3.459 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx\)

Optimal. Leaf size=79 \[ -\frac{a^6}{6 x^6}-\frac{3 a^5 b}{2 x^4}-\frac{15 a^4 b^2}{2 x^2}+20 a^3 b^3 \log (x)+\frac{15}{2} a^2 b^4 x^2+\frac{3}{2} a b^5 x^4+\frac{b^6 x^6}{6} \]

[Out]

-a^6/(6*x^6) - (3*a^5*b)/(2*x^4) - (15*a^4*b^2)/(2*x^2) + (15*a^2*b^4*x^2)/2 + (
3*a*b^5*x^4)/2 + (b^6*x^6)/6 + 20*a^3*b^3*Log[x]

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Rubi [A]  time = 0.129644, antiderivative size = 79, 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 \[ -\frac{a^6}{6 x^6}-\frac{3 a^5 b}{2 x^4}-\frac{15 a^4 b^2}{2 x^2}+20 a^3 b^3 \log (x)+\frac{15}{2} a^2 b^4 x^2+\frac{3}{2} a b^5 x^4+\frac{b^6 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^7,x]

[Out]

-a^6/(6*x^6) - (3*a^5*b)/(2*x^4) - (15*a^4*b^2)/(2*x^2) + (15*a^2*b^4*x^2)/2 + (
3*a*b^5*x^4)/2 + (b^6*x^6)/6 + 20*a^3*b^3*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{6}}{6 x^{6}} - \frac{3 a^{5} b}{2 x^{4}} - \frac{15 a^{4} b^{2}}{2 x^{2}} + 10 a^{3} b^{3} \log{\left (x^{2} \right )} + \frac{15 a^{2} b^{4} x^{2}}{2} + 3 a b^{5} \int ^{x^{2}} x\, dx + \frac{b^{6} x^{6}}{6} \]

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**7,x)

[Out]

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

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Mathematica [A]  time = 0.00852243, size = 79, normalized size = 1. \[ -\frac{a^6}{6 x^6}-\frac{3 a^5 b}{2 x^4}-\frac{15 a^4 b^2}{2 x^2}+20 a^3 b^3 \log (x)+\frac{15}{2} a^2 b^4 x^2+\frac{3}{2} a b^5 x^4+\frac{b^6 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^7,x]

[Out]

-a^6/(6*x^6) - (3*a^5*b)/(2*x^4) - (15*a^4*b^2)/(2*x^2) + (15*a^2*b^4*x^2)/2 + (
3*a*b^5*x^4)/2 + (b^6*x^6)/6 + 20*a^3*b^3*Log[x]

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Maple [A]  time = 0.01, size = 68, normalized size = 0.9 \[ -{\frac{{a}^{6}}{6\,{x}^{6}}}-{\frac{3\,{a}^{5}b}{2\,{x}^{4}}}-{\frac{15\,{a}^{4}{b}^{2}}{2\,{x}^{2}}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{2}}{2}}+{\frac{3\,a{b}^{5}{x}^{4}}{2}}+{\frac{{b}^{6}{x}^{6}}{6}}+20\,{a}^{3}{b}^{3}\ln \left ( x \right ) \]

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^7,x)

[Out]

-1/6*a^6/x^6-3/2*a^5*b/x^4-15/2*a^4*b^2/x^2+15/2*a^2*b^4*x^2+3/2*a*b^5*x^4+1/6*b
^6*x^6+20*a^3*b^3*ln(x)

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Maxima [A]  time = 0.688635, size = 95, normalized size = 1.2 \[ \frac{1}{6} \, b^{6} x^{6} + \frac{3}{2} \, a b^{5} x^{4} + \frac{15}{2} \, a^{2} b^{4} x^{2} + 10 \, a^{3} b^{3} \log \left (x^{2}\right ) - \frac{45 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{6 \, x^{6}} \]

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^7,x, algorithm="maxima")

[Out]

1/6*b^6*x^6 + 3/2*a*b^5*x^4 + 15/2*a^2*b^4*x^2 + 10*a^3*b^3*log(x^2) - 1/6*(45*a
^4*b^2*x^4 + 9*a^5*b*x^2 + a^6)/x^6

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Fricas [A]  time = 0.260625, size = 96, normalized size = 1.22 \[ \frac{b^{6} x^{12} + 9 \, a b^{5} x^{10} + 45 \, a^{2} b^{4} x^{8} + 120 \, a^{3} b^{3} x^{6} \log \left (x\right ) - 45 \, a^{4} b^{2} x^{4} - 9 \, a^{5} b x^{2} - a^{6}}{6 \, x^{6}} \]

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^7,x, algorithm="fricas")

[Out]

1/6*(b^6*x^12 + 9*a*b^5*x^10 + 45*a^2*b^4*x^8 + 120*a^3*b^3*x^6*log(x) - 45*a^4*
b^2*x^4 - 9*a^5*b*x^2 - a^6)/x^6

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Sympy [A]  time = 1.65704, size = 75, normalized size = 0.95 \[ 20 a^{3} b^{3} \log{\left (x \right )} + \frac{15 a^{2} b^{4} x^{2}}{2} + \frac{3 a b^{5} x^{4}}{2} + \frac{b^{6} x^{6}}{6} - \frac{a^{6} + 9 a^{5} b x^{2} + 45 a^{4} b^{2} x^{4}}{6 x^{6}} \]

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**7,x)

[Out]

20*a**3*b**3*log(x) + 15*a**2*b**4*x**2/2 + 3*a*b**5*x**4/2 + b**6*x**6/6 - (a**
6 + 9*a**5*b*x**2 + 45*a**4*b**2*x**4)/(6*x**6)

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GIAC/XCAS [A]  time = 0.271227, size = 109, normalized size = 1.38 \[ \frac{1}{6} \, b^{6} x^{6} + \frac{3}{2} \, a b^{5} x^{4} + \frac{15}{2} \, a^{2} b^{4} x^{2} + 10 \, a^{3} b^{3}{\rm ln}\left (x^{2}\right ) - \frac{110 \, a^{3} b^{3} x^{6} + 45 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{6 \, x^{6}} \]

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^7,x, algorithm="giac")

[Out]

1/6*b^6*x^6 + 3/2*a*b^5*x^4 + 15/2*a^2*b^4*x^2 + 10*a^3*b^3*ln(x^2) - 1/6*(110*a
^3*b^3*x^6 + 45*a^4*b^2*x^4 + 9*a^5*b*x^2 + a^6)/x^6