3.608 \(\int x^6 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{b^5 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac{a b^4 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{a^5 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{5 a^4 b x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{10 a^3 b^2 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )} \]

[Out]

(a^5*x^7*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*(a + b*x^2)) + (5*a^4*b*x^9*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4])/(9*(a + b*x^2)) + (10*a^3*b^2*x^11*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4])/(11*(a + b*x^2)) + (10*a^2*b^3*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4
])/(13*(a + b*x^2)) + (a*b^4*x^15*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2
)) + (b^5*x^17*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(17*(a + b*x^2))

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Rubi [A]  time = 0.184916, antiderivative size = 255, 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 \[ \frac{b^5 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac{a b^4 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{a^5 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{5 a^4 b x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{10 a^3 b^2 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(a^5*x^7*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*(a + b*x^2)) + (5*a^4*b*x^9*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4])/(9*(a + b*x^2)) + (10*a^3*b^2*x^11*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4])/(11*(a + b*x^2)) + (10*a^2*b^3*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4
])/(13*(a + b*x^2)) + (a*b^4*x^15*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2
)) + (b^5*x^17*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(17*(a + b*x^2))

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Rubi in Sympy [A]  time = 26.4365, size = 207, normalized size = 0.81 \[ \frac{256 a^{5} x^{7} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{153153 \left (a + b x^{2}\right )} + \frac{128 a^{4} x^{7} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{21879} + \frac{32 a^{3} x^{7} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2431} + \frac{16 a^{2} x^{7} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{663} + \frac{2 a x^{7} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{51} + \frac{x^{7} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{17} \]

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)**(5/2),x)

[Out]

256*a**5*x**7*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(153153*(a + b*x**2)) + 128*a*
*4*x**7*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/21879 + 32*a**3*x**7*(a + b*x**2)*sq
rt(a**2 + 2*a*b*x**2 + b**2*x**4)/2431 + 16*a**2*x**7*(a**2 + 2*a*b*x**2 + b**2*
x**4)**(3/2)/663 + 2*a*x**7*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/
51 + x**7*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/17

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Mathematica [A]  time = 0.0359517, size = 83, normalized size = 0.33 \[ \frac{x^7 \sqrt{\left (a+b x^2\right )^2} \left (21879 a^5+85085 a^4 b x^2+139230 a^3 b^2 x^4+117810 a^2 b^3 x^6+51051 a b^4 x^8+9009 b^5 x^{10}\right )}{153153 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^7*Sqrt[(a + b*x^2)^2]*(21879*a^5 + 85085*a^4*b*x^2 + 139230*a^3*b^2*x^4 + 117
810*a^2*b^3*x^6 + 51051*a*b^4*x^8 + 9009*b^5*x^10))/(153153*(a + b*x^2))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.3 \[{\frac{{x}^{7} \left ( 9009\,{b}^{5}{x}^{10}+51051\,a{b}^{4}{x}^{8}+117810\,{a}^{2}{b}^{3}{x}^{6}+139230\,{a}^{3}{b}^{2}{x}^{4}+85085\,{a}^{4}b{x}^{2}+21879\,{a}^{5} \right ) }{153153\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

1/153153*x^7*(9009*b^5*x^10+51051*a*b^4*x^8+117810*a^2*b^3*x^6+139230*a^3*b^2*x^
4+85085*a^4*b*x^2+21879*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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Maxima [A]  time = 0.697638, size = 77, normalized size = 0.3 \[ \frac{1}{17} \, b^{5} x^{17} + \frac{1}{3} \, a b^{4} x^{15} + \frac{10}{13} \, a^{2} b^{3} x^{13} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{9} \, a^{4} b x^{9} + \frac{1}{7} \, a^{5} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^6,x, algorithm="maxima")

[Out]

1/17*b^5*x^17 + 1/3*a*b^4*x^15 + 10/13*a^2*b^3*x^13 + 10/11*a^3*b^2*x^11 + 5/9*a
^4*b*x^9 + 1/7*a^5*x^7

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Fricas [A]  time = 0.260199, size = 77, normalized size = 0.3 \[ \frac{1}{17} \, b^{5} x^{17} + \frac{1}{3} \, a b^{4} x^{15} + \frac{10}{13} \, a^{2} b^{3} x^{13} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{9} \, a^{4} b x^{9} + \frac{1}{7} \, a^{5} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^6,x, algorithm="fricas")

[Out]

1/17*b^5*x^17 + 1/3*a*b^4*x^15 + 10/13*a^2*b^3*x^13 + 10/11*a^3*b^2*x^11 + 5/9*a
^4*b*x^9 + 1/7*a^5*x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{6} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

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)**(5/2),x)

[Out]

Integral(x**6*((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.271728, size = 142, normalized size = 0.56 \[ \frac{1}{17} \, b^{5} x^{17}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{3} \, a b^{4} x^{15}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{13} \, a^{2} b^{3} x^{13}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{11} \, a^{3} b^{2} x^{11}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{9} \, a^{4} b x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{7} \, a^{5} x^{7}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^6,x, algorithm="giac")

[Out]

1/17*b^5*x^17*sign(b*x^2 + a) + 1/3*a*b^4*x^15*sign(b*x^2 + a) + 10/13*a^2*b^3*x
^13*sign(b*x^2 + a) + 10/11*a^3*b^2*x^11*sign(b*x^2 + a) + 5/9*a^4*b*x^9*sign(b*
x^2 + a) + 1/7*a^5*x^7*sign(b*x^2 + a)