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

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 = 0.142724, 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, Rules used = {28, 266, 43} \[ \frac{a^7}{10 b^8 \left (a+b x^2\right )^5}-\frac{7 a^6}{8 b^8 \left (a+b x^2\right )^4}+\frac{7 a^5}{2 b^8 \left (a+b x^2\right )^3}-\frac{35 a^4}{4 b^8 \left (a+b x^2\right )^2}+\frac{35 a^3}{2 b^8 \left (a+b x^2\right )}+\frac{21 a^2 \log \left (a+b x^2\right )}{2 b^8}-\frac{3 a x^2}{b^7}+\frac{x^4}{4 b^6} \]

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)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^{15}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \frac{x^7}{\left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^6 \operatorname{Subst}\left (\int \left (-\frac{6 a}{b^{13}}+\frac{x}{b^{12}}-\frac{a^7}{b^{13} (a+b x)^6}+\frac{7 a^6}{b^{13} (a+b x)^5}-\frac{21 a^5}{b^{13} (a+b x)^4}+\frac{35 a^4}{b^{13} (a+b x)^3}-\frac{35 a^3}{b^{13} (a+b x)^2}+\frac{21 a^2}{b^{13} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a x^2}{b^7}+\frac{x^4}{4 b^6}+\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}\\ \end{align*}

Mathematica [A]  time = 0.0243096, size = 114, normalized size = 0.86 \[ \frac{-500 a^2 b^5 x^{10}-400 a^3 b^4 x^8+1300 a^4 b^3 x^6+2700 a^5 b^2 x^4+1875 a^6 b x^2+420 a^2 \left (a+b x^2\right )^5 \log \left (a+b x^2\right )+459 a^7-70 a b^6 x^{12}+10 b^7 x^{14}}{40 b^8 \left (a+b x^2\right )^5} \]

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 = 0.055, size = 120, normalized size = 0.9 \begin{align*} -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}}} \end{align*}

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 = 1.17093, size = 193, normalized size = 1.45 \begin{align*} \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}} \end{align*}

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 + 459*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 = 1.63616, size = 450, normalized size = 3.38 \begin{align*} \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 )}} \end{align*}

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 = 1.46852, size = 150, normalized size = 1.13 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.15741, size = 153, normalized size = 1.15 \begin{align*} \frac{21 \, a^{2} \log \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}} \end{align*}

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*log(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)