3.209 \(\int \frac {x^{24}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=231 \[ -\frac {7436429 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{25/2}}+\frac {7436429 a^2 x}{65536 b^{12}}-\frac {7436429 a x^3}{196608 b^{11}}-\frac {1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac {1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}+\frac {7436429 x^5}{327680 b^{10}} \]

[Out]

7436429/65536*a^2*x/b^12-7436429/196608*a*x^3/b^11+7436429/327680*x^5/b^10-1/18*x^23/b/(b*x^2+a)^9-23/288*x^21
/b^2/(b*x^2+a)^8-23/192*x^19/b^3/(b*x^2+a)^7-437/2304*x^17/b^4/(b*x^2+a)^6-7429/23040*x^15/b^5/(b*x^2+a)^5-742
9/12288*x^13/b^6/(b*x^2+a)^4-96577/73728*x^11/b^7/(b*x^2+a)^3-1062347/294912*x^9/b^8/(b*x^2+a)^2-1062347/65536
*x^7/b^9/(b*x^2+a)-7436429/65536*a^(5/2)*arctan(x*b^(1/2)/a^(1/2))/b^(25/2)

________________________________________________________________________________________

Rubi [A]  time = 0.17, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac {7436429 a^2 x}{65536 b^{12}}-\frac {7436429 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{25/2}}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac {1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac {1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac {7436429 a x^3}{196608 b^{11}}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}+\frac {7436429 x^5}{327680 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^24/(a + b*x^2)^10,x]

[Out]

(7436429*a^2*x)/(65536*b^12) - (7436429*a*x^3)/(196608*b^11) + (7436429*x^5)/(327680*b^10) - x^23/(18*b*(a + b
*x^2)^9) - (23*x^21)/(288*b^2*(a + b*x^2)^8) - (23*x^19)/(192*b^3*(a + b*x^2)^7) - (437*x^17)/(2304*b^4*(a + b
*x^2)^6) - (7429*x^15)/(23040*b^5*(a + b*x^2)^5) - (7429*x^13)/(12288*b^6*(a + b*x^2)^4) - (96577*x^11)/(73728
*b^7*(a + b*x^2)^3) - (1062347*x^9)/(294912*b^8*(a + b*x^2)^2) - (1062347*x^7)/(65536*b^9*(a + b*x^2)) - (7436
429*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*b^(25/2))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin {align*} \int \frac {x^{24}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}+\frac {23 \int \frac {x^{22}}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}+\frac {161 \int \frac {x^{20}}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}+\frac {437 \int \frac {x^{18}}{\left (a+b x^2\right )^7} \, dx}{192 b^3}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}+\frac {7429 \int \frac {x^{16}}{\left (a+b x^2\right )^6} \, dx}{2304 b^4}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}+\frac {7429 \int \frac {x^{14}}{\left (a+b x^2\right )^5} \, dx}{1536 b^5}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}+\frac {96577 \int \frac {x^{12}}{\left (a+b x^2\right )^4} \, dx}{12288 b^6}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}+\frac {1062347 \int \frac {x^{10}}{\left (a+b x^2\right )^3} \, dx}{73728 b^7}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac {1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}+\frac {1062347 \int \frac {x^8}{\left (a+b x^2\right )^2} \, dx}{32768 b^8}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac {1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac {1062347 x^7}{65536 b^9 \left (a+b x^2\right )}+\frac {7436429 \int \frac {x^6}{a+b x^2} \, dx}{65536 b^9}\\ &=-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac {1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac {1062347 x^7}{65536 b^9 \left (a+b x^2\right )}+\frac {7436429 \int \left (\frac {a^2}{b^3}-\frac {a x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx}{65536 b^9}\\ &=\frac {7436429 a^2 x}{65536 b^{12}}-\frac {7436429 a x^3}{196608 b^{11}}+\frac {7436429 x^5}{327680 b^{10}}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac {1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac {1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac {\left (7436429 a^3\right ) \int \frac {1}{a+b x^2} \, dx}{65536 b^{12}}\\ &=\frac {7436429 a^2 x}{65536 b^{12}}-\frac {7436429 a x^3}{196608 b^{11}}+\frac {7436429 x^5}{327680 b^{10}}-\frac {x^{23}}{18 b \left (a+b x^2\right )^9}-\frac {23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac {23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac {437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac {7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac {1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac {1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac {7436429 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{25/2}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 166, normalized size = 0.72 \[ \frac {\frac {\sqrt {b} x \left (334639305 a^{11}[phone] a^{10} b x^2+11110024926 a^9 b^2 x^4+24648575094 a^8 b^3 x^6+34810986496 a^7 b^4 x^8+32314857354 a^6 b^5 x^{10}[phone] a^5 b^6 x^{12}[phone] a^4 b^7 x^{14}[phone] a^3 b^8 x^{16}[phone] a^2 b^9 x^{18}-4521984 a b^{10} x^{20}+589824 b^{11} x^{22}\right )}{\left (a+b x^2\right )^9}-334639305 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2949120 b^{25/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^24/(a + b*x^2)^10,x]

[Out]

((Sqrt[b]*x*(334639305*a^11 + 2900207310*a^10*b*x^2 + 11110024926*a^9*b^2*x^4 + 24648575094*a^8*b^3*x^6 + 3481
0986496*a^7*b^4*x^8 + 32314857354*a^6*b^5*x^10 + 19562592546*a^5*b^6*x^12 + 7323998514*a^4*b^7*x^14 + 14696323
11*a^3*b^8*x^16 + 94961664*a^2*b^9*x^18 - 4521984*a*b^10*x^20 + 589824*b^11*x^22))/(a + b*x^2)^9 - 334639305*a
^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2949120*b^(25/2))

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fricas [A]  time = 0.69, size = 718, normalized size = 3.11 \[ \left [\frac {1179648 \, b^{11} x^{23} - 9043968 \, a b^{10} x^{21} + 189923328 \, a^{2} b^{9} x^{19} + 2939264622 \, a^{3} b^{8} x^{17} + 14647997028 \, a^{4} b^{7} x^{15} + 39125185092 \, a^{5} b^{6} x^{13} + 64629714708 \, a^{6} b^{5} x^{11} + 69621972992 \, a^{7} b^{4} x^{9} + 49297150188 \, a^{8} b^{3} x^{7} + 22220049852 \, a^{9} b^{2} x^{5} + 5800414620 \, a^{10} b x^{3} + 669278610 \, a^{11} x + 334639305 \, {\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 )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{5898240 \, {\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 {589824 \, b^{11} x^{23} - 4521984 \, a b^{10} x^{21} + 94961664 \, a^{2} b^{9} x^{19} + 1469632311 \, a^{3} b^{8} x^{17} + 7323998514 \, a^{4} b^{7} x^{15} + 19562592546 \, a^{5} b^{6} x^{13} + 32314857354 \, a^{6} b^{5} x^{11} + 34810986496 \, a^{7} b^{4} x^{9} + 24648575094 \, a^{8} b^{3} x^{7} + 11110024926 \, a^{9} b^{2} x^{5} + 2900207310 \, a^{10} b x^{3} + 334639305 \, a^{11} x - 334639305 \, {\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 )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{2949120 \, {\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 )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[1/5898240*(1179648*b^11*x^23 - 9043968*a*b^10*x^21 + 189923328*a^2*b^9*x^19 + 2939264622*a^3*b^8*x^17 + 14647
997028*a^4*b^7*x^15 + 39125185092*a^5*b^6*x^13 + 64629714708*a^6*b^5*x^11 + 69621972992*a^7*b^4*x^9 + 49297150
188*a^8*b^3*x^7 + 22220049852*a^9*b^2*x^5 + 5800414620*a^10*b*x^3 + 669278610*a^11*x + 334639305*(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)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(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), 1/2949120*(589824*b^11*x^23 - 4521984*a*b^10*x^21 + 94961664*a
^2*b^9*x^19 + 1469632311*a^3*b^8*x^17 + 7323998514*a^4*b^7*x^15 + 19562592546*a^5*b^6*x^13 + 32314857354*a^6*b
^5*x^11 + 34810986496*a^7*b^4*x^9 + 24648575094*a^8*b^3*x^7 + 11110024926*a^9*b^2*x^5 + 2900207310*a^10*b*x^3
+ 334639305*a^11*x - 334639305*(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)*sqrt(a/b)*arctan(b*x*sqrt(a/
b)/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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giac [A]  time = 0.59, size = 162, normalized size = 0.70 \[ -\frac {7436429 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{12}} + \frac {314167095 \, a^{3} b^{8} x^{17} + 2236176690 \, a^{4} b^{7} x^{15} + 7101970722 \, a^{5} b^{6} x^{13} + 13066540938 \, a^{6} b^{5} x^{11} + 15178104832 \, a^{7} b^{4} x^{9} + 11372226678 \, a^{8} b^{3} x^{7} + 5358651102 \, a^{9} b^{2} x^{5} + 1450223310 \, a^{10} b x^{3} + 172437705 \, a^{11} x}{2949120 \, {\left (b x^{2} + a\right )}^{9} b^{12}} + \frac {3 \, b^{40} x^{5} - 50 \, a b^{39} x^{3} + 825 \, a^{2} b^{38} x}{15 \, b^{50}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-7436429/65536*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^12) + 1/2949120*(314167095*a^3*b^8*x^17 + 2236176690*a^4
*b^7*x^15 + 7101970722*a^5*b^6*x^13 + 13066540938*a^6*b^5*x^11 + 15178104832*a^7*b^4*x^9 + 11372226678*a^8*b^3
*x^7 + 5358651102*a^9*b^2*x^5 + 1450223310*a^10*b*x^3 + 172437705*a^11*x)/((b*x^2 + a)^9*b^12) + 1/15*(3*b^40*
x^5 - 50*a*b^39*x^3 + 825*a^2*b^38*x)/b^50

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maple [A]  time = 0.02, size = 228, normalized size = 0.99 \[ \frac {6981491 a^{3} x^{17}}{65536 \left (b \,x^{2}+a \right )^{9} b^{4}}+\frac {74539223 a^{4} x^{15}}{98304 \left (b \,x^{2}+a \right )^{9} b^{5}}+\frac {394553929 a^{5} x^{13}}{163840 \left (b \,x^{2}+a \right )^{9} b^{6}}+\frac {725918941 a^{6} x^{11}}{163840 \left (b \,x^{2}+a \right )^{9} b^{7}}+\frac {463199 a^{7} x^{9}}{90 \left (b \,x^{2}+a \right )^{9} b^{8}}+\frac {631790371 a^{8} x^{7}}{163840 \left (b \,x^{2}+a \right )^{9} b^{9}}+\frac {297702839 a^{9} x^{5}}{163840 \left (b \,x^{2}+a \right )^{9} b^{10}}+\frac {48340777 a^{10} x^{3}}{98304 \left (b \,x^{2}+a \right )^{9} b^{11}}+\frac {3831949 a^{11} x}{65536 \left (b \,x^{2}+a \right )^{9} b^{12}}+\frac {x^{5}}{5 b^{10}}-\frac {10 a \,x^{3}}{3 b^{11}}-\frac {7436429 a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, b^{12}}+\frac {55 a^{2} x}{b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^24/(b*x^2+a)^10,x)

[Out]

1/5*x^5/b^10-10/3*a*x^3/b^11+55*a^2*x/b^12+3831949/65536/b^12*a^11/(b*x^2+a)^9*x+48340777/98304/b^11*a^10/(b*x
^2+a)^9*x^3+297702839/163840/b^10*a^9/(b*x^2+a)^9*x^5+631790371/163840/b^9*a^8/(b*x^2+a)^9*x^7+463199/90/b^8*a
^7/(b*x^2+a)^9*x^9+725918941/163840/b^7*a^6/(b*x^2+a)^9*x^11+394553929/163840/b^6*a^5/(b*x^2+a)^9*x^13+7453922
3/98304/b^5*a^4/(b*x^2+a)^9*x^15+6981491/65536/b^4*a^3/(b*x^2+a)^9*x^17-7436429/65536/b^12*a^3/(a*b)^(1/2)*arc
tan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.39, size = 248, normalized size = 1.07 \[ \frac {314167095 \, a^{3} b^{8} x^{17} + 2236176690 \, a^{4} b^{7} x^{15} + 7101970722 \, a^{5} b^{6} x^{13} + 13066540938 \, a^{6} b^{5} x^{11} + 15178104832 \, a^{7} b^{4} x^{9} + 11372226678 \, a^{8} b^{3} x^{7} + 5358651102 \, a^{9} b^{2} x^{5} + 1450223310 \, a^{10} b x^{3} + 172437705 \, a^{11} x}{2949120 \, {\left (b^{21} x^{18} + 9 \, a b^{20} x^{16} + 36 \, a^{2} b^{19} x^{14} + 84 \, a^{3} b^{18} x^{12} + 126 \, a^{4} b^{17} x^{10} + 126 \, a^{5} b^{16} x^{8} + 84 \, a^{6} b^{15} x^{6} + 36 \, a^{7} b^{14} x^{4} + 9 \, a^{8} b^{13} x^{2} + a^{9} b^{12}\right )}} - \frac {7436429 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{12}} + \frac {3 \, b^{2} x^{5} - 50 \, a b x^{3} + 825 \, a^{2} x}{15 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^24/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

1/2949120*(314167095*a^3*b^8*x^17 + 2236176690*a^4*b^7*x^15 + 7101970722*a^5*b^6*x^13 + 13066540938*a^6*b^5*x^
11 + 15178104832*a^7*b^4*x^9 + 11372226678*a^8*b^3*x^7 + 5358651102*a^9*b^2*x^5 + 1450223310*a^10*b*x^3 + 1724
37705*a^11*x)/(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) - 7436429/65536*a^3*arctan(b*x/sqrt(a
*b))/(sqrt(a*b)*b^12) + 1/15*(3*b^2*x^5 - 50*a*b*x^3 + 825*a^2*x)/b^12

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mupad [B]  time = 4.90, size = 241, normalized size = 1.04 \[ \frac {\frac {3831949\,a^{11}\,x}{65536}+\frac {48340777\,a^{10}\,b\,x^3}{98304}+\frac {297702839\,a^9\,b^2\,x^5}{163840}+\frac {631790371\,a^8\,b^3\,x^7}{163840}+\frac {463199\,a^7\,b^4\,x^9}{90}+\frac {725918941\,a^6\,b^5\,x^{11}}{163840}+\frac {394553929\,a^5\,b^6\,x^{13}}{163840}+\frac {74539223\,a^4\,b^7\,x^{15}}{98304}+\frac {6981491\,a^3\,b^8\,x^{17}}{65536}}{a^9\,b^{12}+9\,a^8\,b^{13}\,x^2+36\,a^7\,b^{14}\,x^4+84\,a^6\,b^{15}\,x^6+126\,a^5\,b^{16}\,x^8+126\,a^4\,b^{17}\,x^{10}+84\,a^3\,b^{18}\,x^{12}+36\,a^2\,b^{19}\,x^{14}+9\,a\,b^{20}\,x^{16}+b^{21}\,x^{18}}+\frac {x^5}{5\,b^{10}}-\frac {10\,a\,x^3}{3\,b^{11}}+\frac {55\,a^2\,x}{b^{12}}-\frac {7436429\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,b^{25/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^24/(a + b*x^2)^10,x)

[Out]

((3831949*a^11*x)/65536 + (48340777*a^10*b*x^3)/98304 + (297702839*a^9*b^2*x^5)/163840 + (631790371*a^8*b^3*x^
7)/163840 + (463199*a^7*b^4*x^9)/90 + (725918941*a^6*b^5*x^11)/163840 + (394553929*a^5*b^6*x^13)/163840 + (745
39223*a^4*b^7*x^15)/98304 + (6981491*a^3*b^8*x^17)/65536)/(a^9*b^12 + b^21*x^18 + 9*a*b^20*x^16 + 9*a^8*b^13*x
^2 + 36*a^7*b^14*x^4 + 84*a^6*b^15*x^6 + 126*a^5*b^16*x^8 + 126*a^4*b^17*x^10 + 84*a^3*b^18*x^12 + 36*a^2*b^19
*x^14) + x^5/(5*b^10) - (10*a*x^3)/(3*b^11) + (55*a^2*x)/b^12 - (7436429*a^(5/2)*atan((b^(1/2)*x)/a^(1/2)))/(6
5536*b^(25/2))

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sympy [A]  time = 2.04, size = 314, normalized size = 1.36 \[ \frac {55 a^{2} x}{b^{12}} - \frac {10 a x^{3}}{3 b^{11}} + \frac {7436429 \sqrt {- \frac {a^{5}}{b^{25}}} \log {\left (x - \frac {b^{12} \sqrt {- \frac {a^{5}}{b^{25}}}}{a^{2}} \right )}}{131072} - \frac {7436429 \sqrt {- \frac {a^{5}}{b^{25}}} \log {\left (x + \frac {b^{12} \sqrt {- \frac {a^{5}}{b^{25}}}}{a^{2}} \right )}}{131072} + \frac {172437705 a^{11} x + 1450223310 a^{10} b x^{3} + 5358651102 a^{9} b^{2} x^{5} + 11372226678 a^{8} b^{3} x^{7} + 15178104832 a^{7} b^{4} x^{9} + 13066540938 a^{6} b^{5} x^{11} + 7101970722 a^{5} b^{6} x^{13} + 2236176690 a^{4} b^{7} x^{15} + 314167095 a^{3} b^{8} x^{17}}{2949120 a^{9} b^{12} + 26542080 a^{8} b^{13} x^{2} + 106168320 a^{7} b^{14} x^{4} + 247726080 a^{6} b^{15} x^{6} + 371589120 a^{5} b^{16} x^{8} + 371589120 a^{4} b^{17} x^{10} + 247726080 a^{3} b^{18} x^{12} + 106168320 a^{2} b^{19} x^{14} + 26542080 a b^{20} x^{16} + 2949120 b^{21} x^{18}} + \frac {x^{5}}{5 b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**24/(b*x**2+a)**10,x)

[Out]

55*a**2*x/b**12 - 10*a*x**3/(3*b**11) + 7436429*sqrt(-a**5/b**25)*log(x - b**12*sqrt(-a**5/b**25)/a**2)/131072
- 7436429*sqrt(-a**5/b**25)*log(x + b**12*sqrt(-a**5/b**25)/a**2)/131072 + (172437705*a**11*x + 1450223310*a*
*10*b*x**3 + 5358651102*a**9*b**2*x**5 + 11372226678*a**8*b**3*x**7 + 15178104832*a**7*b**4*x**9 + 13066540938
*a**6*b**5*x**11 + 7101970722*a**5*b**6*x**13 + 2236176690*a**4*b**7*x**15 + 314167095*a**3*b**8*x**17)/(29491
20*a**9*b**12 + 26542080*a**8*b**13*x**2 + 106168320*a**7*b**14*x**4 + 247726080*a**6*b**15*x**6 + 371589120*a
**5*b**16*x**8 + 371589120*a**4*b**17*x**10 + 247726080*a**3*b**18*x**12 + 106168320*a**2*b**19*x**14 + 265420
80*a*b**20*x**16 + 2949120*b**21*x**18) + x**5/(5*b**10)

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