3.587 \(\int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\)

Optimal. Leaf size=201 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^9}{20 b^5}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{8 b^5}+\frac{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^5}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{7 b^5} \]

[Out]

(a^4*(a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*b^5) - (2*a^3*(a + b*x^2
)^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*b^5) + (3*a^2*(a + b*x^2)^7*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4])/(8*b^5) - (2*a*(a + b*x^2)^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4
])/(9*b^5) + ((a + b*x^2)^9*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(20*b^5)

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Rubi [A]  time = 0.317753, antiderivative size = 201, 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 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^9}{20 b^5}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{8 b^5}+\frac{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^5}-\frac{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{7 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(a^4*(a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*b^5) - (2*a^3*(a + b*x^2
)^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*b^5) + (3*a^2*(a + b*x^2)^7*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4])/(8*b^5) - (2*a*(a + b*x^2)^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4
])/(9*b^5) + ((a + b*x^2)^9*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(20*b^5)

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Rubi in Sympy [A]  time = 37.3168, size = 196, normalized size = 0.98 \[ \frac{a^{4} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{720 b^{5}} - \frac{a^{3} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{7}{2}}}{420 b^{5}} + \frac{a^{2} x^{4} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{240 b^{3}} - \frac{a x^{6} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{90 b^{2}} + \frac{x^{8} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{40 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

a**4*(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(720*b**5) - a**3*(
a**2 + 2*a*b*x**2 + b**2*x**4)**(7/2)/(420*b**5) + a**2*x**4*(2*a + 2*b*x**2)*(a
**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(240*b**3) - a*x**6*(2*a + 2*b*x**2)*(a**2
+ 2*a*b*x**2 + b**2*x**4)**(5/2)/(90*b**2) + x**8*(2*a + 2*b*x**2)*(a**2 + 2*a*b
*x**2 + b**2*x**4)**(5/2)/(40*b)

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Mathematica [A]  time = 0.03989, size = 83, normalized size = 0.41 \[ \frac{x^{10} \sqrt{\left (a+b x^2\right )^2} \left (252 a^5+1050 a^4 b x^2+1800 a^3 b^2 x^4+1575 a^2 b^3 x^6+700 a b^4 x^8+126 b^5 x^{10}\right )}{2520 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^10*Sqrt[(a + b*x^2)^2]*(252*a^5 + 1050*a^4*b*x^2 + 1800*a^3*b^2*x^4 + 1575*a^
2*b^3*x^6 + 700*a*b^4*x^8 + 126*b^5*x^10))/(2520*(a + b*x^2))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.4 \[{\frac{{x}^{10} \left ( 126\,{b}^{5}{x}^{10}+700\,a{b}^{4}{x}^{8}+1575\,{a}^{2}{b}^{3}{x}^{6}+1800\,{a}^{3}{b}^{2}{x}^{4}+1050\,{a}^{4}b{x}^{2}+252\,{a}^{5} \right ) }{2520\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/2520*x^10*(126*b^5*x^10+700*a*b^4*x^8+1575*a^2*b^3*x^6+1800*a^3*b^2*x^4+1050*a
^4*b*x^2+252*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257458, size = 77, normalized size = 0.38 \[ \frac{1}{20} \, b^{5} x^{20} + \frac{5}{18} \, a b^{4} x^{18} + \frac{5}{8} \, a^{2} b^{3} x^{16} + \frac{5}{7} \, a^{3} b^{2} x^{14} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{10} \, a^{5} x^{10} \]

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^9,x, algorithm="fricas")

[Out]

1/20*b^5*x^20 + 5/18*a*b^4*x^18 + 5/8*a^2*b^3*x^16 + 5/7*a^3*b^2*x^14 + 5/12*a^4
*b*x^12 + 1/10*a^5*x^10

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{9} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**9*((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.270341, size = 142, normalized size = 0.71 \[ \frac{1}{20} \, b^{5} x^{20}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{18} \, a b^{4} x^{18}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{8} \, a^{2} b^{3} x^{16}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{7} \, a^{3} b^{2} x^{14}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{12} \, a^{4} b x^{12}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{10} \, a^{5} x^{10}{\rm sign}\left (b x^{2} + a\right ) \]

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^9,x, algorithm="giac")

[Out]

1/20*b^5*x^20*sign(b*x^2 + a) + 5/18*a*b^4*x^18*sign(b*x^2 + a) + 5/8*a^2*b^3*x^
16*sign(b*x^2 + a) + 5/7*a^3*b^2*x^14*sign(b*x^2 + a) + 5/12*a^4*b*x^12*sign(b*x
^2 + a) + 1/10*a^5*x^10*sign(b*x^2 + a)