3.1002 \(\int \frac{x^2}{(1-a x)^7 (1+a x)^4} \, dx\)

Optimal. Leaf size=28 \[ -\frac{1-3 a x}{24 a^3 (1-a x)^6 (a x+1)^3} \]

[Out]

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

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Rubi [A]  time = 0.0045999, 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} \[ -\frac{1-3 a x}{24 a^3 (1-a x)^6 (a x+1)^3} \]

Antiderivative was successfully verified.

[In]

Int[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)

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)^7 (1+a x)^4} \, dx &=-\frac{1-3 a x}{24 a^3 (1-a x)^6 (1+a x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0170001, size = 27, normalized size = 0.96 \[ \frac{3 a x-1}{24 a^3 (a x-1)^6 (a x+1)^3} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.012, size = 98, normalized size = 3.5 \begin{align*}{\frac{1}{96\,{a}^{3} \left ( ax-1 \right ) ^{6}}}+{\frac{1}{96\,{a}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{5}{512\,{a}^{3} \left ( ax-1 \right ) ^{2}}}+{\frac{1}{128\,{a}^{3} \left ( ax-1 \right ) }}-{\frac{1}{128\,{a}^{3} \left ( ax-1 \right ) ^{4}}}-{\frac{1}{384\,{a}^{3} \left ( ax+1 \right ) ^{3}}}-{\frac{3}{512\,{a}^{3} \left ( ax+1 \right ) ^{2}}}-{\frac{1}{128\,{a}^{3} \left ( ax+1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96/a^3/(a*x-1)^6+1/96/a^3/(a*x-1)^3-5/512/a^3/(a*x-1)^2+1/128/a^3/(a*x-1)-1/128/a^3/(a*x-1)^4-1/384/a^3/(a*x
+1)^3-3/512/a^3/(a*x+1)^2-1/128/a^3/(a*x+1)

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Maxima [B]  time = 1.0581, size = 90, normalized size = 3.21 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.82889, size = 143, normalized size = 5.11 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 1.81195, size = 68, normalized size = 2.43 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 - 7
2*a**4*x + 24*a**3)

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Giac [B]  time = 1.85335, size = 104, normalized size = 3.71 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)