3.1003 \(\int \frac{x^2}{(1-a x)^{11} (1+a x)^7} \, dx\)

Optimal. Leaf size=28 \[ -\frac{1-4 a x}{60 a^3 (1-a x)^{10} (a x+1)^6} \]

[Out]

-(1 - 4*a*x)/(60*a^3*(1 - a*x)^10*(1 + a*x)^6)

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Rubi [A]  time = 0.0050803, 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} \[ -\frac{1-4 a x}{60 a^3 (1-a x)^{10} (a x+1)^6} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((1 - a*x)^11*(1 + a*x)^7),x]

[Out]

-(1 - 4*a*x)/(60*a^3*(1 - a*x)^10*(1 + a*x)^6)

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)^{11} (1+a x)^7} \, dx &=-\frac{1-4 a x}{60 a^3 (1-a x)^{10} (1+a x)^6}\\ \end{align*}

Mathematica [A]  time = 0.0236483, size = 27, normalized size = 0.96 \[ \frac{4 a x-1}{60 a^3 (a x-1)^{10} (a x+1)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((1 - a*x)^11*(1 + a*x)^7),x]

[Out]

(-1 + 4*a*x)/(60*a^3*(-1 + a*x)^10*(1 + a*x)^6)

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Maple [B]  time = 0.016, size = 182, normalized size = 6.5 \begin{align*}{\frac{1}{1280\,{a}^{3} \left ( ax-1 \right ) ^{10}}}-{\frac{1}{768\,{a}^{3} \left ( ax-1 \right ) ^{9}}}-{\frac{7}{6144\,{a}^{3} \left ( ax-1 \right ) ^{6}}}+{\frac{21}{10240\,{a}^{3} \left ( ax-1 \right ) ^{5}}}-{\frac{21}{8192\,{a}^{3} \left ( ax-1 \right ) ^{4}}}+{\frac{11}{4096\,{a}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{165}{65536\,{a}^{3} \left ( ax-1 \right ) ^{2}}}+{\frac{143}{65536\,{a}^{3} \left ( ax-1 \right ) }}+{\frac{1}{1024\,{a}^{3} \left ( ax-1 \right ) ^{8}}}-{\frac{1}{12288\,{a}^{3} \left ( ax+1 \right ) ^{6}}}-{\frac{7}{20480\,{a}^{3} \left ( ax+1 \right ) ^{5}}}-{\frac{11}{8192\,{a}^{3} \left ( ax+1 \right ) ^{3}}}-{\frac{121}{65536\,{a}^{3} \left ( ax+1 \right ) ^{2}}}-{\frac{143}{65536\,{a}^{3} \left ( ax+1 \right ) }}-{\frac{13}{16384\,{a}^{3} \left ( ax+1 \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-a*x+1)^11/(a*x+1)^7,x)

[Out]

1/1280/a^3/(a*x-1)^10-1/768/a^3/(a*x-1)^9-7/6144/a^3/(a*x-1)^6+21/10240/a^3/(a*x-1)^5-21/8192/a^3/(a*x-1)^4+11
/4096/a^3/(a*x-1)^3-165/65536/a^3/(a*x-1)^2+143/65536/a^3/(a*x-1)+1/1024/a^3/(a*x-1)^8-1/12288/a^3/(a*x+1)^6-7
/20480/a^3/(a*x+1)^5-11/8192/a^3/(a*x+1)^3-121/65536/a^3/(a*x+1)^2-143/65536/a^3/(a*x+1)-13/16384/a^3/(a*x+1)^
4

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Maxima [B]  time = 1.18473, size = 166, normalized size = 5.93 \begin{align*} \frac{4 \, a x - 1}{60 \,{\left (a^{19} x^{16} - 4 \, a^{18} x^{15} + 20 \, a^{16} x^{13} - 20 \, a^{15} x^{12} - 36 \, a^{14} x^{11} + 64 \, a^{13} x^{10} + 20 \, a^{12} x^{9} - 90 \, a^{11} x^{8} + 20 \, a^{10} x^{7} + 64 \, a^{9} x^{6} - 36 \, a^{8} x^{5} - 20 \, a^{7} x^{4} + 20 \, a^{6} x^{3} - 4 \, 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)^11/(a*x+1)^7,x, algorithm="maxima")

[Out]

1/60*(4*a*x - 1)/(a^19*x^16 - 4*a^18*x^15 + 20*a^16*x^13 - 20*a^15*x^12 - 36*a^14*x^11 + 64*a^13*x^10 + 20*a^1
2*x^9 - 90*a^11*x^8 + 20*a^10*x^7 + 64*a^9*x^6 - 36*a^8*x^5 - 20*a^7*x^4 + 20*a^6*x^3 - 4*a^4*x + a^3)

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Fricas [B]  time = 2.10024, size = 289, normalized size = 10.32 \begin{align*} \frac{4 \, a x - 1}{60 \,{\left (a^{19} x^{16} - 4 \, a^{18} x^{15} + 20 \, a^{16} x^{13} - 20 \, a^{15} x^{12} - 36 \, a^{14} x^{11} + 64 \, a^{13} x^{10} + 20 \, a^{12} x^{9} - 90 \, a^{11} x^{8} + 20 \, a^{10} x^{7} + 64 \, a^{9} x^{6} - 36 \, a^{8} x^{5} - 20 \, a^{7} x^{4} + 20 \, a^{6} x^{3} - 4 \, 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)^11/(a*x+1)^7,x, algorithm="fricas")

[Out]

1/60*(4*a*x - 1)/(a^19*x^16 - 4*a^18*x^15 + 20*a^16*x^13 - 20*a^15*x^12 - 36*a^14*x^11 + 64*a^13*x^10 + 20*a^1
2*x^9 - 90*a^11*x^8 + 20*a^10*x^7 + 64*a^9*x^6 - 36*a^8*x^5 - 20*a^7*x^4 + 20*a^6*x^3 - 4*a^4*x + a^3)

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Sympy [B]  time = 106.773, size = 128, normalized size = 4.57 \begin{align*} \frac{4 a x - 1}{60 a^{19} x^{16} - 240 a^{18} x^{15} + 1200 a^{16} x^{13} - 1200 a^{15} x^{12} - 2160 a^{14} x^{11} + 3840 a^{13} x^{10} + 1200 a^{12} x^{9} - 5400 a^{11} x^{8} + 1200 a^{10} x^{7} + 3840 a^{9} x^{6} - 2160 a^{8} x^{5} - 1200 a^{7} x^{4} + 1200 a^{6} x^{3} - 240 a^{4} x + 60 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-a*x+1)**11/(a*x+1)**7,x)

[Out]

(4*a*x - 1)/(60*a**19*x**16 - 240*a**18*x**15 + 1200*a**16*x**13 - 1200*a**15*x**12 - 2160*a**14*x**11 + 3840*
a**13*x**10 + 1200*a**12*x**9 - 5400*a**11*x**8 + 1200*a**10*x**7 + 3840*a**9*x**6 - 2160*a**8*x**5 - 1200*a**
7*x**4 + 1200*a**6*x**3 - 240*a**4*x + 60*a**3)

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Giac [B]  time = 1.98472, size = 180, normalized size = 6.43 \begin{align*} -\frac{2145 \, a^{5} x^{5} + 12540 \, a^{4} x^{4} + 30030 \, a^{3} x^{3} + 37080 \, a^{2} x^{2} + 23841 \, a x + 6476}{983040 \,{\left (a x + 1\right )}^{6} a^{3}} + \frac{2145 \, a^{9} x^{9} - 21780 \, a^{8} x^{8} + 99660 \, a^{7} x^{7} - 270480 \, a^{6} x^{6} + 481446 \, a^{5} x^{5} - 584920 \, a^{4} x^{4} + 486220 \, a^{3} x^{3} - 265680 \, a^{2} x^{2} + 84065 \, a x - 9908}{983040 \,{\left (a x - 1\right )}^{10} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-a*x+1)^11/(a*x+1)^7,x, algorithm="giac")

[Out]

-1/983040*(2145*a^5*x^5 + 12540*a^4*x^4 + 30030*a^3*x^3 + 37080*a^2*x^2 + 23841*a*x + 6476)/((a*x + 1)^6*a^3)
+ 1/983040*(2145*a^9*x^9 - 21780*a^8*x^8 + 99660*a^7*x^7 - 270480*a^6*x^6 + 481446*a^5*x^5 - 584920*a^4*x^4 +
486220*a^3*x^3 - 265680*a^2*x^2 + 84065*a*x - 9908)/((a*x - 1)^10*a^3)