3.558 \(\int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=167 \[ \frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{a^2 b x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{b^3 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac{a^3 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )} \]

[Out]

(a^3*x^10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(10*(a + b*x^2)) + (a^2*b*x^12*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(4*(a + b*x^2)) + (3*a*b^2*x^14*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(14*(a + b*x^2)) + (b^3*x^16*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*(a
+ b*x^2))

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Rubi [A]  time = 0.281651, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{a^2 b x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{b^3 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac{a^3 x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(a^3*x^10*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(10*(a + b*x^2)) + (a^2*b*x^12*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(4*(a + b*x^2)) + (3*a*b^2*x^14*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(14*(a + b*x^2)) + (b^3*x^16*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(16*(a
+ b*x^2))

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Rubi in Sympy [A]  time = 17.1633, size = 133, normalized size = 0.8 \[ \frac{a^{3} x^{10} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{560 \left (a + b x^{2}\right )} + \frac{a^{2} x^{10} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{112} + \frac{3 a x^{10} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{112} + \frac{x^{10} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{16} \]

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)**(3/2),x)

[Out]

a**3*x**10*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(560*(a + b*x**2)) + a**2*x**10*s
qrt(a**2 + 2*a*b*x**2 + b**2*x**4)/112 + 3*a*x**10*(a + b*x**2)*sqrt(a**2 + 2*a*
b*x**2 + b**2*x**4)/112 + x**10*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/16

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Mathematica [A]  time = 0.0323573, size = 61, normalized size = 0.37 \[ \frac{x^{10} \sqrt{\left (a+b x^2\right )^2} \left (56 a^3+140 a^2 b x^2+120 a b^2 x^4+35 b^3 x^6\right )}{560 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(x^10*Sqrt[(a + b*x^2)^2]*(56*a^3 + 140*a^2*b*x^2 + 120*a*b^2*x^4 + 35*b^3*x^6))
/(560*(a + b*x^2))

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Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[{\frac{{x}^{10} \left ( 35\,{b}^{3}{x}^{6}+120\,a{b}^{2}{x}^{4}+140\,{a}^{2}b{x}^{2}+56\,{a}^{3} \right ) }{560\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

1/560*x^10*(35*b^3*x^6+120*a*b^2*x^4+140*a^2*b*x^2+56*a^3)*((b*x^2+a)^2)^(3/2)/(
b*x^2+a)^3

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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)^(3/2)*x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266724, size = 47, normalized size = 0.28 \[ \frac{1}{16} \, b^{3} x^{16} + \frac{3}{14} \, a b^{2} x^{14} + \frac{1}{4} \, a^{2} b x^{12} + \frac{1}{10} \, a^{3} x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^9,x, algorithm="fricas")

[Out]

1/16*b^3*x^16 + 3/14*a*b^2*x^14 + 1/4*a^2*b*x^12 + 1/10*a^3*x^10

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{9} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \]

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)**(3/2),x)

[Out]

Integral(x**9*((a + b*x**2)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.269299, size = 90, normalized size = 0.54 \[ \frac{1}{16} \, b^{3} x^{16}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{14} \, a b^{2} x^{14}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{4} \, a^{2} b x^{12}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{10} \, a^{3} x^{10}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^9,x, algorithm="giac")

[Out]

1/16*b^3*x^16*sign(b*x^2 + a) + 3/14*a*b^2*x^14*sign(b*x^2 + a) + 1/4*a^2*b*x^12
*sign(b*x^2 + a) + 1/10*a^3*x^10*sign(b*x^2 + a)