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

Optimal. Leaf size=196 \[ -\frac{3 a^2}{2 b^5 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a}{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^4}{8 b^5 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a^3}{3 b^5 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(2*a)/(b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - a^4/(8*b^5*(a + b*x^2)^3*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) + (2*a^3)/(3*b^5*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^
2*x^4]) - (3*a^2)/(2*b^5*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((a + b*
x^2)*Log[a + b*x^2])/(2*b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.377918, antiderivative size = 196, 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 \[ -\frac{3 a^2}{2 b^5 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a}{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^4}{8 b^5 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a^3}{3 b^5 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a)/(b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - a^4/(8*b^5*(a + b*x^2)^3*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) + (2*a^3)/(3*b^5*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^
2*x^4]) - (3*a^2)/(2*b^5*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((a + b*
x^2)*Log[a + b*x^2])/(2*b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [A]  time = 28.0512, size = 184, normalized size = 0.94 \[ \frac{a x^{6} \left (a + b x^{2}\right )}{8 b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}} + \frac{a x^{2} \left (a + b x^{2}\right )}{4 b^{4} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} - \frac{7 x^{6}}{24 b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} - \frac{3 x^{2}}{4 b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}} + \frac{\left (a + b x^{2}\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}} \]

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*x**6*(a + b*x**2)/(8*b**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)) + a*x**2*(a
+ b*x**2)/(4*b**4*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)) - 7*x**6/(24*b**2*(a**
2 + 2*a*b*x**2 + b**2*x**4)**(3/2)) - 3*x**2/(4*b**4*sqrt(a**2 + 2*a*b*x**2 + b*
*2*x**4)) + (a + b*x**2)*log(a + b*x**2)/(2*b**5*sqrt(a**2 + 2*a*b*x**2 + b**2*x
**4))

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Mathematica [A]  time = 0.0554598, size = 83, normalized size = 0.42 \[ \frac{a \left (25 a^3+88 a^2 b x^2+108 a b^2 x^4+48 b^3 x^6\right )+12 \left (a+b x^2\right )^4 \log \left (a+b x^2\right )}{24 b^5 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*(25*a^3 + 88*a^2*b*x^2 + 108*a*b^2*x^4 + 48*b^3*x^6) + 12*(a + b*x^2)^4*Log[a
+ b*x^2])/(24*b^5*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.024, size = 141, normalized size = 0.7 \[{\frac{ \left ( 12\,\ln \left ( b{x}^{2}+a \right ){x}^{8}{b}^{4}+48\,\ln \left ( b{x}^{2}+a \right ){x}^{6}a{b}^{3}+48\,a{b}^{3}{x}^{6}+72\,\ln \left ( b{x}^{2}+a \right ){x}^{4}{a}^{2}{b}^{2}+108\,{a}^{2}{b}^{2}{x}^{4}+48\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{3}b+88\,{a}^{3}b{x}^{2}+12\,\ln \left ( b{x}^{2}+a \right ){a}^{4}+25\,{a}^{4} \right ) \left ( b{x}^{2}+a \right ) }{24\,{b}^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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]

1/24*(12*ln(b*x^2+a)*x^8*b^4+48*ln(b*x^2+a)*x^6*a*b^3+48*a*b^3*x^6+72*ln(b*x^2+a
)*x^4*a^2*b^2+108*a^2*b^2*x^4+48*ln(b*x^2+a)*x^2*a^3*b+88*a^3*b*x^2+12*ln(b*x^2+
a)*a^4+25*a^4)*(b*x^2+a)/b^5/((b*x^2+a)^2)^(5/2)

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Maxima [A]  time = 0.702838, size = 134, normalized size = 0.68 \[ \frac{48 \, a b^{3} x^{6} + 108 \, a^{2} b^{2} x^{4} + 88 \, a^{3} b x^{2} + 25 \, a^{4}}{24 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{5}} \]

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, algorithm="maxima")

[Out]

1/24*(48*a*b^3*x^6 + 108*a^2*b^2*x^4 + 88*a^3*b*x^2 + 25*a^4)/(b^9*x^8 + 4*a*b^8
*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5) + 1/2*log(b*x^2 + a)/b^5

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Fricas [A]  time = 0.260762, size = 182, normalized size = 0.93 \[ \frac{48 \, a b^{3} x^{6} + 108 \, a^{2} b^{2} x^{4} + 88 \, a^{3} b x^{2} + 25 \, a^{4} + 12 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{24 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \]

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, algorithm="fricas")

[Out]

1/24*(48*a*b^3*x^6 + 108*a^2*b^2*x^4 + 88*a^3*b*x^2 + 25*a^4 + 12*(b^4*x^8 + 4*a
*b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4)*log(b*x^2 + a))/(b^9*x^8 + 4*a*b^8
*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)

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

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 = 0.620414, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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, algorithm="giac")

[Out]

sage0*x