\(\int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx\) [1004]

Optimal result

Integrand size = 19, antiderivative size = 28 \[ \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx=-\frac {1-5 a x}{120 a^3 (1-a x)^{15} (1+a x)^{10}} \]

[Out]

1/120*(5*a*x-1)/a^3/(-a*x+1)^15/(a*x+1)^10

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {82} \[ \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx=-\frac {1-5 a x}{120 a^3 (1-a x)^{15} (a x+1)^{10}} \]

[In]

Int[x^2/((1 - a*x)^16*(1 + a*x)^11),x]

[Out]

-1/120*(1 - 5*a*x)/(a^3*(1 - a*x)^15*(1 + a*x)^10)

Rule 82

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x
)^(n + 1)*(e + f*x)^(p + 1)*((2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x)/(d^2*
f^2*(n + p + 2)*(n + p + 3))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3,
0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1)
+ c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1-5 a x}{120 a^3 (1-a x)^{15} (1+a x)^{10}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx=\frac {1-5 a x}{120 a^3 (-1+a x)^{15} (1+a x)^{10}} \]

[In]

Integrate[x^2/((1 - a*x)^16*(1 + a*x)^11),x]

[Out]

(1 - 5*a*x)/(120*a^3*(-1 + a*x)^15*(1 + a*x)^10)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {5 a x -1}{120 \left (a x -1\right )^{15} a^{3} \left (a x +1\right )^{10}}\)	\(26\)
risch	\(\frac {\frac {1}{120 a^{3}}-\frac {x}{24 a^{2}}}{\left (a x -1\right )^{15} \left (a x +1\right )^{10}}\)	\(28\)
norman	\(\frac {-\frac {1}{3} x^{3}+\frac {5}{12} a \,x^{4}+\frac {21}{20} a^{2} x^{5}-\frac {7}{3} a^{3} x^{6}-\frac {4}{3} a^{4} x^{7}+\frac {51}{8} a^{5} x^{8}-\frac {7}{8} a^{6} x^{9}-\frac {52}{5} a^{7} x^{10}+6 a^{8} x^{11}+\frac {21}{2} a^{9} x^{12}-\frac {21}{2} a^{10} x^{13}-6 a^{11} x^{14}+\frac {52}{5} a^{12} x^{15}+\frac {7}{8} a^{13} x^{16}-\frac {51}{8} a^{14} x^{17}+\frac {4}{3} a^{15} x^{18}+\frac {7}{3} a^{16} x^{19}-\frac {21}{20} a^{17} x^{20}-\frac {5}{12} a^{18} x^{21}+\frac {1}{3} a^{19} x^{22}-\frac {1}{24} a^{21} x^{24}+\frac {1}{120} a^{22} x^{25}}{\left (a x -1\right )^{15} \left (a x +1\right )^{10}}\)	\(188\)
parallelrisch	\(-\frac {-a^{22} x^{25}+5 a^{21} x^{24}-40 a^{19} x^{22}+50 a^{18} x^{21}+126 a^{17} x^{20}-280 a^{16} x^{19}-160 a^{15} x^{18}+765 a^{14} x^{17}-105 a^{13} x^{16}-1248 a^{12} x^{15}+720 a^{11} x^{14}+1260 a^{10} x^{13}-1260 a^{9} x^{12}-720 a^{8} x^{11}+1248 a^{7} x^{10}+105 a^{6} x^{9}-765 a^{5} x^{8}+160 a^{4} x^{7}+280 a^{3} x^{6}-126 a^{2} x^{5}-50 a \,x^{4}+40 x^{3}}{120 \left (a x -1\right )^{15} \left (a x +1\right )^{10}}\)	\(189\)
default	\(-\frac {1}{30720 a^{3} \left (a x -1\right )^{15}}+\frac {1}{8192 a^{3} \left (a x -1\right )^{14}}+\frac {11}{32768 a^{3} \left (a x -1\right )^{12}}-\frac {11}{32768 a^{3} \left (a x -1\right )^{11}}+\frac {143}{655360 a^{3} \left (a x -1\right )^{10}}-\frac {143}{524288 a^{3} \left (a x -1\right )^{8}}+\frac {143}{262144 a^{3} \left (a x -1\right )^{7}}-\frac {2431}{3145728 a^{3} \left (a x -1\right )^{6}}+\frac {2431}{2621440 a^{3} \left (a x -1\right )^{5}}-\frac {4199}{4194304 a^{3} \left (a x -1\right )^{4}}+\frac {4199}{4194304 a^{3} \left (a x -1\right )^{3}}-\frac {15827}{16777216 a^{3} \left (a x -1\right )^{2}}+\frac {3553}{4194304 a^{3} \left (a x -1\right )}-\frac {1}{4096 a^{3} \left (a x -1\right )^{13}}-\frac {1}{655360 a^{3} \left (a x +1\right )^{10}}-\frac {1}{98304 a^{3} \left (a x +1\right )^{9}}-\frac {3}{32768 a^{3} \left (a x +1\right )^{7}}-\frac {289}{1572864 a^{3} \left (a x +1\right )^{6}}-\frac {51}{163840 a^{3} \left (a x +1\right )^{5}}-\frac {969}{2097152 a^{3} \left (a x +1\right )^{4}}-\frac {323}{524288 a^{3} \left (a x +1\right )^{3}}-\frac {12597}{16777216 a^{3} \left (a x +1\right )^{2}}-\frac {3553}{4194304 a^{3} \left (a x +1\right )}-\frac {19}{524288 a^{3} \left (a x +1\right )^{8}}\)	\(290\)

[In]

int(x^2/(-a*x+1)^16/(a*x+1)^11,x,method=_RETURNVERBOSE)

[Out]

-1/120*(5*a*x-1)/(a*x-1)^15/a^3/(a*x+1)^10

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (25) = 50\).

Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.04 \[ \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx=-\frac {5 \, a x - 1}{120 \, {\left (a^{28} x^{25} - 5 \, a^{27} x^{24} + 40 \, a^{25} x^{22} - 50 \, a^{24} x^{21} - 126 \, a^{23} x^{20} + 280 \, a^{22} x^{19} + 160 \, a^{21} x^{18} - 765 \, a^{20} x^{17} + 105 \, a^{19} x^{16} + 1248 \, a^{18} x^{15} - 720 \, a^{17} x^{14} - 1260 \, a^{16} x^{13} + 1260 \, a^{15} x^{12} + 720 \, a^{14} x^{11} - 1248 \, a^{13} x^{10} - 105 \, a^{12} x^{9} + 765 \, a^{11} x^{8} - 160 \, a^{10} x^{7} - 280 \, a^{9} x^{6} + 126 \, a^{8} x^{5} + 50 \, a^{7} x^{4} - 40 \, a^{6} x^{3} + 5 \, a^{4} x - a^{3}\right )}} \]

[In]

integrate(x^2/(-a*x+1)^16/(a*x+1)^11,x, algorithm="fricas")

[Out]

-1/120*(5*a*x - 1)/(a^28*x^25 - 5*a^27*x^24 + 40*a^25*x^22 - 50*a^24*x^21 - 126*a^23*x^20 + 280*a^22*x^19 + 16
0*a^21*x^18 - 765*a^20*x^17 + 105*a^19*x^16 + 1248*a^18*x^15 - 720*a^17*x^14 - 1260*a^16*x^13 + 1260*a^15*x^12
+ 720*a^14*x^11 - 1248*a^13*x^10 - 105*a^12*x^9 + 765*a^11*x^8 - 160*a^10*x^7 - 280*a^9*x^6 + 126*a^8*x^5 + 5
0*a^7*x^4 - 40*a^6*x^3 + 5*a^4*x - a^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (24) = 48\).

Time = 0.99 (sec) , antiderivative size = 204, normalized size of antiderivative = 7.29 \[ \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx=\frac {- 5 a x + 1}{120 a^{28} x^{25} - 600 a^{27} x^{24} + 4800 a^{25} x^{22} - 6000 a^{24} x^{21} - 15120 a^{23} x^{20} + 33600 a^{22} x^{19} + 19200 a^{21} x^{18} - 91800 a^{20} x^{17} + 12600 a^{19} x^{16} + 149760 a^{18} x^{15} - 86400 a^{17} x^{14} - 151200 a^{16} x^{13} + 151200 a^{15} x^{12} + 86400 a^{14} x^{11} - 149760 a^{13} x^{10} - 12600 a^{12} x^{9} + 91800 a^{11} x^{8} - 19200 a^{10} x^{7} - 33600 a^{9} x^{6} + 15120 a^{8} x^{5} + 6000 a^{7} x^{4} - 4800 a^{6} x^{3} + 600 a^{4} x - 120 a^{3}} \]

[In]

integrate(x**2/(-a*x+1)**16/(a*x+1)**11,x)

[Out]

(-5*a*x + 1)/(120*a**28*x**25 - 600*a**27*x**24 + 4800*a**25*x**22 - 6000*a**24*x**21 - 15120*a**23*x**20 + 33
600*a**22*x**19 + 19200*a**21*x**18 - 91800*a**20*x**17 + 12600*a**19*x**16 + 149760*a**18*x**15 - 86400*a**17
*x**14 - 151200*a**16*x**13 + 151200*a**15*x**12 + 86400*a**14*x**11 - 149760*a**13*x**10 - 12600*a**12*x**9 +
91800*a**11*x**8 - 19200*a**10*x**7 - 33600*a**9*x**6 + 15120*a**8*x**5 + 6000*a**7*x**4 - 4800*a**6*x**3 + 6
00*a**4*x - 120*a**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (25) = 50\).

Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.04 \[ \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx=-\frac {5 \, a x - 1}{120 \, {\left (a^{28} x^{25} - 5 \, a^{27} x^{24} + 40 \, a^{25} x^{22} - 50 \, a^{24} x^{21} - 126 \, a^{23} x^{20} + 280 \, a^{22} x^{19} + 160 \, a^{21} x^{18} - 765 \, a^{20} x^{17} + 105 \, a^{19} x^{16} + 1248 \, a^{18} x^{15} - 720 \, a^{17} x^{14} - 1260 \, a^{16} x^{13} + 1260 \, a^{15} x^{12} + 720 \, a^{14} x^{11} - 1248 \, a^{13} x^{10} - 105 \, a^{12} x^{9} + 765 \, a^{11} x^{8} - 160 \, a^{10} x^{7} - 280 \, a^{9} x^{6} + 126 \, a^{8} x^{5} + 50 \, a^{7} x^{4} - 40 \, a^{6} x^{3} + 5 \, a^{4} x - a^{3}\right )}} \]

[In]

integrate(x^2/(-a*x+1)^16/(a*x+1)^11,x, algorithm="maxima")

[Out]

-1/120*(5*a*x - 1)/(a^28*x^25 - 5*a^27*x^24 + 40*a^25*x^22 - 50*a^24*x^21 - 126*a^23*x^20 + 280*a^22*x^19 + 16
0*a^21*x^18 - 765*a^20*x^17 + 105*a^19*x^16 + 1248*a^18*x^15 - 720*a^17*x^14 - 1260*a^16*x^13 + 1260*a^15*x^12
+ 720*a^14*x^11 - 1248*a^13*x^10 - 105*a^12*x^9 + 765*a^11*x^8 - 160*a^10*x^7 - 280*a^9*x^6 + 126*a^8*x^5 + 5
0*a^7*x^4 - 40*a^6*x^3 + 5*a^4*x - a^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (25) = 50\).

Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 7.32 \[ \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx=-\frac {213180 \, a^{9} x^{9} + 2107575 \, a^{8} x^{8} + 9341160 \, a^{7} x^{7} + 24399420 \, a^{6} x^{6} + 41474016 \, a^{5} x^{5} + 47696050 \, a^{4} x^{4} + 37231960 \, a^{3} x^{3} + 19104300 \, a^{2} x^{2} + 5879780 \, a x + 833135}{251658240 \, {\left (a x + 1\right )}^{10} a^{3}} + \frac {213180 \, a^{14} x^{14} - 3221925 \, a^{13} x^{13} + 22737585 \, a^{12} x^{12} - 99390330 \, a^{11} x^{11} + 300923766 \, a^{10} x^{10} - 668342675 \, a^{9} x^{9} + 1124389695 \, a^{8} x^{8} - 1457870700 \, a^{7} x^{7} + 1466424960 \, a^{6} x^{6} - 1140648795 \, a^{5} x^{5} + 676154655 \, a^{4} x^{4} - 295952250 \, a^{3} x^{3} + 89819310 \, a^{2} x^{2} - 16508685 \, a x + 1264017}{251658240 \, {\left (a x - 1\right )}^{15} a^{3}} \]

[In]

integrate(x^2/(-a*x+1)^16/(a*x+1)^11,x, algorithm="giac")

[Out]

-1/251658240*(213180*a^9*x^9 + 2107575*a^8*x^8 + 9341160*a^7*x^7 + 24399420*a^6*x^6 + 41474016*a^5*x^5 + 47696
050*a^4*x^4 + 37231960*a^3*x^3 + 19104300*a^2*x^2 + 5879780*a*x + 833135)/((a*x + 1)^10*a^3) + 1/251658240*(21
3180*a^14*x^14 - 3221925*a^13*x^13 + 22737585*a^12*x^12 - 99390330*a^11*x^11 + 300923766*a^10*x^10 - 668342675
*a^9*x^9 + 1124389695*a^8*x^8 - 1457870700*a^7*x^7 + 1466424960*a^6*x^6 - 1140648795*a^5*x^5 + 676154655*a^4*x
^4 - 295952250*a^3*x^3 + 89819310*a^2*x^2 - 16508685*a*x + 1264017)/((a*x - 1)^15*a^3)

Mupad [B] (verification not implemented)

Time = 96.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.04 \[ \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx=-\frac {\frac {x}{24\,a^2}-\frac {1}{120\,a^3}}{a^{25}\,x^{25}-5\,a^{24}\,x^{24}+40\,a^{22}\,x^{22}-50\,a^{21}\,x^{21}-126\,a^{20}\,x^{20}+280\,a^{19}\,x^{19}+160\,a^{18}\,x^{18}-765\,a^{17}\,x^{17}+105\,a^{16}\,x^{16}+1248\,a^{15}\,x^{15}-720\,a^{14}\,x^{14}-1260\,a^{13}\,x^{13}+1260\,a^{12}\,x^{12}+720\,a^{11}\,x^{11}-1248\,a^{10}\,x^{10}-105\,a^9\,x^9+765\,a^8\,x^8-160\,a^7\,x^7-280\,a^6\,x^6+126\,a^5\,x^5+50\,a^4\,x^4-40\,a^3\,x^3+5\,a\,x-1} \]

[In]

int(x^2/((a*x - 1)^16*(a*x + 1)^11),x)

[Out]

-(x/(24*a^2) - 1/(120*a^3))/(5*a*x - 40*a^3*x^3 + 50*a^4*x^4 + 126*a^5*x^5 - 280*a^6*x^6 - 160*a^7*x^7 + 765*a
^8*x^8 - 105*a^9*x^9 - 1248*a^10*x^10 + 720*a^11*x^11 + 1260*a^12*x^12 - 1260*a^13*x^13 - 720*a^14*x^14 + 1248
*a^15*x^15 + 105*a^16*x^16 - 765*a^17*x^17 + 160*a^18*x^18 + 280*a^19*x^19 - 126*a^20*x^20 - 50*a^21*x^21 + 40
*a^22*x^22 - 5*a^24*x^24 + a^25*x^25 - 1)