3.193 \(\int \frac{x^{23}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=205 \[ \frac{a^{11}}{18 b^{12} \left (a+b x^2\right )^9}-\frac{11 a^{10}}{16 b^{12} \left (a+b x^2\right )^8}+\frac{55 a^9}{14 b^{12} \left (a+b x^2\right )^7}-\frac{55 a^8}{4 b^{12} \left (a+b x^2\right )^6}+\frac{33 a^7}{b^{12} \left (a+b x^2\right )^5}-\frac{231 a^6}{4 b^{12} \left (a+b x^2\right )^4}+\frac{77 a^5}{b^{12} \left (a+b x^2\right )^3}-\frac{165 a^4}{2 b^{12} \left (a+b x^2\right )^2}+\frac{165 a^3}{2 b^{12} \left (a+b x^2\right )}+\frac{55 a^2 \log \left (a+b x^2\right )}{2 b^{12}}-\frac{5 a x^2}{b^{11}}+\frac{x^4}{4 b^{10}} \]

[Out]

(-5*a*x^2)/b^11 + x^4/(4*b^10) + a^11/(18*b^12*(a + b*x^2)^9) - (11*a^10)/(16*b^12*(a + b*x^2)^8) + (55*a^9)/(
14*b^12*(a + b*x^2)^7) - (55*a^8)/(4*b^12*(a + b*x^2)^6) + (33*a^7)/(b^12*(a + b*x^2)^5) - (231*a^6)/(4*b^12*(
a + b*x^2)^4) + (77*a^5)/(b^12*(a + b*x^2)^3) - (165*a^4)/(2*b^12*(a + b*x^2)^2) + (165*a^3)/(2*b^12*(a + b*x^
2)) + (55*a^2*Log[a + b*x^2])/(2*b^12)

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Rubi [A]  time = 0.21444, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^{11}}{18 b^{12} \left (a+b x^2\right )^9}-\frac{11 a^{10}}{16 b^{12} \left (a+b x^2\right )^8}+\frac{55 a^9}{14 b^{12} \left (a+b x^2\right )^7}-\frac{55 a^8}{4 b^{12} \left (a+b x^2\right )^6}+\frac{33 a^7}{b^{12} \left (a+b x^2\right )^5}-\frac{231 a^6}{4 b^{12} \left (a+b x^2\right )^4}+\frac{77 a^5}{b^{12} \left (a+b x^2\right )^3}-\frac{165 a^4}{2 b^{12} \left (a+b x^2\right )^2}+\frac{165 a^3}{2 b^{12} \left (a+b x^2\right )}+\frac{55 a^2 \log \left (a+b x^2\right )}{2 b^{12}}-\frac{5 a x^2}{b^{11}}+\frac{x^4}{4 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^23/(a + b*x^2)^10,x]

[Out]

(-5*a*x^2)/b^11 + x^4/(4*b^10) + a^11/(18*b^12*(a + b*x^2)^9) - (11*a^10)/(16*b^12*(a + b*x^2)^8) + (55*a^9)/(
14*b^12*(a + b*x^2)^7) - (55*a^8)/(4*b^12*(a + b*x^2)^6) + (33*a^7)/(b^12*(a + b*x^2)^5) - (231*a^6)/(4*b^12*(
a + b*x^2)^4) + (77*a^5)/(b^12*(a + b*x^2)^3) - (165*a^4)/(2*b^12*(a + b*x^2)^2) + (165*a^3)/(2*b^12*(a + b*x^
2)) + (55*a^2*Log[a + b*x^2])/(2*b^12)

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^{23}}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^{11}}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{10 a}{b^{11}}+\frac{x}{b^{10}}-\frac{a^{11}}{b^{11} (a+b x)^{10}}+\frac{11 a^{10}}{b^{11} (a+b x)^9}-\frac{55 a^9}{b^{11} (a+b x)^8}+\frac{165 a^8}{b^{11} (a+b x)^7}-\frac{330 a^7}{b^{11} (a+b x)^6}+\frac{462 a^6}{b^{11} (a+b x)^5}-\frac{462 a^5}{b^{11} (a+b x)^4}+\frac{330 a^4}{b^{11} (a+b x)^3}-\frac{165 a^3}{b^{11} (a+b x)^2}+\frac{55 a^2}{b^{11} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{5 a x^2}{b^{11}}+\frac{x^4}{4 b^{10}}+\frac{a^{11}}{18 b^{12} \left (a+b x^2\right )^9}-\frac{11 a^{10}}{16 b^{12} \left (a+b x^2\right )^8}+\frac{55 a^9}{14 b^{12} \left (a+b x^2\right )^7}-\frac{55 a^8}{4 b^{12} \left (a+b x^2\right )^6}+\frac{33 a^7}{b^{12} \left (a+b x^2\right )^5}-\frac{231 a^6}{4 b^{12} \left (a+b x^2\right )^4}+\frac{77 a^5}{b^{12} \left (a+b x^2\right )^3}-\frac{165 a^4}{2 b^{12} \left (a+b x^2\right )^2}+\frac{165 a^3}{2 b^{12} \left (a+b x^2\right )}+\frac{55 a^2 \log \left (a+b x^2\right )}{2 b^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0282518, size = 158, normalized size = 0.77 \[ \frac{-36288 a^2 b^9 x^{18}-77112 a^3 b^8 x^{16}+190512 a^4 b^7 x^{14}+1220688 a^5 b^6 x^{12}+2704212 a^6 b^5 x^{10}+3402756 a^7 b^4 x^8+2656584 a^8 b^3 x^6+1281096 a^9 b^2 x^4+351459 a^{10} b x^2+27720 a^2 \left (a+b x^2\right )^9 \log \left (a+b x^2\right )+42131 a^{11}-2772 a b^{10} x^{20}+252 b^{11} x^{22}}{1008 b^{12} \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^23/(a + b*x^2)^10,x]

[Out]

(42131*a^11 + 351459*a^10*b*x^2 + 1281096*a^9*b^2*x^4 + 2656584*a^8*b^3*x^6 + 3402756*a^7*b^4*x^8 + 2704212*a^
6*b^5*x^10 + 1220688*a^5*b^6*x^12 + 190512*a^4*b^7*x^14 - 77112*a^3*b^8*x^16 - 36288*a^2*b^9*x^18 - 2772*a*b^1
0*x^20 + 252*b^11*x^22 + 27720*a^2*(a + b*x^2)^9*Log[a + b*x^2])/(1008*b^12*(a + b*x^2)^9)

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Maple [A]  time = 0.018, size = 188, normalized size = 0.9 \begin{align*} -5\,{\frac{a{x}^{2}}{{b}^{11}}}+{\frac{{x}^{4}}{4\,{b}^{10}}}+{\frac{{a}^{11}}{18\,{b}^{12} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{11\,{a}^{10}}{16\,{b}^{12} \left ( b{x}^{2}+a \right ) ^{8}}}+{\frac{55\,{a}^{9}}{14\,{b}^{12} \left ( b{x}^{2}+a \right ) ^{7}}}-{\frac{55\,{a}^{8}}{4\,{b}^{12} \left ( b{x}^{2}+a \right ) ^{6}}}+33\,{\frac{{a}^{7}}{{b}^{12} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{231\,{a}^{6}}{4\,{b}^{12} \left ( b{x}^{2}+a \right ) ^{4}}}+77\,{\frac{{a}^{5}}{{b}^{12} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{165\,{a}^{4}}{2\,{b}^{12} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{165\,{a}^{3}}{2\,{b}^{12} \left ( b{x}^{2}+a \right ) }}+{\frac{55\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^23/(b*x^2+a)^10,x)

[Out]

-5*a*x^2/b^11+1/4*x^4/b^10+1/18*a^11/b^12/(b*x^2+a)^9-11/16*a^10/b^12/(b*x^2+a)^8+55/14*a^9/b^12/(b*x^2+a)^7-5
5/4*a^8/b^12/(b*x^2+a)^6+33*a^7/b^12/(b*x^2+a)^5-231/4*a^6/b^12/(b*x^2+a)^4+77*a^5/b^12/(b*x^2+a)^3-165/2*a^4/
b^12/(b*x^2+a)^2+165/2*a^3/b^12/(b*x^2+a)+55/2*a^2*ln(b*x^2+a)/b^12

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Maxima [A]  time = 2.69169, size = 312, normalized size = 1.52 \begin{align*} \frac{83160 \, a^{3} b^{8} x^{16} + 582120 \, a^{4} b^{7} x^{14} + 1823976 \, a^{5} b^{6} x^{12} + 3318084 \, a^{6} b^{5} x^{10} + 3817044 \, a^{7} b^{4} x^{8} + 2835756 \, a^{8} b^{3} x^{6} + 1326204 \, a^{9} b^{2} x^{4} + 356499 \, a^{10} b x^{2} + 42131 \, a^{11}}{1008 \,{\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}} + \frac{55 \, a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{12}} + \frac{b x^{4} - 20 \, a x^{2}}{4 \, b^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

1/1008*(83160*a^3*b^8*x^16 + 582120*a^4*b^7*x^14 + 1823976*a^5*b^6*x^12 + 3318084*a^6*b^5*x^10 + 3817044*a^7*b
^4*x^8 + 2835756*a^8*b^3*x^6 + 1326204*a^9*b^2*x^4 + 356499*a^10*b*x^2 + 42131*a^11)/(b^21*x^18 + 9*a*b^20*x^1
6 + 36*a^2*b^19*x^14 + 84*a^3*b^18*x^12 + 126*a^4*b^17*x^10 + 126*a^5*b^16*x^8 + 84*a^6*b^15*x^6 + 36*a^7*b^14
*x^4 + 9*a^8*b^13*x^2 + a^9*b^12) + 55/2*a^2*log(b*x^2 + a)/b^12 + 1/4*(b*x^4 - 20*a*x^2)/b^11

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Fricas [A]  time = 1.22048, size = 817, normalized size = 3.99 \begin{align*} \frac{252 \, b^{11} x^{22} - 2772 \, a b^{10} x^{20} - 36288 \, a^{2} b^{9} x^{18} - 77112 \, a^{3} b^{8} x^{16} + 190512 \, a^{4} b^{7} x^{14} + 1220688 \, a^{5} b^{6} x^{12} + 2704212 \, a^{6} b^{5} x^{10} + 3402756 \, a^{7} b^{4} x^{8} + 2656584 \, a^{8} b^{3} x^{6} + 1281096 \, a^{9} b^{2} x^{4} + 351459 \, a^{10} b x^{2} + 42131 \, a^{11} + 27720 \,{\left (a^{2} b^{9} x^{18} + 9 \, a^{3} b^{8} x^{16} + 36 \, a^{4} b^{7} x^{14} + 84 \, a^{5} b^{6} x^{12} + 126 \, a^{6} b^{5} x^{10} + 126 \, a^{7} b^{4} x^{8} + 84 \, a^{8} b^{3} x^{6} + 36 \, a^{9} b^{2} x^{4} + 9 \, a^{10} b x^{2} + a^{11}\right )} \log \left (b x^{2} + a\right )}{1008 \,{\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

1/1008*(252*b^11*x^22 - 2772*a*b^10*x^20 - 36288*a^2*b^9*x^18 - 77112*a^3*b^8*x^16 + 190512*a^4*b^7*x^14 + 122
0688*a^5*b^6*x^12 + 2704212*a^6*b^5*x^10 + 3402756*a^7*b^4*x^8 + 2656584*a^8*b^3*x^6 + 1281096*a^9*b^2*x^4 + 3
51459*a^10*b*x^2 + 42131*a^11 + 27720*(a^2*b^9*x^18 + 9*a^3*b^8*x^16 + 36*a^4*b^7*x^14 + 84*a^5*b^6*x^12 + 126
*a^6*b^5*x^10 + 126*a^7*b^4*x^8 + 84*a^8*b^3*x^6 + 36*a^9*b^2*x^4 + 9*a^10*b*x^2 + a^11)*log(b*x^2 + a))/(b^21
*x^18 + 9*a*b^20*x^16 + 36*a^2*b^19*x^14 + 84*a^3*b^18*x^12 + 126*a^4*b^17*x^10 + 126*a^5*b^16*x^8 + 84*a^6*b^
15*x^6 + 36*a^7*b^14*x^4 + 9*a^8*b^13*x^2 + a^9*b^12)

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Sympy [A]  time = 8.4607, size = 245, normalized size = 1.2 \begin{align*} \frac{55 a^{2} \log{\left (a + b x^{2} \right )}}{2 b^{12}} - \frac{5 a x^{2}}{b^{11}} + \frac{42131 a^{11} + 356499 a^{10} b x^{2} + 1326204 a^{9} b^{2} x^{4} + 2835756 a^{8} b^{3} x^{6} + 3817044 a^{7} b^{4} x^{8} + 3318084 a^{6} b^{5} x^{10} + 1823976 a^{5} b^{6} x^{12} + 582120 a^{4} b^{7} x^{14} + 83160 a^{3} b^{8} x^{16}}{1008 a^{9} b^{12} + 9072 a^{8} b^{13} x^{2} + 36288 a^{7} b^{14} x^{4} + 84672 a^{6} b^{15} x^{6} + 127008 a^{5} b^{16} x^{8} + 127008 a^{4} b^{17} x^{10} + 84672 a^{3} b^{18} x^{12} + 36288 a^{2} b^{19} x^{14} + 9072 a b^{20} x^{16} + 1008 b^{21} x^{18}} + \frac{x^{4}}{4 b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**23/(b*x**2+a)**10,x)

[Out]

55*a**2*log(a + b*x**2)/(2*b**12) - 5*a*x**2/b**11 + (42131*a**11 + 356499*a**10*b*x**2 + 1326204*a**9*b**2*x*
*4 + 2835756*a**8*b**3*x**6 + 3817044*a**7*b**4*x**8 + 3318084*a**6*b**5*x**10 + 1823976*a**5*b**6*x**12 + 582
120*a**4*b**7*x**14 + 83160*a**3*b**8*x**16)/(1008*a**9*b**12 + 9072*a**8*b**13*x**2 + 36288*a**7*b**14*x**4 +
84672*a**6*b**15*x**6 + 127008*a**5*b**16*x**8 + 127008*a**4*b**17*x**10 + 84672*a**3*b**18*x**12 + 36288*a**
2*b**19*x**14 + 9072*a*b**20*x**16 + 1008*b**21*x**18) + x**4/(4*b**10)

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Giac [A]  time = 3.20379, size = 212, normalized size = 1.03 \begin{align*} \frac{55 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{12}} + \frac{b^{10} x^{4} - 20 \, a b^{9} x^{2}}{4 \, b^{20}} - \frac{78419 \, a^{2} b^{9} x^{18} + 622611 \, a^{3} b^{8} x^{16} + 2240964 \, a^{4} b^{7} x^{14} + 4763220 \, a^{5} b^{6} x^{12} + 6562710 \, a^{6} b^{5} x^{10} + 6063750 \, a^{7} b^{4} x^{8} + 3751440 \, a^{8} b^{3} x^{6} + 1496880 \, a^{9} b^{2} x^{4} + 349272 \, a^{10} b x^{2} + 36288 \, a^{11}}{1008 \,{\left (b x^{2} + a\right )}^{9} b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(b*x^2+a)^10,x, algorithm="giac")

[Out]

55/2*a^2*log(abs(b*x^2 + a))/b^12 + 1/4*(b^10*x^4 - 20*a*b^9*x^2)/b^20 - 1/1008*(78419*a^2*b^9*x^18 + 622611*a
^3*b^8*x^16 + 2240964*a^4*b^7*x^14 + 4763220*a^5*b^6*x^12 + 6562710*a^6*b^5*x^10 + 6063750*a^7*b^4*x^8 + 37514
40*a^8*b^3*x^6 + 1496880*a^9*b^2*x^4 + 349272*a^10*b*x^2 + 36288*a^11)/((b*x^2 + a)^9*b^12)