3.465 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{a^6}{12 x^{12}}-\frac{3 a^5 b}{5 x^{10}}-\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]

[Out]

-a^6/(12*x^12) - (3*a^5*b)/(5*x^10) - (15*a^4*b^2)/(8*x^8) - (10*a^3*b^3)/(3*x^6
) - (15*a^2*b^4)/(4*x^4) - (3*a*b^5)/x^2 + b^6*Log[x]

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Rubi [A]  time = 0.119613, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^6}{12 x^{12}}-\frac{3 a^5 b}{5 x^{10}}-\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^13,x]

[Out]

-a^6/(12*x^12) - (3*a^5*b)/(5*x^10) - (15*a^4*b^2)/(8*x^8) - (10*a^3*b^3)/(3*x^6
) - (15*a^2*b^4)/(4*x^4) - (3*a*b^5)/x^2 + b^6*Log[x]

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Rubi in Sympy [A]  time = 27.2467, size = 80, normalized size = 1.05 \[ - \frac{a^{6}}{12 x^{12}} - \frac{3 a^{5} b}{5 x^{10}} - \frac{15 a^{4} b^{2}}{8 x^{8}} - \frac{10 a^{3} b^{3}}{3 x^{6}} - \frac{15 a^{2} b^{4}}{4 x^{4}} - \frac{3 a b^{5}}{x^{2}} + \frac{b^{6} \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**13,x)

[Out]

-a**6/(12*x**12) - 3*a**5*b/(5*x**10) - 15*a**4*b**2/(8*x**8) - 10*a**3*b**3/(3*
x**6) - 15*a**2*b**4/(4*x**4) - 3*a*b**5/x**2 + b**6*log(x**2)/2

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Mathematica [A]  time = 0.00848403, size = 76, normalized size = 1. \[ -\frac{a^6}{12 x^{12}}-\frac{3 a^5 b}{5 x^{10}}-\frac{15 a^4 b^2}{8 x^8}-\frac{10 a^3 b^3}{3 x^6}-\frac{15 a^2 b^4}{4 x^4}-\frac{3 a b^5}{x^2}+b^6 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^13,x]

[Out]

-a^6/(12*x^12) - (3*a^5*b)/(5*x^10) - (15*a^4*b^2)/(8*x^8) - (10*a^3*b^3)/(3*x^6
) - (15*a^2*b^4)/(4*x^4) - (3*a*b^5)/x^2 + b^6*Log[x]

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Maple [A]  time = 0.01, size = 67, normalized size = 0.9 \[ -{\frac{{a}^{6}}{12\,{x}^{12}}}-{\frac{3\,{a}^{5}b}{5\,{x}^{10}}}-{\frac{15\,{a}^{4}{b}^{2}}{8\,{x}^{8}}}-{\frac{10\,{a}^{3}{b}^{3}}{3\,{x}^{6}}}-{\frac{15\,{a}^{2}{b}^{4}}{4\,{x}^{4}}}-3\,{\frac{a{b}^{5}}{{x}^{2}}}+{b}^{6}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^3/x^13,x)

[Out]

-1/12*a^6/x^12-3/5*a^5*b/x^10-15/8*a^4*b^2/x^8-10/3*a^3*b^3/x^6-15/4*a^2*b^4/x^4
-3*a*b^5/x^2+b^6*ln(x)

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Maxima [A]  time = 0.684928, size = 97, normalized size = 1.28 \[ \frac{1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac{360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3/x^13,x, algorithm="maxima")

[Out]

1/2*b^6*log(x^2) - 1/120*(360*a*b^5*x^10 + 450*a^2*b^4*x^8 + 400*a^3*b^3*x^6 + 2
25*a^4*b^2*x^4 + 72*a^5*b*x^2 + 10*a^6)/x^12

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Fricas [A]  time = 0.255739, size = 97, normalized size = 1.28 \[ \frac{120 \, b^{6} x^{12} \log \left (x\right ) - 360 \, a b^{5} x^{10} - 450 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 225 \, a^{4} b^{2} x^{4} - 72 \, a^{5} b x^{2} - 10 \, a^{6}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3/x^13,x, algorithm="fricas")

[Out]

1/120*(120*b^6*x^12*log(x) - 360*a*b^5*x^10 - 450*a^2*b^4*x^8 - 400*a^3*b^3*x^6
- 225*a^4*b^2*x^4 - 72*a^5*b*x^2 - 10*a^6)/x^12

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Sympy [A]  time = 2.52048, size = 71, normalized size = 0.93 \[ b^{6} \log{\left (x \right )} - \frac{10 a^{6} + 72 a^{5} b x^{2} + 225 a^{4} b^{2} x^{4} + 400 a^{3} b^{3} x^{6} + 450 a^{2} b^{4} x^{8} + 360 a b^{5} x^{10}}{120 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**13,x)

[Out]

b**6*log(x) - (10*a**6 + 72*a**5*b*x**2 + 225*a**4*b**2*x**4 + 400*a**3*b**3*x**
6 + 450*a**2*b**4*x**8 + 360*a*b**5*x**10)/(120*x**12)

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GIAC/XCAS [A]  time = 0.270636, size = 108, normalized size = 1.42 \[ \frac{1}{2} \, b^{6}{\rm ln}\left (x^{2}\right ) - \frac{147 \, b^{6} x^{12} + 360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3/x^13,x, algorithm="giac")

[Out]

1/2*b^6*ln(x^2) - 1/120*(147*b^6*x^12 + 360*a*b^5*x^10 + 450*a^2*b^4*x^8 + 400*a
^3*b^3*x^6 + 225*a^4*b^2*x^4 + 72*a^5*b*x^2 + 10*a^6)/x^12