3.512 \(\int \frac{x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=133 \[ \frac{a^7}{10 b^8 \left (a+b x^2\right )^5}-\frac{7 a^6}{8 b^8 \left (a+b x^2\right )^4}+\frac{7 a^5}{2 b^8 \left (a+b x^2\right )^3}-\frac{35 a^4}{4 b^8 \left (a+b x^2\right )^2}+\frac{35 a^3}{2 b^8 \left (a+b x^2\right )}+\frac{21 a^2 \log \left (a+b x^2\right )}{2 b^8}-\frac{3 a x^2}{b^7}+\frac{x^4}{4 b^6} \]

[Out]

(-3*a*x^2)/b^7 + x^4/(4*b^6) + a^7/(10*b^8*(a + b*x^2)^5) - (7*a^6)/(8*b^8*(a +
b*x^2)^4) + (7*a^5)/(2*b^8*(a + b*x^2)^3) - (35*a^4)/(4*b^8*(a + b*x^2)^2) + (35
*a^3)/(2*b^8*(a + b*x^2)) + (21*a^2*Log[a + b*x^2])/(2*b^8)

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Rubi [A]  time = 0.308286, antiderivative size = 133, 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 \[ \frac{a^7}{10 b^8 \left (a+b x^2\right )^5}-\frac{7 a^6}{8 b^8 \left (a+b x^2\right )^4}+\frac{7 a^5}{2 b^8 \left (a+b x^2\right )^3}-\frac{35 a^4}{4 b^8 \left (a+b x^2\right )^2}+\frac{35 a^3}{2 b^8 \left (a+b x^2\right )}+\frac{21 a^2 \log \left (a+b x^2\right )}{2 b^8}-\frac{3 a x^2}{b^7}+\frac{x^4}{4 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^15/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(-3*a*x^2)/b^7 + x^4/(4*b^6) + a^7/(10*b^8*(a + b*x^2)^5) - (7*a^6)/(8*b^8*(a +
b*x^2)^4) + (7*a^5)/(2*b^8*(a + b*x^2)^3) - (35*a^4)/(4*b^8*(a + b*x^2)^2) + (35
*a^3)/(2*b^8*(a + b*x^2)) + (21*a^2*Log[a + b*x^2])/(2*b^8)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{7}}{10 b^{8} \left (a + b x^{2}\right )^{5}} - \frac{7 a^{6}}{8 b^{8} \left (a + b x^{2}\right )^{4}} + \frac{7 a^{5}}{2 b^{8} \left (a + b x^{2}\right )^{3}} - \frac{35 a^{4}}{4 b^{8} \left (a + b x^{2}\right )^{2}} + \frac{35 a^{3}}{2 b^{8} \left (a + b x^{2}\right )} + \frac{21 a^{2} \log{\left (a + b x^{2} \right )}}{2 b^{8}} - \frac{3 a x^{2}}{b^{7}} + \frac{\int ^{x^{2}} x\, dx}{2 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**15/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**7/(10*b**8*(a + b*x**2)**5) - 7*a**6/(8*b**8*(a + b*x**2)**4) + 7*a**5/(2*b**
8*(a + b*x**2)**3) - 35*a**4/(4*b**8*(a + b*x**2)**2) + 35*a**3/(2*b**8*(a + b*x
**2)) + 21*a**2*log(a + b*x**2)/(2*b**8) - 3*a*x**2/b**7 + Integral(x, (x, x**2)
)/(2*b**6)

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Mathematica [A]  time = 0.0462715, size = 114, normalized size = 0.86 \[ \frac{459 a^7+1875 a^6 b x^2+2700 a^5 b^2 x^4+1300 a^4 b^3 x^6-400 a^3 b^4 x^8-500 a^2 b^5 x^{10}+420 a^2 \left (a+b x^2\right )^5 \log \left (a+b x^2\right )-70 a b^6 x^{12}+10 b^7 x^{14}}{40 b^8 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^15/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(459*a^7 + 1875*a^6*b*x^2 + 2700*a^5*b^2*x^4 + 1300*a^4*b^3*x^6 - 400*a^3*b^4*x^
8 - 500*a^2*b^5*x^10 - 70*a*b^6*x^12 + 10*b^7*x^14 + 420*a^2*(a + b*x^2)^5*Log[a
+ b*x^2])/(40*b^8*(a + b*x^2)^5)

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Maple [A]  time = 0.024, size = 120, normalized size = 0.9 \[ -3\,{\frac{a{x}^{2}}{{b}^{7}}}+{\frac{{x}^{4}}{4\,{b}^{6}}}+{\frac{{a}^{7}}{10\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{7\,{a}^{6}}{8\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{7\,{a}^{5}}{2\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{35\,{a}^{4}}{4\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{a}^{3}}{2\,{b}^{8} \left ( b{x}^{2}+a \right ) }}+{\frac{21\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^15/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-3*a*x^2/b^7+1/4*x^4/b^6+1/10*a^7/b^8/(b*x^2+a)^5-7/8*a^6/b^8/(b*x^2+a)^4+7/2*a^
5/b^8/(b*x^2+a)^3-35/4*a^4/b^8/(b*x^2+a)^2+35/2*a^3/b^8/(b*x^2+a)+21/2*a^2*ln(b*
x^2+a)/b^8

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Maxima [A]  time = 0.714016, size = 193, normalized size = 1.45 \[ \frac{700 \, a^{3} b^{4} x^{8} + 2450 \, a^{4} b^{3} x^{6} + 3290 \, a^{5} b^{2} x^{4} + 1995 \, a^{6} b x^{2} + 459 \, a^{7}}{40 \,{\left (b^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}} + \frac{21 \, a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{8}} + \frac{b x^{4} - 12 \, a x^{2}}{4 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")

[Out]

1/40*(700*a^3*b^4*x^8 + 2450*a^4*b^3*x^6 + 3290*a^5*b^2*x^4 + 1995*a^6*b*x^2 + 4
59*a^7)/(b^13*x^10 + 5*a*b^12*x^8 + 10*a^2*b^11*x^6 + 10*a^3*b^10*x^4 + 5*a^4*b^
9*x^2 + a^5*b^8) + 21/2*a^2*log(b*x^2 + a)/b^8 + 1/4*(b*x^4 - 12*a*x^2)/b^7

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Fricas [A]  time = 0.265403, size = 274, normalized size = 2.06 \[ \frac{10 \, b^{7} x^{14} - 70 \, a b^{6} x^{12} - 500 \, a^{2} b^{5} x^{10} - 400 \, a^{3} b^{4} x^{8} + 1300 \, a^{4} b^{3} x^{6} + 2700 \, a^{5} b^{2} x^{4} + 1875 \, a^{6} b x^{2} + 459 \, a^{7} + 420 \,{\left (a^{2} b^{5} x^{10} + 5 \, a^{3} b^{4} x^{8} + 10 \, a^{4} b^{3} x^{6} + 10 \, a^{5} b^{2} x^{4} + 5 \, a^{6} b x^{2} + a^{7}\right )} \log \left (b x^{2} + a\right )}{40 \,{\left (b^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

1/40*(10*b^7*x^14 - 70*a*b^6*x^12 - 500*a^2*b^5*x^10 - 400*a^3*b^4*x^8 + 1300*a^
4*b^3*x^6 + 2700*a^5*b^2*x^4 + 1875*a^6*b*x^2 + 459*a^7 + 420*(a^2*b^5*x^10 + 5*
a^3*b^4*x^8 + 10*a^4*b^3*x^6 + 10*a^5*b^2*x^4 + 5*a^6*b*x^2 + a^7)*log(b*x^2 + a
))/(b^13*x^10 + 5*a*b^12*x^8 + 10*a^2*b^11*x^6 + 10*a^3*b^10*x^4 + 5*a^4*b^9*x^2
+ a^5*b^8)

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Sympy [A]  time = 5.28281, size = 150, normalized size = 1.13 \[ \frac{21 a^{2} \log{\left (a + b x^{2} \right )}}{2 b^{8}} - \frac{3 a x^{2}}{b^{7}} + \frac{459 a^{7} + 1995 a^{6} b x^{2} + 3290 a^{5} b^{2} x^{4} + 2450 a^{4} b^{3} x^{6} + 700 a^{3} b^{4} x^{8}}{40 a^{5} b^{8} + 200 a^{4} b^{9} x^{2} + 400 a^{3} b^{10} x^{4} + 400 a^{2} b^{11} x^{6} + 200 a b^{12} x^{8} + 40 b^{13} x^{10}} + \frac{x^{4}}{4 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**15/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

21*a**2*log(a + b*x**2)/(2*b**8) - 3*a*x**2/b**7 + (459*a**7 + 1995*a**6*b*x**2
+ 3290*a**5*b**2*x**4 + 2450*a**4*b**3*x**6 + 700*a**3*b**4*x**8)/(40*a**5*b**8
+ 200*a**4*b**9*x**2 + 400*a**3*b**10*x**4 + 400*a**2*b**11*x**6 + 200*a*b**12*x
**8 + 40*b**13*x**10) + x**4/(4*b**6)

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GIAC/XCAS [A]  time = 0.272786, size = 153, normalized size = 1.15 \[ \frac{21 \, a^{2}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{8}} + \frac{b^{6} x^{4} - 12 \, a b^{5} x^{2}}{4 \, b^{12}} - \frac{959 \, a^{2} b^{5} x^{10} + 4095 \, a^{3} b^{4} x^{8} + 7140 \, a^{4} b^{3} x^{6} + 6300 \, a^{5} b^{2} x^{4} + 2800 \, a^{6} b x^{2} + 500 \, a^{7}}{40 \,{\left (b x^{2} + a\right )}^{5} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^15/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

21/2*a^2*ln(abs(b*x^2 + a))/b^8 + 1/4*(b^6*x^4 - 12*a*b^5*x^2)/b^12 - 1/40*(959*
a^2*b^5*x^10 + 4095*a^3*b^4*x^8 + 7140*a^4*b^3*x^6 + 6300*a^5*b^2*x^4 + 2800*a^6
*b*x^2 + 500*a^7)/((b*x^2 + a)^5*b^8)