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

Optimal. Leaf size=41 \[ \frac{x^8}{8 a \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

x^8/(8*a*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.119058, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{x^8}{8 a \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

x^8/(8*a*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [A]  time = 8.45338, size = 37, normalized size = 0.9 \[ \frac{x^{8} \left (2 a + 2 b x^{2}\right )}{16 a \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}} \]

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]

x**8*(2*a + 2*b*x**2)/(16*a*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2))

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Mathematica [A]  time = 0.030128, size = 61, normalized size = 1.49 \[ \frac{-a^3-4 a^2 b x^2-6 a b^2 x^4-4 b^3 x^6}{8 b^4 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-a^3 - 4*a^2*b*x^2 - 6*a*b^2*x^4 - 4*b^3*x^6)/(8*b^4*(a + b*x^2)^3*Sqrt[(a + b*
x^2)^2])

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Maple [A]  time = 0.011, size = 54, normalized size = 1.3 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 4\,{b}^{3}{x}^{6}+6\,a{x}^{4}{b}^{2}+4\,{a}^{2}b{x}^{2}+{a}^{3} \right ) }{8\,{b}^{4}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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]

-1/8*(b*x^2+a)*(4*b^3*x^6+6*a*b^2*x^4+4*a^2*b*x^2+a^3)/b^4/((b*x^2+a)^2)^(5/2)

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Maxima [A]  time = 0.698362, size = 197, normalized size = 4.8 \[ -\frac{x^{4}}{2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{a^{2}}{3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{3}{2}} b^{4}} + \frac{a^{2}}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{3}} - \frac{a}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2} b} - \frac{a^{3} b}{8 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{4}} + \frac{a^{3}}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{4} b^{3}} \]

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, algorithm="maxima")

[Out]

-1/2*x^4/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*b^2) - 1/3*a^2/((b^2*x^4 + 2*a*b*x^2
+ a^2)^(3/2)*b^4) + 1/3*a^2/((b^2)^(7/2)*(x^2 + a/b)^3) - 1/4*a/((b^2)^(5/2)*(x
^2 + a/b)^2*b) - 1/8*a^3*b/((b^2)^(9/2)*(x^2 + a/b)^4) + 1/4*a^3/((b^2)^(5/2)*(x
^2 + a/b)^4*b^3)

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Fricas [A]  time = 0.258123, size = 108, normalized size = 2.63 \[ -\frac{4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} \]

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, algorithm="fricas")

[Out]

-1/8*(4*b^3*x^6 + 6*a*b^2*x^4 + 4*a^2*b*x^2 + a^3)/(b^8*x^8 + 4*a*b^7*x^6 + 6*a^
2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)

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

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 = 0.63871, size = 4, normalized size = 0.1 \[ \mathit{sage}_{0} x \]

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, algorithm="giac")

[Out]

sage0*x