3.588 \(\int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\)

Optimal. Leaf size=160 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{18 b^4}-\frac{3 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{14 b^4}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^4} \]

[Out]

-(a^3*(a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*b^4) + (3*a^2*(a + b*x^
2)^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*b^4) - (3*a*(a + b*x^2)^7*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4])/(16*b^4) + ((a + b*x^2)^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/
(18*b^4)

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Rubi [A]  time = 0.292674, antiderivative size = 160, 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 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{18 b^4}-\frac{3 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{14 b^4}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

-(a^3*(a + b*x^2)^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*b^4) + (3*a^2*(a + b*x^
2)^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*b^4) - (3*a*(a + b*x^2)^7*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4])/(16*b^4) + ((a + b*x^2)^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/
(18*b^4)

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Rubi in Sympy [A]  time = 28.4076, size = 151, normalized size = 0.94 \[ - \frac{a^{3} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{288 b^{4}} + \frac{a^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{7}{2}}}{168 b^{4}} - \frac{a x^{4} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{96 b^{2}} + \frac{x^{6} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{36 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

-a**3*(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(288*b**4) + a**2*
(a**2 + 2*a*b*x**2 + b**2*x**4)**(7/2)/(168*b**4) - a*x**4*(2*a + 2*b*x**2)*(a**
2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(96*b**2) + x**6*(2*a + 2*b*x**2)*(a**2 + 2*a
*b*x**2 + b**2*x**4)**(5/2)/(36*b)

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Mathematica [A]  time = 0.0348257, size = 83, normalized size = 0.52 \[ \frac{x^8 \sqrt{\left (a+b x^2\right )^2} \left (126 a^5+504 a^4 b x^2+840 a^3 b^2 x^4+720 a^2 b^3 x^6+315 a b^4 x^8+56 b^5 x^{10}\right )}{1008 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^8*Sqrt[(a + b*x^2)^2]*(126*a^5 + 504*a^4*b*x^2 + 840*a^3*b^2*x^4 + 720*a^2*b^
3*x^6 + 315*a*b^4*x^8 + 56*b^5*x^10))/(1008*(a + b*x^2))

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Maple [A]  time = 0.009, size = 80, normalized size = 0.5 \[{\frac{{x}^{8} \left ( 56\,{b}^{5}{x}^{10}+315\,a{b}^{4}{x}^{8}+720\,{a}^{2}{b}^{3}{x}^{6}+840\,{a}^{3}{b}^{2}{x}^{4}+504\,{a}^{4}b{x}^{2}+126\,{a}^{5} \right ) }{1008\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/1008*x^8*(56*b^5*x^10+315*a*b^4*x^8+720*a^2*b^3*x^6+840*a^3*b^2*x^4+504*a^4*b*
x^2+126*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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258676, size = 77, normalized size = 0.48 \[ \frac{1}{18} \, b^{5} x^{18} + \frac{5}{16} \, a b^{4} x^{16} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{8} \, a^{5} x^{8} \]

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^7,x, algorithm="fricas")

[Out]

1/18*b^5*x^18 + 5/16*a*b^4*x^16 + 5/7*a^2*b^3*x^14 + 5/6*a^3*b^2*x^12 + 1/2*a^4*
b*x^10 + 1/8*a^5*x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{7} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**7*((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.270583, size = 142, normalized size = 0.89 \[ \frac{1}{18} \, b^{5} x^{18}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{16} \, a b^{4} x^{16}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{7} \, a^{2} b^{3} x^{14}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{6} \, a^{3} b^{2} x^{12}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{2} \, a^{4} b x^{10}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{8} \, a^{5} x^{8}{\rm sign}\left (b x^{2} + a\right ) \]

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^7,x, algorithm="giac")

[Out]

1/18*b^5*x^18*sign(b*x^2 + a) + 5/16*a*b^4*x^16*sign(b*x^2 + a) + 5/7*a^2*b^3*x^
14*sign(b*x^2 + a) + 5/6*a^3*b^2*x^12*sign(b*x^2 + a) + 1/2*a^4*b*x^10*sign(b*x^
2 + a) + 1/8*a^5*x^8*sign(b*x^2 + a)