\(\int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx\) [1002]

Optimal result

Integrand size = 19, antiderivative size = 28 \[ \int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx=-\frac {1-3 a x}{24 a^3 (1-a x)^6 (1+a x)^3} \]

[Out]

1/24*(3*a*x-1)/a^3/(-a*x+1)^6/(a*x+1)^3

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} \[ \int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx=-\frac {1-3 a x}{24 a^3 (1-a x)^6 (a x+1)^3} \]

[In]

Int[x^2/((1 - a*x)^7*(1 + a*x)^4),x]

[Out]

-1/24*(1 - 3*a*x)/(a^3*(1 - a*x)^6*(1 + a*x)^3)

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 {1-3 a x}{24 a^3 (1-a x)^6 (1+a x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx=\frac {-1+3 a x}{24 a^3 (-1+a x)^6 (1+a x)^3} \]

[In]

Integrate[x^2/((1 - a*x)^7*(1 + a*x)^4),x]

[Out]

(-1 + 3*a*x)/(24*a^3*(-1 + a*x)^6*(1 + a*x)^3)

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

method	result	size
gosper	\(\frac {3 a x -1}{24 \left (a x -1\right )^{6} a^{3} \left (a x +1\right )^{3}}\)	\(26\)
risch	\(\frac {\frac {x}{8 a^{2}}-\frac {1}{24 a^{3}}}{\left (a x -1\right )^{6} \left (a x +1\right )^{3}}\)	\(28\)
norman	\(\frac {\frac {1}{3} x^{3}-\frac {1}{4} a \,x^{4}-\frac {1}{4} a^{2} x^{5}+\frac {1}{3} a^{3} x^{6}-\frac {1}{8} a^{5} x^{8}+\frac {1}{24} a^{6} x^{9}}{\left (a x -1\right )^{6} \left (a x +1\right )^{3}}\)	\(60\)
parallelrisch	\(\frac {a^{6} x^{9}-3 a^{5} x^{8}+8 a^{3} x^{6}-6 a^{2} x^{5}-6 a \,x^{4}+8 x^{3}}{24 \left (a x -1\right )^{6} \left (a x +1\right )^{3}}\)	\(60\)
default	\(\frac {1}{96 a^{3} \left (a x -1\right )^{6}}+\frac {1}{96 a^{3} \left (a x -1\right )^{3}}-\frac {5}{512 a^{3} \left (a x -1\right )^{2}}+\frac {1}{128 a^{3} \left (a x -1\right )}-\frac {1}{128 a^{3} \left (a x -1\right )^{4}}-\frac {1}{384 a^{3} \left (a x +1\right )^{3}}-\frac {3}{512 a^{3} \left (a x +1\right )^{2}}-\frac {1}{128 a^{3} \left (a x +1\right )}\)	\(98\)

[In]

int(x^2/(-a*x+1)^7/(a*x+1)^4,x,method=_RETURNVERBOSE)

[Out]

1/24*(3*a*x-1)/(a*x-1)^6/a^3/(a*x+1)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (25) = 50\).

Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx=\frac {3 \, a x - 1}{24 \, {\left (a^{12} x^{9} - 3 \, a^{11} x^{8} + 8 \, a^{9} x^{6} - 6 \, a^{8} x^{5} - 6 \, a^{7} x^{4} + 8 \, a^{6} x^{3} - 3 \, a^{4} x + a^{3}\right )}} \]

[In]

integrate(x^2/(-a*x+1)^7/(a*x+1)^4,x, algorithm="fricas")

[Out]

1/24*(3*a*x - 1)/(a^12*x^9 - 3*a^11*x^8 + 8*a^9*x^6 - 6*a^8*x^5 - 6*a^7*x^4 + 8*a^6*x^3 - 3*a^4*x + a^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).

Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx=- \frac {- 3 a x + 1}{24 a^{12} x^{9} - 72 a^{11} x^{8} + 192 a^{9} x^{6} - 144 a^{8} x^{5} - 144 a^{7} x^{4} + 192 a^{6} x^{3} - 72 a^{4} x + 24 a^{3}} \]

[In]

integrate(x**2/(-a*x+1)**7/(a*x+1)**4,x)

[Out]

-(-3*a*x + 1)/(24*a**12*x**9 - 72*a**11*x**8 + 192*a**9*x**6 - 144*a**8*x**5 - 144*a**7*x**4 + 192*a**6*x**3 -
72*a**4*x + 24*a**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (25) = 50\).

Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx=\frac {3 \, a x - 1}{24 \, {\left (a^{12} x^{9} - 3 \, a^{11} x^{8} + 8 \, a^{9} x^{6} - 6 \, a^{8} x^{5} - 6 \, a^{7} x^{4} + 8 \, a^{6} x^{3} - 3 \, a^{4} x + a^{3}\right )}} \]

[In]

integrate(x^2/(-a*x+1)^7/(a*x+1)^4,x, algorithm="maxima")

[Out]

1/24*(3*a*x - 1)/(a^12*x^9 - 3*a^11*x^8 + 8*a^9*x^6 - 6*a^8*x^5 - 6*a^7*x^4 + 8*a^6*x^3 - 3*a^4*x + a^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (25) = 50\).

Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.75 \[ \int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx=-\frac {12 \, a^{2} x^{2} + 33 \, a x + 25}{1536 \, {\left (a x + 1\right )}^{3} a^{3}} + \frac {12 \, a^{5} x^{5} - 75 \, a^{4} x^{4} + 196 \, a^{3} x^{3} - 270 \, a^{2} x^{2} + 192 \, a x - 39}{1536 \, {\left (a x - 1\right )}^{6} a^{3}} \]

[In]

integrate(x^2/(-a*x+1)^7/(a*x+1)^4,x, algorithm="giac")

[Out]

-1/1536*(12*a^2*x^2 + 33*a*x + 25)/((a*x + 1)^3*a^3) + 1/1536*(12*a^5*x^5 - 75*a^4*x^4 + 196*a^3*x^3 - 270*a^2
*x^2 + 192*a*x - 39)/((a*x - 1)^6*a^3)

Mupad [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {x^2}{(1-a x)^7 (1+a x)^4} \, dx=-\frac {\frac {x}{8\,a^2}-\frac {1}{24\,a^3}}{-a^9\,x^9+3\,a^8\,x^8-8\,a^6\,x^6+6\,a^5\,x^5+6\,a^4\,x^4-8\,a^3\,x^3+3\,a\,x-1} \]

[In]

int(-x^2/((a*x - 1)^7*(a*x + 1)^4),x)

[Out]

-(x/(8*a^2) - 1/(24*a^3))/(3*a*x - 8*a^3*x^3 + 6*a^4*x^4 + 6*a^5*x^5 - 8*a^6*x^6 + 3*a^8*x^8 - a^9*x^9 - 1)