*Sqrt[a
^2 + 2*a*b*x^3 + b^2*x^6])/(10*(a + b*x^3)) + (3*a*b^2*x^13*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(13*(a + b*x^3)) + (b^3*x^16*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(16*(
a + b*x^3))

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Rubi [A]  time = 0.1208, 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 \[ \frac{3 a b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left (a+b x^3\right )}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac{b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left (a+b x^3\right )}+\frac{a^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*x^7*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3)) + (3*a^2*b*x^10*Sqrt[a
^2 + 2*a*b*x^3 + b^2*x^6])/(10*(a + b*x^3)) + (3*a*b^2*x^13*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6])/(13*(a + b*x^3)) + (b^3*x^16*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(16*(
a + b*x^3))

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Rubi in Sympy [A]  time = 17.9684, size = 136, normalized size = 0.81 \[ \frac{81 a^{3} x^{7} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{7280 \left (a + b x^{3}\right )} + \frac{27 a^{2} x^{7} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{1040} + \frac{9 a x^{7} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{208} + \frac{x^{7} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81*a**3*x**7*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(7280*(a + b*x**3)) + 27*a**2*x
**7*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/1040 + 9*a*x**7*(a + b*x**3)*sqrt(a**2 +
2*a*b*x**3 + b**2*x**6)/208 + x**7*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/16

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Mathematica [A]  time = 0.0324748, size = 61, normalized size = 0.37 \[ \frac{x^7 \sqrt{\left (a+b x^3\right )^2} \left (1040 a^3+2184 a^2 b x^3+1680 a b^2 x^6+455 b^3 x^9\right )}{7280 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^7*Sqrt[(a + b*x^3)^2]*(1040*a^3 + 2184*a^2*b*x^3 + 1680*a*b^2*x^6 + 455*b^3*x
^9))/(7280*(a + b*x^3))

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Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[{\frac{{x}^{7} \left ( 455\,{b}^{3}{x}^{9}+1680\,a{b}^{2}{x}^{6}+2184\,{a}^{2}b{x}^{3}+1040\,{a}^{3} \right ) }{7280\, \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7280*x^7*(455*b^3*x^9+1680*a*b^2*x^6+2184*a^2*b*x^3+1040*a^3)*((b*x^3+a)^2)^(3
/2)/(b*x^3+a)^3

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Maxima [A]  time = 0.762293, size = 47, normalized size = 0.28 \[ \frac{1}{16} \, b^{3} x^{16} + \frac{3}{13} \, a b^{2} x^{13} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{7} \, a^{3} x^{7} \]

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^6,x, algorithm="maxima")

[Out]

1/16*b^3*x^16 + 3/13*a*b^2*x^13 + 3/10*a^2*b*x^10 + 1/7*a^3*x^7

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Fricas [A]  time = 0.256258, size = 47, normalized size = 0.28 \[ \frac{1}{16} \, b^{3} x^{16} + \frac{3}{13} \, a b^{2} x^{13} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{7} \, a^{3} x^{7} \]

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^6,x, algorithm="fricas")

[Out]

1/16*b^3*x^16 + 3/13*a*b^2*x^13 + 3/10*a^2*b*x^10 + 1/7*a^3*x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{6} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**6*((a + b*x**3)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.27162, size = 90, normalized size = 0.54 \[ \frac{1}{16} \, b^{3} x^{16}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{13} \, a b^{2} x^{13}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{10} \, a^{2} b x^{10}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{7} \, a^{3} x^{7}{\rm sign}\left (b x^{3} + a\right ) \]

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^6,x, algorithm="giac")

[Out]

1/16*b^3*x^16*sign(b*x^3 + a) + 3/13*a*b^2*x^13*sign(b*x^3 + a) + 3/10*a^2*b*x^1
0*sign(b*x^3 + a) + 1/7*a^3*x^7*sign(b*x^3 + a)