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

Optimal. Leaf size=28 \[ -\frac {1-5 a x}{120 a^3 (1-a x)^{15} (a x+1)^{10}} \]

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {81} \begin {gather*} -\frac {1-5 a x}{120 a^3 (1-a x)^{15} (a x+1)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[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)

Rule 81

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*} \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}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 27, normalized size = 0.96 \begin {gather*} \frac {1-5 a x}{120 a^3 (a x-1)^{15} (a x+1)^{10}} \end {gather*}

Antiderivative was successfully verified.

[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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.26, size = 197, normalized size = 7.04 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 1.21, size = 205, normalized size = 7.32 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.02, size = 290, normalized size = 10.36 \begin {gather*} -\frac {1}{655360 \left (a x +1\right )^{10} a^{3}}-\frac {1}{98304 \left (a x +1\right )^{9} a^{3}}-\frac {3}{32768 \left (a x +1\right )^{7} a^{3}}-\frac {289}{1572864 \left (a x +1\right )^{6} a^{3}}-\frac {51}{163840 \left (a x +1\right )^{5} a^{3}}-\frac {969}{2097152 \left (a x +1\right )^{4} a^{3}}-\frac {323}{524288 \left (a x +1\right )^{3} a^{3}}-\frac {12597}{16777216 \left (a x +1\right )^{2} a^{3}}-\frac {3553}{4194304 \left (a x +1\right ) a^{3}}-\frac {19}{524288 \left (a x +1\right )^{8} a^{3}}-\frac {1}{30720 \left (a x -1\right )^{15} a^{3}}+\frac {1}{8192 \left (a x -1\right )^{14} a^{3}}+\frac {11}{32768 \left (a x -1\right )^{12} a^{3}}-\frac {11}{32768 \left (a x -1\right )^{11} a^{3}}+\frac {143}{655360 \left (a x -1\right )^{10} a^{3}}-\frac {143}{524288 \left (a x -1\right )^{8} a^{3}}+\frac {143}{262144 \left (a x -1\right )^{7} a^{3}}-\frac {2431}{3145728 \left (a x -1\right )^{6} a^{3}}+\frac {2431}{2621440 \left (a x -1\right )^{5} a^{3}}-\frac {4199}{4194304 \left (a x -1\right )^{4} a^{3}}+\frac {4199}{4194304 \left (a x -1\right )^{3} a^{3}}-\frac {15827}{16777216 \left (a x -1\right )^{2} a^{3}}+\frac {3553}{4194304 \left (a x -1\right ) a^{3}}-\frac {1}{4096 \left (a x -1\right )^{13} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/655360/a^3/(a*x+1)^10-1/98304/a^3/(a*x+1)^9-3/32768/a^3/(a*x+1)^7-289/1572864/(a*x+1)^6/a^3-51/163840/(a*x+
1)^5/a^3-969/2097152/(a*x+1)^4/a^3-323/524288/(a*x+1)^3/a^3-12597/16777216/(a*x+1)^2/a^3-3553/4194304/(a*x+1)/
a^3-19/524288/a^3/(a*x+1)^8-1/30720/a^3/(a*x-1)^15+1/8192/a^3/(a*x-1)^14+11/32768/a^3/(a*x-1)^12-11/32768/a^3/
(a*x-1)^11+143/655360/(a*x-1)^10/a^3-143/524288/(a*x-1)^8/a^3+143/262144/a^3/(a*x-1)^7-2431/3145728/(a*x-1)^6/
a^3+2431/2621440/(a*x-1)^5/a^3-4199/4194304/(a*x-1)^4/a^3+4199/4194304/(a*x-1)^3/a^3-15827/16777216/(a*x-1)^2/
a^3+3553/4194304/(a*x-1)/a^3-1/4096/a^3/(a*x-1)^13

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maxima [B]  time = 0.99, size = 197, normalized size = 7.04 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 88.87, size = 197, normalized size = 7.04 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 2.02, size = 204, normalized size = 7.29 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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