\(\int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx\) [1003]

Optimal result

Integrand size = 19, antiderivative size = 28 \[ \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx=-\frac {1-4 a x}{60 a^3 (1-a x)^{10} (1+a x)^6} \]

[Out]

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

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} \[ \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx=-\frac {1-4 a x}{60 a^3 (1-a x)^{10} (a x+1)^6} \]

[In]

Int[x^2/((1 - a*x)^11*(1 + a*x)^7),x]

[Out]

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

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 {1-4 a x}{60 a^3 (1-a x)^{10} (1+a x)^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx=\frac {-1+4 a x}{60 a^3 (-1+a x)^{10} (1+a x)^6} \]

[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)

Maple [A] (verified)

Time = 2.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

method	result	size
gosper	\(\frac {4 a x -1}{60 \left (a x -1\right )^{10} a^{3} \left (a x +1\right )^{6}}\)	\(26\)
risch	\(\frac {-\frac {1}{60 a^{3}}+\frac {x}{15 a^{2}}}{\left (a x -1\right )^{10} \left (a x +1\right )^{6}}\)	\(28\)
norman	\(\frac {\frac {1}{3} x^{3}-\frac {1}{3} a \,x^{4}-\frac {3}{5} a^{2} x^{5}+\frac {16}{15} a^{3} x^{6}+\frac {1}{3} a^{4} x^{7}-\frac {3}{2} a^{5} x^{8}+\frac {1}{3} a^{6} x^{9}+\frac {16}{15} a^{7} x^{10}-\frac {3}{5} a^{8} x^{11}-\frac {1}{3} a^{9} x^{12}+\frac {1}{3} a^{10} x^{13}-\frac {1}{15} a^{12} x^{15}+\frac {1}{60} a^{13} x^{16}}{\left (a x -1\right )^{10} \left (a x +1\right )^{6}}\)	\(116\)
parallelrisch	\(\frac {a^{13} x^{16}-4 a^{12} x^{15}+20 a^{10} x^{13}-20 a^{9} x^{12}-36 a^{8} x^{11}+64 a^{7} x^{10}+20 a^{6} x^{9}-90 a^{5} x^{8}+20 a^{4} x^{7}+64 a^{3} x^{6}-36 a^{2} x^{5}-20 a \,x^{4}+20 x^{3}}{60 \left (a x -1\right )^{10} \left (a x +1\right )^{6}}\)	\(116\)
default	\(\frac {1}{1280 a^{3} \left (a x -1\right )^{10}}-\frac {1}{768 a^{3} \left (a x -1\right )^{9}}-\frac {7}{6144 a^{3} \left (a x -1\right )^{6}}+\frac {21}{10240 a^{3} \left (a x -1\right )^{5}}-\frac {21}{8192 a^{3} \left (a x -1\right )^{4}}+\frac {11}{4096 a^{3} \left (a x -1\right )^{3}}-\frac {165}{65536 a^{3} \left (a x -1\right )^{2}}+\frac {143}{65536 a^{3} \left (a x -1\right )}+\frac {1}{1024 a^{3} \left (a x -1\right )^{8}}-\frac {1}{12288 a^{3} \left (a x +1\right )^{6}}-\frac {7}{20480 a^{3} \left (a x +1\right )^{5}}-\frac {11}{8192 a^{3} \left (a x +1\right )^{3}}-\frac {121}{65536 a^{3} \left (a x +1\right )^{2}}-\frac {143}{65536 a^{3} \left (a x +1\right )}-\frac {13}{16384 a^{3} \left (a x +1\right )^{4}}\)	\(182\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (25) = 50\).

Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.39 \[ \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx=\frac {4 \, a x - 1}{60 \, {\left (a^{19} x^{16} - 4 \, a^{18} x^{15} + 20 \, a^{16} x^{13} - 20 \, a^{15} x^{12} - 36 \, a^{14} x^{11} + 64 \, a^{13} x^{10} + 20 \, a^{12} x^{9} - 90 \, a^{11} x^{8} + 20 \, a^{10} x^{7} + 64 \, a^{9} x^{6} - 36 \, a^{8} x^{5} - 20 \, a^{7} x^{4} + 20 \, a^{6} x^{3} - 4 \, a^{4} x + a^{3}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (24) = 48\).

Time = 0.59 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.61 \[ \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx=- \frac {- 4 a x + 1}{60 a^{19} x^{16} - 240 a^{18} x^{15} + 1200 a^{16} x^{13} - 1200 a^{15} x^{12} - 2160 a^{14} x^{11} + 3840 a^{13} x^{10} + 1200 a^{12} x^{9} - 5400 a^{11} x^{8} + 1200 a^{10} x^{7} + 3840 a^{9} x^{6} - 2160 a^{8} x^{5} - 1200 a^{7} x^{4} + 1200 a^{6} x^{3} - 240 a^{4} x + 60 a^{3}} \]

[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 + 384
0*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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (25) = 50\).

Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.39 \[ \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx=\frac {4 \, a x - 1}{60 \, {\left (a^{19} x^{16} - 4 \, a^{18} x^{15} + 20 \, a^{16} x^{13} - 20 \, a^{15} x^{12} - 36 \, a^{14} x^{11} + 64 \, a^{13} x^{10} + 20 \, a^{12} x^{9} - 90 \, a^{11} x^{8} + 20 \, a^{10} x^{7} + 64 \, a^{9} x^{6} - 36 \, a^{8} x^{5} - 20 \, a^{7} x^{4} + 20 \, a^{6} x^{3} - 4 \, a^{4} x + a^{3}\right )}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (25) = 50\).

Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.75 \[ \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx=-\frac {2145 \, a^{5} x^{5} + 12540 \, a^{4} x^{4} + 30030 \, a^{3} x^{3} + 37080 \, a^{2} x^{2} + 23841 \, a x + 6476}{983040 \, {\left (a x + 1\right )}^{6} a^{3}} + \frac {2145 \, a^{9} x^{9} - 21780 \, a^{8} x^{8} + 99660 \, a^{7} x^{7} - 270480 \, a^{6} x^{6} + 481446 \, a^{5} x^{5} - 584920 \, a^{4} x^{4} + 486220 \, a^{3} x^{3} - 265680 \, a^{2} x^{2} + 84065 \, a x - 9908}{983040 \, {\left (a x - 1\right )}^{10} a^{3}} \]

[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)

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.43 \[ \int \frac {x^2}{(1-a x)^{11} (1+a x)^7} \, dx=\frac {\frac {x}{15\,a^2}-\frac {1}{60\,a^3}}{a^{16}\,x^{16}-4\,a^{15}\,x^{15}+20\,a^{13}\,x^{13}-20\,a^{12}\,x^{12}-36\,a^{11}\,x^{11}+64\,a^{10}\,x^{10}+20\,a^9\,x^9-90\,a^8\,x^8+20\,a^7\,x^7+64\,a^6\,x^6-36\,a^5\,x^5-20\,a^4\,x^4+20\,a^3\,x^3-4\,a\,x+1} \]

[In]

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

[Out]

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