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

Optimal. Leaf size=167 \[ \frac{3 a b^2 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{a^2 b x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{b^3 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{a^3 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )} \]

[Out]

(a^3*x^7*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*(a + b*x^2)) + (a^2*b*x^9*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2)) + (3*a*b^2*x^11*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])/(11*(a + b*x^2)) + (b^3*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*(a +
b*x^2))

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Rubi [A]  time = 0.132912, 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 \[ \frac{3 a b^2 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{a^2 b x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{b^3 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{a^3 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*x^7*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*(a + b*x^2)) + (a^2*b*x^9*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2)) + (3*a*b^2*x^11*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])/(11*(a + b*x^2)) + (b^3*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*(a +
b*x^2))

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Rubi in Sympy [A]  time = 17.2463, size = 136, normalized size = 0.81 \[ \frac{16 a^{3} x^{7} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{3003 \left (a + b x^{2}\right )} + \frac{8 a^{2} x^{7} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{429} + \frac{6 a x^{7} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{143} + \frac{x^{7} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*a**3*x**7*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(3003*(a + b*x**2)) + 8*a**2*x*
*7*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/429 + 6*a*x**7*(a + b*x**2)*sqrt(a**2 + 2
*a*b*x**2 + b**2*x**4)/143 + x**7*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/13

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Mathematica [A]  time = 0.0298419, size = 61, normalized size = 0.37 \[ \frac{x^7 \sqrt{\left (a+b x^2\right )^2} \left (429 a^3+1001 a^2 b x^2+819 a b^2 x^4+231 b^3 x^6\right )}{3003 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^7*Sqrt[(a + b*x^2)^2]*(429*a^3 + 1001*a^2*b*x^2 + 819*a*b^2*x^4 + 231*b^3*x^6
))/(3003*(a + b*x^2))

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Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[{\frac{{x}^{7} \left ( 231\,{b}^{3}{x}^{6}+819\,a{x}^{4}{b}^{2}+1001\,{a}^{2}b{x}^{2}+429\,{a}^{3} \right ) }{3003\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3003*x^7*(231*b^3*x^6+819*a*b^2*x^4+1001*a^2*b*x^2+429*a^3)*((b*x^2+a)^2)^(3/2
)/(b*x^2+a)^3

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Maxima [A]  time = 0.702122, size = 47, normalized size = 0.28 \[ \frac{1}{13} \, b^{3} x^{13} + \frac{3}{11} \, a b^{2} x^{11} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{7} \, a^{3} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^6,x, algorithm="maxima")

[Out]

1/13*b^3*x^13 + 3/11*a*b^2*x^11 + 1/3*a^2*b*x^9 + 1/7*a^3*x^7

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Fricas [A]  time = 0.256437, size = 47, normalized size = 0.28 \[ \frac{1}{13} \, b^{3} x^{13} + \frac{3}{11} \, a b^{2} x^{11} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{7} \, a^{3} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^6,x, algorithm="fricas")

[Out]

1/13*b^3*x^13 + 3/11*a*b^2*x^11 + 1/3*a^2*b*x^9 + 1/7*a^3*x^7

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270074, size = 90, normalized size = 0.54 \[ \frac{1}{13} \, b^{3} x^{13}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{11} \, a b^{2} x^{11}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{3} \, a^{2} b x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{7} \, a^{3} x^{7}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^6,x, algorithm="giac")

[Out]

1/13*b^3*x^13*sign(b*x^2 + a) + 3/11*a*b^2*x^11*sign(b*x^2 + a) + 1/3*a^2*b*x^9*
sign(b*x^2 + a) + 1/7*a^3*x^7*sign(b*x^2 + a)