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

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

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Rubi [A]  time = 0.14, antiderivative size = 133, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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])

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]]

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*}

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Mathematica [A]  time = 0.03, size = 114, normalized size = 0.86 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.88, size = 203, normalized size = 1.53 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.16, size = 113, normalized size = 0.85 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.02, size = 120, normalized size = 0.90 \begin {gather*} \frac {a^{7}}{10 \left (b \,x^{2}+a \right )^{5} b^{8}}-\frac {7 a^{6}}{8 \left (b \,x^{2}+a \right )^{4} b^{8}}+\frac {x^{4}}{4 b^{6}}+\frac {7 a^{5}}{2 \left (b \,x^{2}+a \right )^{3} b^{8}}-\frac {35 a^{4}}{4 \left (b \,x^{2}+a \right )^{2} b^{8}}-\frac {3 a \,x^{2}}{b^{7}}+\frac {35 a^{3}}{2 \left (b \,x^{2}+a \right ) b^{8}}+\frac {21 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{8}} \end {gather*}

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 = 1.42, size = 143, normalized size = 1.08 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.13, size = 142, normalized size = 1.07 \begin {gather*} \frac {\frac {459\,a^7}{40\,b}+\frac {399\,a^6\,x^2}{8}+\frac {329\,a^5\,b\,x^4}{4}+\frac {245\,a^4\,b^2\,x^6}{4}+\frac {35\,a^3\,b^3\,x^8}{2}}{a^5\,b^7+5\,a^4\,b^8\,x^2+10\,a^3\,b^9\,x^4+10\,a^2\,b^{10}\,x^6+5\,a\,b^{11}\,x^8+b^{12}\,x^{10}}+\frac {x^4}{4\,b^6}-\frac {3\,a\,x^2}{b^7}+\frac {21\,a^2\,\ln \left (b\,x^2+a\right )}{2\,b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((459*a^7)/(40*b) + (399*a^6*x^2)/8 + (329*a^5*b*x^4)/4 + (245*a^4*b^2*x^6)/4 + (35*a^3*b^3*x^8)/2)/(a^5*b^7 +
b^12*x^10 + 5*a*b^11*x^8 + 5*a^4*b^8*x^2 + 10*a^3*b^9*x^4 + 10*a^2*b^10*x^6) + x^4/(4*b^6) - (3*a*x^2)/b^7 +
(21*a^2*log(a + b*x^2))/(2*b^8)

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sympy [A]  time = 0.99, size = 150, normalized size = 1.13 \begin {gather*} \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 {gather*}

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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