3.1524 \(\int \frac {1}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=170 \[ -\frac {10 b^2 e^3 \log (a+b x)}{(b d-a e)^6}+\frac {10 b^2 e^3 \log (d+e x)}{(b d-a e)^6}-\frac {6 b^2 e^2}{(a+b x) (b d-a e)^5}+\frac {3 b^2 e}{2 (a+b x)^2 (b d-a e)^4}-\frac {b^2}{3 (a+b x)^3 (b d-a e)^3}-\frac {4 b e^3}{(d+e x) (b d-a e)^5}-\frac {e^3}{2 (d+e x)^2 (b d-a e)^4} \]

[Out]

-1/3*b^2/(-a*e+b*d)^3/(b*x+a)^3+3/2*b^2*e/(-a*e+b*d)^4/(b*x+a)^2-6*b^2*e^2/(-a*e+b*d)^5/(b*x+a)-1/2*e^3/(-a*e+
b*d)^4/(e*x+d)^2-4*b*e^3/(-a*e+b*d)^5/(e*x+d)-10*b^2*e^3*ln(b*x+a)/(-a*e+b*d)^6+10*b^2*e^3*ln(e*x+d)/(-a*e+b*d
)^6

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 44} \[ -\frac {6 b^2 e^2}{(a+b x) (b d-a e)^5}-\frac {10 b^2 e^3 \log (a+b x)}{(b d-a e)^6}+\frac {10 b^2 e^3 \log (d+e x)}{(b d-a e)^6}+\frac {3 b^2 e}{2 (a+b x)^2 (b d-a e)^4}-\frac {b^2}{3 (a+b x)^3 (b d-a e)^3}-\frac {4 b e^3}{(d+e x) (b d-a e)^5}-\frac {e^3}{2 (d+e x)^2 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-b^2/(3*(b*d - a*e)^3*(a + b*x)^3) + (3*b^2*e)/(2*(b*d - a*e)^4*(a + b*x)^2) - (6*b^2*e^2)/((b*d - a*e)^5*(a +
b*x)) - e^3/(2*(b*d - a*e)^4*(d + e*x)^2) - (4*b*e^3)/((b*d - a*e)^5*(d + e*x)) - (10*b^2*e^3*Log[a + b*x])/(
b*d - a*e)^6 + (10*b^2*e^3*Log[d + e*x])/(b*d - a*e)^6

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^4 (d+e x)^3} \, dx\\ &=\int \left (\frac {b^3}{(b d-a e)^3 (a+b x)^4}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)^3}+\frac {6 b^3 e^2}{(b d-a e)^5 (a+b x)^2}-\frac {10 b^3 e^3}{(b d-a e)^6 (a+b x)}+\frac {e^4}{(b d-a e)^4 (d+e x)^3}+\frac {4 b e^4}{(b d-a e)^5 (d+e x)^2}+\frac {10 b^2 e^4}{(b d-a e)^6 (d+e x)}\right ) \, dx\\ &=-\frac {b^2}{3 (b d-a e)^3 (a+b x)^3}+\frac {3 b^2 e}{2 (b d-a e)^4 (a+b x)^2}-\frac {6 b^2 e^2}{(b d-a e)^5 (a+b x)}-\frac {e^3}{2 (b d-a e)^4 (d+e x)^2}-\frac {4 b e^3}{(b d-a e)^5 (d+e x)}-\frac {10 b^2 e^3 \log (a+b x)}{(b d-a e)^6}+\frac {10 b^2 e^3 \log (d+e x)}{(b d-a e)^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 154, normalized size = 0.91 \[ -\frac {\frac {36 b^2 e^2 (b d-a e)}{a+b x}-\frac {9 b^2 e (b d-a e)^2}{(a+b x)^2}+\frac {2 b^2 (b d-a e)^3}{(a+b x)^3}+60 b^2 e^3 \log (a+b x)+\frac {24 b e^3 (b d-a e)}{d+e x}+\frac {3 e^3 (b d-a e)^2}{(d+e x)^2}-60 b^2 e^3 \log (d+e x)}{6 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-1/6*((2*b^2*(b*d - a*e)^3)/(a + b*x)^3 - (9*b^2*e*(b*d - a*e)^2)/(a + b*x)^2 + (36*b^2*e^2*(b*d - a*e))/(a +
b*x) + (3*e^3*(b*d - a*e)^2)/(d + e*x)^2 + (24*b*e^3*(b*d - a*e))/(d + e*x) + 60*b^2*e^3*Log[a + b*x] - 60*b^2
*e^3*Log[d + e*x])/(b*d - a*e)^6

________________________________________________________________________________________

fricas [B]  time = 0.73, size = 1151, normalized size = 6.77 \[ -\frac {2 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 30 \, a^{4} b d e^{4} + 3 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 30 \, {\left (3 \, b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 5 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} + 21 \, a b^{4} d^{2} e^{3} - 12 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} + 32 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} e^{5} x^{5} + a^{3} b^{2} d^{2} e^{3} + {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + {\left (3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{3} d^{2} e^{3} + 2 \, a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (b x + a\right ) - 60 \, {\left (b^{5} e^{5} x^{5} + a^{3} b^{2} d^{2} e^{3} + {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + {\left (3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{3} d^{2} e^{3} + 2 \, a^{3} b^{2} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (a^{3} b^{6} d^{8} - 6 \, a^{4} b^{5} d^{7} e + 15 \, a^{5} b^{4} d^{6} e^{2} - 20 \, a^{6} b^{3} d^{5} e^{3} + 15 \, a^{7} b^{2} d^{4} e^{4} - 6 \, a^{8} b d^{3} e^{5} + a^{9} d^{2} e^{6} + {\left (b^{9} d^{6} e^{2} - 6 \, a b^{8} d^{5} e^{3} + 15 \, a^{2} b^{7} d^{4} e^{4} - 20 \, a^{3} b^{6} d^{3} e^{5} + 15 \, a^{4} b^{5} d^{2} e^{6} - 6 \, a^{5} b^{4} d e^{7} + a^{6} b^{3} e^{8}\right )} x^{5} + {\left (2 \, b^{9} d^{7} e - 9 \, a b^{8} d^{6} e^{2} + 12 \, a^{2} b^{7} d^{5} e^{3} + 5 \, a^{3} b^{6} d^{4} e^{4} - 30 \, a^{4} b^{5} d^{3} e^{5} + 33 \, a^{5} b^{4} d^{2} e^{6} - 16 \, a^{6} b^{3} d e^{7} + 3 \, a^{7} b^{2} e^{8}\right )} x^{4} + {\left (b^{9} d^{8} - 18 \, a^{2} b^{7} d^{6} e^{2} + 52 \, a^{3} b^{6} d^{5} e^{3} - 60 \, a^{4} b^{5} d^{4} e^{4} + 24 \, a^{5} b^{4} d^{3} e^{5} + 10 \, a^{6} b^{3} d^{2} e^{6} - 12 \, a^{7} b^{2} d e^{7} + 3 \, a^{8} b e^{8}\right )} x^{3} + {\left (3 \, a b^{8} d^{8} - 12 \, a^{2} b^{7} d^{7} e + 10 \, a^{3} b^{6} d^{6} e^{2} + 24 \, a^{4} b^{5} d^{5} e^{3} - 60 \, a^{5} b^{4} d^{4} e^{4} + 52 \, a^{6} b^{3} d^{3} e^{5} - 18 \, a^{7} b^{2} d^{2} e^{6} + a^{9} e^{8}\right )} x^{2} + {\left (3 \, a^{2} b^{7} d^{8} - 16 \, a^{3} b^{6} d^{7} e + 33 \, a^{4} b^{5} d^{6} e^{2} - 30 \, a^{5} b^{4} d^{5} e^{3} + 5 \, a^{6} b^{3} d^{4} e^{4} + 12 \, a^{7} b^{2} d^{3} e^{5} - 9 \, a^{8} b d^{2} e^{6} + 2 \, a^{9} d e^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

-1/6*(2*b^5*d^5 - 15*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 30*a^4*b*d*e^4 + 3*a^5*e^5 + 60*(
b^5*d*e^4 - a*b^4*e^5)*x^4 + 30*(3*b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 5*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 + 21*a
*b^4*d^2*e^3 - 12*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 12*a*b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3 +
32*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x + 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + (b^
5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + (3*a^
2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*log(b*x + a) - 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*
e^5)*x^4 + (b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^
5)*x^2 + (3*a^2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*log(e*x + d))/(a^3*b^6*d^8 - 6*a^4*b^5*d^7*e + 15*a^5*b^4*d^
6*e^2 - 20*a^6*b^3*d^5*e^3 + 15*a^7*b^2*d^4*e^4 - 6*a^8*b*d^3*e^5 + a^9*d^2*e^6 + (b^9*d^6*e^2 - 6*a*b^8*d^5*e
^3 + 15*a^2*b^7*d^4*e^4 - 20*a^3*b^6*d^3*e^5 + 15*a^4*b^5*d^2*e^6 - 6*a^5*b^4*d*e^7 + a^6*b^3*e^8)*x^5 + (2*b^
9*d^7*e - 9*a*b^8*d^6*e^2 + 12*a^2*b^7*d^5*e^3 + 5*a^3*b^6*d^4*e^4 - 30*a^4*b^5*d^3*e^5 + 33*a^5*b^4*d^2*e^6 -
16*a^6*b^3*d*e^7 + 3*a^7*b^2*e^8)*x^4 + (b^9*d^8 - 18*a^2*b^7*d^6*e^2 + 52*a^3*b^6*d^5*e^3 - 60*a^4*b^5*d^4*e
^4 + 24*a^5*b^4*d^3*e^5 + 10*a^6*b^3*d^2*e^6 - 12*a^7*b^2*d*e^7 + 3*a^8*b*e^8)*x^3 + (3*a*b^8*d^8 - 12*a^2*b^7
*d^7*e + 10*a^3*b^6*d^6*e^2 + 24*a^4*b^5*d^5*e^3 - 60*a^5*b^4*d^4*e^4 + 52*a^6*b^3*d^3*e^5 - 18*a^7*b^2*d^2*e^
6 + a^9*e^8)*x^2 + (3*a^2*b^7*d^8 - 16*a^3*b^6*d^7*e + 33*a^4*b^5*d^6*e^2 - 30*a^5*b^4*d^5*e^3 + 5*a^6*b^3*d^4
*e^4 + 12*a^7*b^2*d^3*e^5 - 9*a^8*b*d^2*e^6 + 2*a^9*d*e^7)*x)

________________________________________________________________________________________

giac [B]  time = 0.18, size = 435, normalized size = 2.56 \[ -\frac {10 \, b^{3} e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac {10 \, b^{2} e^{4} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac {2 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 30 \, a^{4} b d e^{4} + 3 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 30 \, {\left (3 \, b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 5 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} + 21 \, a b^{4} d^{2} e^{3} - 12 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} + 32 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x}{6 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{3} {\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

-10*b^3*e^3*log(abs(b*x + a))/(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*
d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6) + 10*b^2*e^4*log(abs(x*e + d))/(b^6*d^6*e - 6*a*b^5*d^5*e^2 + 15*a^2*b^
4*d^4*e^3 - 20*a^3*b^3*d^3*e^4 + 15*a^4*b^2*d^2*e^5 - 6*a^5*b*d*e^6 + a^6*e^7) - 1/6*(2*b^5*d^5 - 15*a*b^4*d^4
*e + 60*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 30*a^4*b*d*e^4 + 3*a^5*e^5 + 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 3
0*(3*b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 5*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 + 21*a*b^4*d^2*e^3 - 12*a^2*b^3*d*e^
4 - 11*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 12*a*b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3 + 32*a^3*b^2*d*e^4 + 3*a^4*b*e^
5)*x)/((b*d - a*e)^6*(b*x + a)^3*(x*e + d)^2)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 165, normalized size = 0.97 \[ -\frac {10 b^{2} e^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}+\frac {10 b^{2} e^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}+\frac {6 b^{2} e^{2}}{\left (a e -b d \right )^{5} \left (b x +a \right )}+\frac {4 b \,e^{3}}{\left (a e -b d \right )^{5} \left (e x +d \right )}+\frac {3 b^{2} e}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}-\frac {e^{3}}{2 \left (a e -b d \right )^{4} \left (e x +d \right )^{2}}+\frac {b^{2}}{3 \left (a e -b d \right )^{3} \left (b x +a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/3*b^2/(a*e-b*d)^3/(b*x+a)^3-10*b^2/(a*e-b*d)^6*e^3*ln(b*x+a)+6*b^2/(a*e-b*d)^5*e^2/(b*x+a)+3/2*b^2/(a*e-b*d)
^4*e/(b*x+a)^2-1/2*e^3/(a*e-b*d)^4/(e*x+d)^2+10*b^2/(a*e-b*d)^6*e^3*ln(e*x+d)+4*e^3/(a*e-b*d)^5*b/(e*x+d)

________________________________________________________________________________________

maxima [B]  time = 2.25, size = 889, normalized size = 5.23 \[ -\frac {10 \, b^{2} e^{3} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {10 \, b^{2} e^{3} \log \left (e x + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} - \frac {60 \, b^{4} e^{4} x^{4} + 2 \, b^{4} d^{4} - 13 \, a b^{3} d^{3} e + 47 \, a^{2} b^{2} d^{2} e^{2} + 27 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 30 \, {\left (3 \, b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (2 \, b^{4} d^{2} e^{2} + 23 \, a b^{3} d e^{3} + 11 \, a^{2} b^{2} e^{4}\right )} x^{2} - 5 \, {\left (b^{4} d^{3} e - 11 \, a b^{3} d^{2} e^{2} - 35 \, a^{2} b^{2} d e^{3} - 3 \, a^{3} b e^{4}\right )} x}{6 \, {\left (a^{3} b^{5} d^{7} - 5 \, a^{4} b^{4} d^{6} e + 10 \, a^{5} b^{3} d^{5} e^{2} - 10 \, a^{6} b^{2} d^{4} e^{3} + 5 \, a^{7} b d^{3} e^{4} - a^{8} d^{2} e^{5} + {\left (b^{8} d^{5} e^{2} - 5 \, a b^{7} d^{4} e^{3} + 10 \, a^{2} b^{6} d^{3} e^{4} - 10 \, a^{3} b^{5} d^{2} e^{5} + 5 \, a^{4} b^{4} d e^{6} - a^{5} b^{3} e^{7}\right )} x^{5} + {\left (2 \, b^{8} d^{6} e - 7 \, a b^{7} d^{5} e^{2} + 5 \, a^{2} b^{6} d^{4} e^{3} + 10 \, a^{3} b^{5} d^{3} e^{4} - 20 \, a^{4} b^{4} d^{2} e^{5} + 13 \, a^{5} b^{3} d e^{6} - 3 \, a^{6} b^{2} e^{7}\right )} x^{4} + {\left (b^{8} d^{7} + a b^{7} d^{6} e - 17 \, a^{2} b^{6} d^{5} e^{2} + 35 \, a^{3} b^{5} d^{4} e^{3} - 25 \, a^{4} b^{4} d^{3} e^{4} - a^{5} b^{3} d^{2} e^{5} + 9 \, a^{6} b^{2} d e^{6} - 3 \, a^{7} b e^{7}\right )} x^{3} + {\left (3 \, a b^{7} d^{7} - 9 \, a^{2} b^{6} d^{6} e + a^{3} b^{5} d^{5} e^{2} + 25 \, a^{4} b^{4} d^{4} e^{3} - 35 \, a^{5} b^{3} d^{3} e^{4} + 17 \, a^{6} b^{2} d^{2} e^{5} - a^{7} b d e^{6} - a^{8} e^{7}\right )} x^{2} + {\left (3 \, a^{2} b^{6} d^{7} - 13 \, a^{3} b^{5} d^{6} e + 20 \, a^{4} b^{4} d^{5} e^{2} - 10 \, a^{5} b^{3} d^{4} e^{3} - 5 \, a^{6} b^{2} d^{3} e^{4} + 7 \, a^{7} b d^{2} e^{5} - 2 \, a^{8} d e^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

-10*b^2*e^3*log(b*x + a)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e
^4 - 6*a^5*b*d*e^5 + a^6*e^6) + 10*b^2*e^3*log(e*x + d)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3
*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) - 1/6*(60*b^4*e^4*x^4 + 2*b^4*d^4 - 13*a*b^3*d^3*
e + 47*a^2*b^2*d^2*e^2 + 27*a^3*b*d*e^3 - 3*a^4*e^4 + 30*(3*b^4*d*e^3 + 5*a*b^3*e^4)*x^3 + 10*(2*b^4*d^2*e^2 +
23*a*b^3*d*e^3 + 11*a^2*b^2*e^4)*x^2 - 5*(b^4*d^3*e - 11*a*b^3*d^2*e^2 - 35*a^2*b^2*d*e^3 - 3*a^3*b*e^4)*x)/(
a^3*b^5*d^7 - 5*a^4*b^4*d^6*e + 10*a^5*b^3*d^5*e^2 - 10*a^6*b^2*d^4*e^3 + 5*a^7*b*d^3*e^4 - a^8*d^2*e^5 + (b^8
*d^5*e^2 - 5*a*b^7*d^4*e^3 + 10*a^2*b^6*d^3*e^4 - 10*a^3*b^5*d^2*e^5 + 5*a^4*b^4*d*e^6 - a^5*b^3*e^7)*x^5 + (2
*b^8*d^6*e - 7*a*b^7*d^5*e^2 + 5*a^2*b^6*d^4*e^3 + 10*a^3*b^5*d^3*e^4 - 20*a^4*b^4*d^2*e^5 + 13*a^5*b^3*d*e^6
- 3*a^6*b^2*e^7)*x^4 + (b^8*d^7 + a*b^7*d^6*e - 17*a^2*b^6*d^5*e^2 + 35*a^3*b^5*d^4*e^3 - 25*a^4*b^4*d^3*e^4 -
a^5*b^3*d^2*e^5 + 9*a^6*b^2*d*e^6 - 3*a^7*b*e^7)*x^3 + (3*a*b^7*d^7 - 9*a^2*b^6*d^6*e + a^3*b^5*d^5*e^2 + 25*
a^4*b^4*d^4*e^3 - 35*a^5*b^3*d^3*e^4 + 17*a^6*b^2*d^2*e^5 - a^7*b*d*e^6 - a^8*e^7)*x^2 + (3*a^2*b^6*d^7 - 13*a
^3*b^5*d^6*e + 20*a^4*b^4*d^5*e^2 - 10*a^5*b^3*d^4*e^3 - 5*a^6*b^2*d^3*e^4 + 7*a^7*b*d^2*e^5 - 2*a^8*d*e^6)*x)

________________________________________________________________________________________

mupad [B]  time = 1.00, size = 797, normalized size = 4.69 \[ \frac {\frac {-3\,a^4\,e^4+27\,a^3\,b\,d\,e^3+47\,a^2\,b^2\,d^2\,e^2-13\,a\,b^3\,d^3\,e+2\,b^4\,d^4}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e\,x\,\left (3\,a^3\,b\,e^3+35\,a^2\,b^2\,d\,e^2+11\,a\,b^3\,d^2\,e-b^4\,d^3\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {10\,b^4\,e^4\,x^4}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {5\,e^2\,x^2\,\left (11\,a^2\,b^2\,e^2+23\,a\,b^3\,d\,e+2\,b^4\,d^2\right )}{3\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,b\,e^2\,x^3\,\left (3\,d\,b^3\,e+5\,a\,b^2\,e^2\right )}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}}{x^2\,\left (a^3\,e^2+6\,a^2\,b\,d\,e+3\,a\,b^2\,d^2\right )+x^3\,\left (3\,a^2\,b\,e^2+6\,a\,b^2\,d\,e+b^3\,d^2\right )+x\,\left (2\,e\,a^3\,d+3\,b\,a^2\,d^2\right )+x^4\,\left (2\,d\,b^3\,e+3\,a\,b^2\,e^2\right )+a^3\,d^2+b^3\,e^2\,x^5}-\frac {20\,b^2\,e^3\,\mathrm {atanh}\left (\frac {a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6}{{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,e\,x\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^6}\right )}{{\left (a\,e-b\,d\right )}^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^2),x)

[Out]

((2*b^4*d^4 - 3*a^4*e^4 + 47*a^2*b^2*d^2*e^2 - 13*a*b^3*d^3*e + 27*a^3*b*d*e^3)/(6*(a^5*e^5 - b^5*d^5 - 10*a^2
*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)) + (5*e*x*(3*a^3*b*e^3 - b^4*d^3 + 35*a^2*b
^2*d*e^2 + 11*a*b^3*d^2*e))/(6*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e -
5*a^4*b*d*e^4)) + (10*b^4*e^4*x^4)/(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*
e - 5*a^4*b*d*e^4) + (5*e^2*x^2*(2*b^4*d^2 + 11*a^2*b^2*e^2 + 23*a*b^3*d*e))/(3*(a^5*e^5 - b^5*d^5 - 10*a^2*b^
3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)) + (5*b*e^2*x^3*(5*a*b^2*e^2 + 3*b^3*d*e))/(a^
5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4))/(x^2*(a^3*e^2 + 3*
a*b^2*d^2 + 6*a^2*b*d*e) + x^3*(b^3*d^2 + 3*a^2*b*e^2 + 6*a*b^2*d*e) + x*(3*a^2*b*d^2 + 2*a^3*d*e) + x^4*(3*a*
b^2*e^2 + 2*b^3*d*e) + a^3*d^2 + b^3*e^2*x^5) - (20*b^2*e^3*atanh((a^6*e^6 - b^6*d^6 - 5*a^2*b^4*d^4*e^2 + 5*a
^4*b^2*d^2*e^4 + 4*a*b^5*d^5*e - 4*a^5*b*d*e^5)/(a*e - b*d)^6 + (2*b*e*x*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e
^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4))/(a*e - b*d)^6))/(a*e - b*d)^6

________________________________________________________________________________________

sympy [B]  time = 3.53, size = 1217, normalized size = 7.16 \[ \frac {10 b^{2} e^{3} \log {\left (x + \frac {- \frac {10 a^{7} b^{2} e^{10}}{\left (a e - b d\right )^{6}} + \frac {70 a^{6} b^{3} d e^{9}}{\left (a e - b d\right )^{6}} - \frac {210 a^{5} b^{4} d^{2} e^{8}}{\left (a e - b d\right )^{6}} + \frac {350 a^{4} b^{5} d^{3} e^{7}}{\left (a e - b d\right )^{6}} - \frac {350 a^{3} b^{6} d^{4} e^{6}}{\left (a e - b d\right )^{6}} + \frac {210 a^{2} b^{7} d^{5} e^{5}}{\left (a e - b d\right )^{6}} - \frac {70 a b^{8} d^{6} e^{4}}{\left (a e - b d\right )^{6}} + 10 a b^{2} e^{4} + \frac {10 b^{9} d^{7} e^{3}}{\left (a e - b d\right )^{6}} + 10 b^{3} d e^{3}}{20 b^{3} e^{4}} \right )}}{\left (a e - b d\right )^{6}} - \frac {10 b^{2} e^{3} \log {\left (x + \frac {\frac {10 a^{7} b^{2} e^{10}}{\left (a e - b d\right )^{6}} - \frac {70 a^{6} b^{3} d e^{9}}{\left (a e - b d\right )^{6}} + \frac {210 a^{5} b^{4} d^{2} e^{8}}{\left (a e - b d\right )^{6}} - \frac {350 a^{4} b^{5} d^{3} e^{7}}{\left (a e - b d\right )^{6}} + \frac {350 a^{3} b^{6} d^{4} e^{6}}{\left (a e - b d\right )^{6}} - \frac {210 a^{2} b^{7} d^{5} e^{5}}{\left (a e - b d\right )^{6}} + \frac {70 a b^{8} d^{6} e^{4}}{\left (a e - b d\right )^{6}} + 10 a b^{2} e^{4} - \frac {10 b^{9} d^{7} e^{3}}{\left (a e - b d\right )^{6}} + 10 b^{3} d e^{3}}{20 b^{3} e^{4}} \right )}}{\left (a e - b d\right )^{6}} + \frac {- 3 a^{4} e^{4} + 27 a^{3} b d e^{3} + 47 a^{2} b^{2} d^{2} e^{2} - 13 a b^{3} d^{3} e + 2 b^{4} d^{4} + 60 b^{4} e^{4} x^{4} + x^{3} \left (150 a b^{3} e^{4} + 90 b^{4} d e^{3}\right ) + x^{2} \left (110 a^{2} b^{2} e^{4} + 230 a b^{3} d e^{3} + 20 b^{4} d^{2} e^{2}\right ) + x \left (15 a^{3} b e^{4} + 175 a^{2} b^{2} d e^{3} + 55 a b^{3} d^{2} e^{2} - 5 b^{4} d^{3} e\right )}{6 a^{8} d^{2} e^{5} - 30 a^{7} b d^{3} e^{4} + 60 a^{6} b^{2} d^{4} e^{3} - 60 a^{5} b^{3} d^{5} e^{2} + 30 a^{4} b^{4} d^{6} e - 6 a^{3} b^{5} d^{7} + x^{5} \left (6 a^{5} b^{3} e^{7} - 30 a^{4} b^{4} d e^{6} + 60 a^{3} b^{5} d^{2} e^{5} - 60 a^{2} b^{6} d^{3} e^{4} + 30 a b^{7} d^{4} e^{3} - 6 b^{8} d^{5} e^{2}\right ) + x^{4} \left (18 a^{6} b^{2} e^{7} - 78 a^{5} b^{3} d e^{6} + 120 a^{4} b^{4} d^{2} e^{5} - 60 a^{3} b^{5} d^{3} e^{4} - 30 a^{2} b^{6} d^{4} e^{3} + 42 a b^{7} d^{5} e^{2} - 12 b^{8} d^{6} e\right ) + x^{3} \left (18 a^{7} b e^{7} - 54 a^{6} b^{2} d e^{6} + 6 a^{5} b^{3} d^{2} e^{5} + 150 a^{4} b^{4} d^{3} e^{4} - 210 a^{3} b^{5} d^{4} e^{3} + 102 a^{2} b^{6} d^{5} e^{2} - 6 a b^{7} d^{6} e - 6 b^{8} d^{7}\right ) + x^{2} \left (6 a^{8} e^{7} + 6 a^{7} b d e^{6} - 102 a^{6} b^{2} d^{2} e^{5} + 210 a^{5} b^{3} d^{3} e^{4} - 150 a^{4} b^{4} d^{4} e^{3} - 6 a^{3} b^{5} d^{5} e^{2} + 54 a^{2} b^{6} d^{6} e - 18 a b^{7} d^{7}\right ) + x \left (12 a^{8} d e^{6} - 42 a^{7} b d^{2} e^{5} + 30 a^{6} b^{2} d^{3} e^{4} + 60 a^{5} b^{3} d^{4} e^{3} - 120 a^{4} b^{4} d^{5} e^{2} + 78 a^{3} b^{5} d^{6} e - 18 a^{2} b^{6} d^{7}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

10*b**2*e**3*log(x + (-10*a**7*b**2*e**10/(a*e - b*d)**6 + 70*a**6*b**3*d*e**9/(a*e - b*d)**6 - 210*a**5*b**4*
d**2*e**8/(a*e - b*d)**6 + 350*a**4*b**5*d**3*e**7/(a*e - b*d)**6 - 350*a**3*b**6*d**4*e**6/(a*e - b*d)**6 + 2
10*a**2*b**7*d**5*e**5/(a*e - b*d)**6 - 70*a*b**8*d**6*e**4/(a*e - b*d)**6 + 10*a*b**2*e**4 + 10*b**9*d**7*e**
3/(a*e - b*d)**6 + 10*b**3*d*e**3)/(20*b**3*e**4))/(a*e - b*d)**6 - 10*b**2*e**3*log(x + (10*a**7*b**2*e**10/(
a*e - b*d)**6 - 70*a**6*b**3*d*e**9/(a*e - b*d)**6 + 210*a**5*b**4*d**2*e**8/(a*e - b*d)**6 - 350*a**4*b**5*d*
*3*e**7/(a*e - b*d)**6 + 350*a**3*b**6*d**4*e**6/(a*e - b*d)**6 - 210*a**2*b**7*d**5*e**5/(a*e - b*d)**6 + 70*
a*b**8*d**6*e**4/(a*e - b*d)**6 + 10*a*b**2*e**4 - 10*b**9*d**7*e**3/(a*e - b*d)**6 + 10*b**3*d*e**3)/(20*b**3
*e**4))/(a*e - b*d)**6 + (-3*a**4*e**4 + 27*a**3*b*d*e**3 + 47*a**2*b**2*d**2*e**2 - 13*a*b**3*d**3*e + 2*b**4
*d**4 + 60*b**4*e**4*x**4 + x**3*(150*a*b**3*e**4 + 90*b**4*d*e**3) + x**2*(110*a**2*b**2*e**4 + 230*a*b**3*d*
e**3 + 20*b**4*d**2*e**2) + x*(15*a**3*b*e**4 + 175*a**2*b**2*d*e**3 + 55*a*b**3*d**2*e**2 - 5*b**4*d**3*e))/(
6*a**8*d**2*e**5 - 30*a**7*b*d**3*e**4 + 60*a**6*b**2*d**4*e**3 - 60*a**5*b**3*d**5*e**2 + 30*a**4*b**4*d**6*e
- 6*a**3*b**5*d**7 + x**5*(6*a**5*b**3*e**7 - 30*a**4*b**4*d*e**6 + 60*a**3*b**5*d**2*e**5 - 60*a**2*b**6*d**
3*e**4 + 30*a*b**7*d**4*e**3 - 6*b**8*d**5*e**2) + x**4*(18*a**6*b**2*e**7 - 78*a**5*b**3*d*e**6 + 120*a**4*b*
*4*d**2*e**5 - 60*a**3*b**5*d**3*e**4 - 30*a**2*b**6*d**4*e**3 + 42*a*b**7*d**5*e**2 - 12*b**8*d**6*e) + x**3*
(18*a**7*b*e**7 - 54*a**6*b**2*d*e**6 + 6*a**5*b**3*d**2*e**5 + 150*a**4*b**4*d**3*e**4 - 210*a**3*b**5*d**4*e
**3 + 102*a**2*b**6*d**5*e**2 - 6*a*b**7*d**6*e - 6*b**8*d**7) + x**2*(6*a**8*e**7 + 6*a**7*b*d*e**6 - 102*a**
6*b**2*d**2*e**5 + 210*a**5*b**3*d**3*e**4 - 150*a**4*b**4*d**4*e**3 - 6*a**3*b**5*d**5*e**2 + 54*a**2*b**6*d*
*6*e - 18*a*b**7*d**7) + x*(12*a**8*d*e**6 - 42*a**7*b*d**2*e**5 + 30*a**6*b**2*d**3*e**4 + 60*a**5*b**3*d**4*
e**3 - 120*a**4*b**4*d**5*e**2 + 78*a**3*b**5*d**6*e - 18*a**2*b**6*d**7))

________________________________________________________________________________________