3.1004 \(\int \frac{x^2}{(1-a x)^{16} (1+a x)^{11}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0049202, antiderivative size = 28, normalized size of antiderivative = 1., 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} \[ -\frac{1-5 a x}{120 a^3 (1-a x)^{15} (a x+1)^{10}} \]

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*}

Mathematica [A]  time = 0.0455704, size = 27, normalized size = 0.96 \[ \frac{1-5 a x}{120 a^3 (a x-1)^{15} (a x+1)^{10}} \]

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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Maple [B]  time = 0.023, size = 290, normalized size = 10.4 \begin{align*} -{\frac{1}{30720\,{a}^{3} \left ( ax-1 \right ) ^{15}}}+{\frac{1}{8192\,{a}^{3} \left ( ax-1 \right ) ^{14}}}+{\frac{11}{32768\,{a}^{3} \left ( ax-1 \right ) ^{12}}}-{\frac{11}{32768\,{a}^{3} \left ( ax-1 \right ) ^{11}}}+{\frac{143}{655360\,{a}^{3} \left ( ax-1 \right ) ^{10}}}-{\frac{143}{524288\,{a}^{3} \left ( ax-1 \right ) ^{8}}}+{\frac{143}{262144\,{a}^{3} \left ( ax-1 \right ) ^{7}}}-{\frac{2431}{3145728\,{a}^{3} \left ( ax-1 \right ) ^{6}}}+{\frac{2431}{2621440\,{a}^{3} \left ( ax-1 \right ) ^{5}}}-{\frac{4199}{4194304\,{a}^{3} \left ( ax-1 \right ) ^{4}}}+{\frac{4199}{4194304\,{a}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{15827}{16777216\,{a}^{3} \left ( ax-1 \right ) ^{2}}}+{\frac{3553}{4194304\,{a}^{3} \left ( ax-1 \right ) }}-{\frac{1}{4096\,{a}^{3} \left ( ax-1 \right ) ^{13}}}-{\frac{1}{655360\,{a}^{3} \left ( ax+1 \right ) ^{10}}}-{\frac{1}{98304\,{a}^{3} \left ( ax+1 \right ) ^{9}}}-{\frac{3}{32768\,{a}^{3} \left ( ax+1 \right ) ^{7}}}-{\frac{289}{1572864\,{a}^{3} \left ( ax+1 \right ) ^{6}}}-{\frac{51}{163840\,{a}^{3} \left ( ax+1 \right ) ^{5}}}-{\frac{969}{2097152\,{a}^{3} \left ( ax+1 \right ) ^{4}}}-{\frac{323}{524288\,{a}^{3} \left ( ax+1 \right ) ^{3}}}-{\frac{12597}{16777216\,{a}^{3} \left ( ax+1 \right ) ^{2}}}-{\frac{3553}{4194304\,{a}^{3} \left ( ax+1 \right ) }}-{\frac{19}{524288\,{a}^{3} \left ( ax+1 \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.48498, size = 266, normalized size = 9.5 \begin{align*} -\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{align*}

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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Fricas [B]  time = 2.64434, size = 501, normalized size = 17.89 \begin{align*} -\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{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 2.73451, size = 277, normalized size = 9.89 \begin{align*} -\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{align*}

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)