3.1536 \(\int \frac {1}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=220 \[ -\frac {21 b^2 e^5 \log (a+b x)}{(b d-a e)^8}+\frac {21 b^2 e^5 \log (d+e x)}{(b d-a e)^8}-\frac {15 b^2 e^4}{(a+b x) (b d-a e)^7}+\frac {5 b^2 e^3}{(a+b x)^2 (b d-a e)^6}-\frac {2 b^2 e^2}{(a+b x)^3 (b d-a e)^5}+\frac {3 b^2 e}{4 (a+b x)^4 (b d-a e)^4}-\frac {b^2}{5 (a+b x)^5 (b d-a e)^3}-\frac {6 b e^5}{(d+e x) (b d-a e)^7}-\frac {e^5}{2 (d+e x)^2 (b d-a e)^6} \]

[Out]

-1/5*b^2/(-a*e+b*d)^3/(b*x+a)^5+3/4*b^2*e/(-a*e+b*d)^4/(b*x+a)^4-2*b^2*e^2/(-a*e+b*d)^5/(b*x+a)^3+5*b^2*e^3/(-
a*e+b*d)^6/(b*x+a)^2-15*b^2*e^4/(-a*e+b*d)^7/(b*x+a)-1/2*e^5/(-a*e+b*d)^6/(e*x+d)^2-6*b*e^5/(-a*e+b*d)^7/(e*x+
d)-21*b^2*e^5*ln(b*x+a)/(-a*e+b*d)^8+21*b^2*e^5*ln(e*x+d)/(-a*e+b*d)^8

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Rubi [A]  time = 0.25, antiderivative size = 220, 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} \[ -\frac {15 b^2 e^4}{(a+b x) (b d-a e)^7}+\frac {5 b^2 e^3}{(a+b x)^2 (b d-a e)^6}-\frac {2 b^2 e^2}{(a+b x)^3 (b d-a e)^5}-\frac {21 b^2 e^5 \log (a+b x)}{(b d-a e)^8}+\frac {21 b^2 e^5 \log (d+e x)}{(b d-a e)^8}+\frac {3 b^2 e}{4 (a+b x)^4 (b d-a e)^4}-\frac {b^2}{5 (a+b x)^5 (b d-a e)^3}-\frac {6 b e^5}{(d+e x) (b d-a e)^7}-\frac {e^5}{2 (d+e x)^2 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^2/(5*(b*d - a*e)^3*(a + b*x)^5) + (3*b^2*e)/(4*(b*d - a*e)^4*(a + b*x)^4) - (2*b^2*e^2)/((b*d - a*e)^5*(a +
b*x)^3) + (5*b^2*e^3)/((b*d - a*e)^6*(a + b*x)^2) - (15*b^2*e^4)/((b*d - a*e)^7*(a + b*x)) - e^5/(2*(b*d - a*
e)^6*(d + e*x)^2) - (6*b*e^5)/((b*d - a*e)^7*(d + e*x)) - (21*b^2*e^5*Log[a + b*x])/(b*d - a*e)^8 + (21*b^2*e^
5*Log[d + e*x])/(b*d - a*e)^8

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 )^3} \, dx &=\int \frac {1}{(a+b x)^6 (d+e x)^3} \, dx\\ &=\int \left (\frac {b^3}{(b d-a e)^3 (a+b x)^6}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)^5}+\frac {6 b^3 e^2}{(b d-a e)^5 (a+b x)^4}-\frac {10 b^3 e^3}{(b d-a e)^6 (a+b x)^3}+\frac {15 b^3 e^4}{(b d-a e)^7 (a+b x)^2}-\frac {21 b^3 e^5}{(b d-a e)^8 (a+b x)}+\frac {e^6}{(b d-a e)^6 (d+e x)^3}+\frac {6 b e^6}{(b d-a e)^7 (d+e x)^2}+\frac {21 b^2 e^6}{(b d-a e)^8 (d+e x)}\right ) \, dx\\ &=-\frac {b^2}{5 (b d-a e)^3 (a+b x)^5}+\frac {3 b^2 e}{4 (b d-a e)^4 (a+b x)^4}-\frac {2 b^2 e^2}{(b d-a e)^5 (a+b x)^3}+\frac {5 b^2 e^3}{(b d-a e)^6 (a+b x)^2}-\frac {15 b^2 e^4}{(b d-a e)^7 (a+b x)}-\frac {e^5}{2 (b d-a e)^6 (d+e x)^2}-\frac {6 b e^5}{(b d-a e)^7 (d+e x)}-\frac {21 b^2 e^5 \log (a+b x)}{(b d-a e)^8}+\frac {21 b^2 e^5 \log (d+e x)}{(b d-a e)^8}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 204, normalized size = 0.93 \[ -\frac {\frac {300 b^2 e^4 (b d-a e)}{a+b x}-\frac {100 b^2 e^3 (b d-a e)^2}{(a+b x)^2}+\frac {40 b^2 e^2 (b d-a e)^3}{(a+b x)^3}-\frac {15 b^2 e (b d-a e)^4}{(a+b x)^4}+\frac {4 b^2 (b d-a e)^5}{(a+b x)^5}+420 b^2 e^5 \log (a+b x)+\frac {120 b e^5 (b d-a e)}{d+e x}+\frac {10 e^5 (b d-a e)^2}{(d+e x)^2}-420 b^2 e^5 \log (d+e x)}{20 (b d-a e)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/20*((4*b^2*(b*d - a*e)^5)/(a + b*x)^5 - (15*b^2*e*(b*d - a*e)^4)/(a + b*x)^4 + (40*b^2*e^2*(b*d - a*e)^3)/(
a + b*x)^3 - (100*b^2*e^3*(b*d - a*e)^2)/(a + b*x)^2 + (300*b^2*e^4*(b*d - a*e))/(a + b*x) + (10*e^5*(b*d - a*
e)^2)/(d + e*x)^2 + (120*b*e^5*(b*d - a*e))/(d + e*x) + 420*b^2*e^5*Log[a + b*x] - 420*b^2*e^5*Log[d + e*x])/(
b*d - a*e)^8

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fricas [B]  time = 1.08, size = 2005, normalized size = 9.11 \[ \text {result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

-1/20*(4*b^7*d^7 - 35*a*b^6*d^6*e + 140*a^2*b^5*d^5*e^2 - 350*a^3*b^4*d^4*e^3 + 700*a^4*b^3*d^3*e^4 - 329*a^5*
b^2*d^2*e^5 - 140*a^6*b*d*e^6 + 10*a^7*e^7 + 420*(b^7*d*e^6 - a*b^6*e^7)*x^6 + 630*(b^7*d^2*e^5 + 2*a*b^6*d*e^
6 - 3*a^2*b^5*e^7)*x^5 + 70*(2*b^7*d^3*e^4 + 39*a*b^6*d^2*e^5 + 6*a^2*b^5*d*e^6 - 47*a^3*b^4*e^7)*x^4 - 35*(b^
7*d^4*e^3 - 20*a*b^6*d^3*e^4 - 126*a^2*b^5*d^2*e^5 + 68*a^3*b^4*d*e^6 + 77*a^4*b^3*e^7)*x^3 + 7*(2*b^7*d^5*e^2
- 25*a*b^6*d^4*e^3 + 200*a^2*b^5*d^3*e^4 + 430*a^3*b^4*d^2*e^5 - 470*a^4*b^3*d*e^6 - 137*a^5*b^2*e^7)*x^2 - 7
*(b^7*d^6*e - 10*a*b^6*d^5*e^2 + 50*a^2*b^5*d^4*e^3 - 200*a^3*b^4*d^3*e^4 - 65*a^4*b^3*d^2*e^5 + 214*a^5*b^2*d
*e^6 + 10*a^6*b*e^7)*x + 420*(b^7*e^7*x^7 + a^5*b^2*d^2*e^5 + (2*b^7*d*e^6 + 5*a*b^6*e^7)*x^6 + (b^7*d^2*e^5 +
10*a*b^6*d*e^6 + 10*a^2*b^5*e^7)*x^5 + 5*(a*b^6*d^2*e^5 + 4*a^2*b^5*d*e^6 + 2*a^3*b^4*e^7)*x^4 + 5*(2*a^2*b^5
*d^2*e^5 + 4*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + (10*a^3*b^4*d^2*e^5 + 10*a^4*b^3*d*e^6 + a^5*b^2*e^7)*x^2 + (5
*a^4*b^3*d^2*e^5 + 2*a^5*b^2*d*e^6)*x)*log(b*x + a) - 420*(b^7*e^7*x^7 + a^5*b^2*d^2*e^5 + (2*b^7*d*e^6 + 5*a*
b^6*e^7)*x^6 + (b^7*d^2*e^5 + 10*a*b^6*d*e^6 + 10*a^2*b^5*e^7)*x^5 + 5*(a*b^6*d^2*e^5 + 4*a^2*b^5*d*e^6 + 2*a^
3*b^4*e^7)*x^4 + 5*(2*a^2*b^5*d^2*e^5 + 4*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + (10*a^3*b^4*d^2*e^5 + 10*a^4*b^3*
d*e^6 + a^5*b^2*e^7)*x^2 + (5*a^4*b^3*d^2*e^5 + 2*a^5*b^2*d*e^6)*x)*log(e*x + d))/(a^5*b^8*d^10 - 8*a^6*b^7*d^
9*e + 28*a^7*b^6*d^8*e^2 - 56*a^8*b^5*d^7*e^3 + 70*a^9*b^4*d^6*e^4 - 56*a^10*b^3*d^5*e^5 + 28*a^11*b^2*d^4*e^6
- 8*a^12*b*d^3*e^7 + a^13*d^2*e^8 + (b^13*d^8*e^2 - 8*a*b^12*d^7*e^3 + 28*a^2*b^11*d^6*e^4 - 56*a^3*b^10*d^5*
e^5 + 70*a^4*b^9*d^4*e^6 - 56*a^5*b^8*d^3*e^7 + 28*a^6*b^7*d^2*e^8 - 8*a^7*b^6*d*e^9 + a^8*b^5*e^10)*x^7 + (2*
b^13*d^9*e - 11*a*b^12*d^8*e^2 + 16*a^2*b^11*d^7*e^3 + 28*a^3*b^10*d^6*e^4 - 140*a^4*b^9*d^5*e^5 + 238*a^5*b^8
*d^4*e^6 - 224*a^6*b^7*d^3*e^7 + 124*a^7*b^6*d^2*e^8 - 38*a^8*b^5*d*e^9 + 5*a^9*b^4*e^10)*x^6 + (b^13*d^10 + 2
*a*b^12*d^9*e - 42*a^2*b^11*d^8*e^2 + 144*a^3*b^10*d^7*e^3 - 210*a^4*b^9*d^6*e^4 + 84*a^5*b^8*d^5*e^5 + 168*a^
6*b^7*d^4*e^6 - 288*a^7*b^6*d^3*e^7 + 201*a^8*b^5*d^2*e^8 - 70*a^9*b^4*d*e^9 + 10*a^10*b^3*e^10)*x^5 + 5*(a*b^
12*d^10 - 4*a^2*b^11*d^9*e - 2*a^3*b^10*d^8*e^2 + 40*a^4*b^9*d^7*e^3 - 98*a^5*b^8*d^6*e^4 + 112*a^6*b^7*d^5*e^
5 - 56*a^7*b^6*d^4*e^6 - 8*a^8*b^5*d^3*e^7 + 25*a^9*b^4*d^2*e^8 - 12*a^10*b^3*d*e^9 + 2*a^11*b^2*e^10)*x^4 + 5
*(2*a^2*b^11*d^10 - 12*a^3*b^10*d^9*e + 25*a^4*b^9*d^8*e^2 - 8*a^5*b^8*d^7*e^3 - 56*a^6*b^7*d^6*e^4 + 112*a^7*
b^6*d^5*e^5 - 98*a^8*b^5*d^4*e^6 + 40*a^9*b^4*d^3*e^7 - 2*a^10*b^3*d^2*e^8 - 4*a^11*b^2*d*e^9 + a^12*b*e^10)*x
^3 + (10*a^3*b^10*d^10 - 70*a^4*b^9*d^9*e + 201*a^5*b^8*d^8*e^2 - 288*a^6*b^7*d^7*e^3 + 168*a^7*b^6*d^6*e^4 +
84*a^8*b^5*d^5*e^5 - 210*a^9*b^4*d^4*e^6 + 144*a^10*b^3*d^3*e^7 - 42*a^11*b^2*d^2*e^8 + 2*a^12*b*d*e^9 + a^13*
e^10)*x^2 + (5*a^4*b^9*d^10 - 38*a^5*b^8*d^9*e + 124*a^6*b^7*d^8*e^2 - 224*a^7*b^6*d^7*e^3 + 238*a^8*b^5*d^6*e
^4 - 140*a^9*b^4*d^5*e^5 + 28*a^10*b^3*d^4*e^6 + 16*a^11*b^2*d^3*e^7 - 11*a^12*b*d^2*e^8 + 2*a^13*d*e^9)*x)

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giac [B]  time = 0.19, size = 672, normalized size = 3.05 \[ -\frac {21 \, b^{3} e^{5} \log \left ({\left | b x + a \right |}\right )}{b^{9} d^{8} - 8 \, a b^{8} d^{7} e + 28 \, a^{2} b^{7} d^{6} e^{2} - 56 \, a^{3} b^{6} d^{5} e^{3} + 70 \, a^{4} b^{5} d^{4} e^{4} - 56 \, a^{5} b^{4} d^{3} e^{5} + 28 \, a^{6} b^{3} d^{2} e^{6} - 8 \, a^{7} b^{2} d e^{7} + a^{8} b e^{8}} + \frac {21 \, b^{2} e^{6} \log \left ({\left | x e + d \right |}\right )}{b^{8} d^{8} e - 8 \, a b^{7} d^{7} e^{2} + 28 \, a^{2} b^{6} d^{6} e^{3} - 56 \, a^{3} b^{5} d^{5} e^{4} + 70 \, a^{4} b^{4} d^{4} e^{5} - 56 \, a^{5} b^{3} d^{3} e^{6} + 28 \, a^{6} b^{2} d^{2} e^{7} - 8 \, a^{7} b d e^{8} + a^{8} e^{9}} - \frac {4 \, b^{7} d^{7} - 35 \, a b^{6} d^{6} e + 140 \, a^{2} b^{5} d^{5} e^{2} - 350 \, a^{3} b^{4} d^{4} e^{3} + 700 \, a^{4} b^{3} d^{3} e^{4} - 329 \, a^{5} b^{2} d^{2} e^{5} - 140 \, a^{6} b d e^{6} + 10 \, a^{7} e^{7} + 420 \, {\left (b^{7} d e^{6} - a b^{6} e^{7}\right )} x^{6} + 630 \, {\left (b^{7} d^{2} e^{5} + 2 \, a b^{6} d e^{6} - 3 \, a^{2} b^{5} e^{7}\right )} x^{5} + 70 \, {\left (2 \, b^{7} d^{3} e^{4} + 39 \, a b^{6} d^{2} e^{5} + 6 \, a^{2} b^{5} d e^{6} - 47 \, a^{3} b^{4} e^{7}\right )} x^{4} - 35 \, {\left (b^{7} d^{4} e^{3} - 20 \, a b^{6} d^{3} e^{4} - 126 \, a^{2} b^{5} d^{2} e^{5} + 68 \, a^{3} b^{4} d e^{6} + 77 \, a^{4} b^{3} e^{7}\right )} x^{3} + 7 \, {\left (2 \, b^{7} d^{5} e^{2} - 25 \, a b^{6} d^{4} e^{3} + 200 \, a^{2} b^{5} d^{3} e^{4} + 430 \, a^{3} b^{4} d^{2} e^{5} - 470 \, a^{4} b^{3} d e^{6} - 137 \, a^{5} b^{2} e^{7}\right )} x^{2} - 7 \, {\left (b^{7} d^{6} e - 10 \, a b^{6} d^{5} e^{2} + 50 \, a^{2} b^{5} d^{4} e^{3} - 200 \, a^{3} b^{4} d^{3} e^{4} - 65 \, a^{4} b^{3} d^{2} e^{5} + 214 \, a^{5} b^{2} d e^{6} + 10 \, a^{6} b e^{7}\right )} x}{20 \, {\left (b d - a e\right )}^{8} {\left (b x + a\right )}^{5} {\left (x e + d\right )}^{2}} \]

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)^3,x, algorithm="giac")

[Out]

-21*b^3*e^5*log(abs(b*x + a))/(b^9*d^8 - 8*a*b^8*d^7*e + 28*a^2*b^7*d^6*e^2 - 56*a^3*b^6*d^5*e^3 + 70*a^4*b^5*
d^4*e^4 - 56*a^5*b^4*d^3*e^5 + 28*a^6*b^3*d^2*e^6 - 8*a^7*b^2*d*e^7 + a^8*b*e^8) + 21*b^2*e^6*log(abs(x*e + d)
)/(b^8*d^8*e - 8*a*b^7*d^7*e^2 + 28*a^2*b^6*d^6*e^3 - 56*a^3*b^5*d^5*e^4 + 70*a^4*b^4*d^4*e^5 - 56*a^5*b^3*d^3
*e^6 + 28*a^6*b^2*d^2*e^7 - 8*a^7*b*d*e^8 + a^8*e^9) - 1/20*(4*b^7*d^7 - 35*a*b^6*d^6*e + 140*a^2*b^5*d^5*e^2
- 350*a^3*b^4*d^4*e^3 + 700*a^4*b^3*d^3*e^4 - 329*a^5*b^2*d^2*e^5 - 140*a^6*b*d*e^6 + 10*a^7*e^7 + 420*(b^7*d*
e^6 - a*b^6*e^7)*x^6 + 630*(b^7*d^2*e^5 + 2*a*b^6*d*e^6 - 3*a^2*b^5*e^7)*x^5 + 70*(2*b^7*d^3*e^4 + 39*a*b^6*d^
2*e^5 + 6*a^2*b^5*d*e^6 - 47*a^3*b^4*e^7)*x^4 - 35*(b^7*d^4*e^3 - 20*a*b^6*d^3*e^4 - 126*a^2*b^5*d^2*e^5 + 68*
a^3*b^4*d*e^6 + 77*a^4*b^3*e^7)*x^3 + 7*(2*b^7*d^5*e^2 - 25*a*b^6*d^4*e^3 + 200*a^2*b^5*d^3*e^4 + 430*a^3*b^4*
d^2*e^5 - 470*a^4*b^3*d*e^6 - 137*a^5*b^2*e^7)*x^2 - 7*(b^7*d^6*e - 10*a*b^6*d^5*e^2 + 50*a^2*b^5*d^4*e^3 - 20
0*a^3*b^4*d^3*e^4 - 65*a^4*b^3*d^2*e^5 + 214*a^5*b^2*d*e^6 + 10*a^6*b*e^7)*x)/((b*d - a*e)^8*(b*x + a)^5*(x*e
+ d)^2)

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maple [A]  time = 0.06, size = 215, normalized size = 0.98 \[ -\frac {21 b^{2} e^{5} \ln \left (b x +a \right )}{\left (a e -b d \right )^{8}}+\frac {21 b^{2} e^{5} \ln \left (e x +d \right )}{\left (a e -b d \right )^{8}}+\frac {15 b^{2} e^{4}}{\left (a e -b d \right )^{7} \left (b x +a \right )}+\frac {6 b \,e^{5}}{\left (a e -b d \right )^{7} \left (e x +d \right )}+\frac {5 b^{2} e^{3}}{\left (a e -b d \right )^{6} \left (b x +a \right )^{2}}-\frac {e^{5}}{2 \left (a e -b d \right )^{6} \left (e x +d \right )^{2}}+\frac {2 b^{2} e^{2}}{\left (a e -b d \right )^{5} \left (b x +a \right )^{3}}+\frac {3 b^{2} e}{4 \left (a e -b d \right )^{4} \left (b x +a \right )^{4}}+\frac {b^{2}}{5 \left (a e -b d \right )^{3} \left (b x +a \right )^{5}} \]

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)^3,x)

[Out]

1/5*b^2/(a*e-b*d)^3/(b*x+a)^5-21*b^2/(a*e-b*d)^8*e^5*ln(b*x+a)+15*b^2/(a*e-b*d)^7*e^4/(b*x+a)+5*b^2/(a*e-b*d)^
6*e^3/(b*x+a)^2+2*b^2/(a*e-b*d)^5*e^2/(b*x+a)^3+3/4*b^2/(a*e-b*d)^4*e/(b*x+a)^4-1/2*e^5/(a*e-b*d)^6/(e*x+d)^2+
21*b^2/(a*e-b*d)^8*e^5*ln(e*x+d)+6*e^5/(a*e-b*d)^7*b/(e*x+d)

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maxima [B]  time = 3.21, size = 1558, normalized size = 7.08 \[ \text {result too large to display} \]

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)^3,x, algorithm="maxima")

[Out]

-21*b^2*e^5*log(b*x + a)/(b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e
^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8) + 21*b^2*e^5*log(e*x + d)/(b^8*d^8 - 8
*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*
d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8) - 1/20*(420*b^6*e^6*x^6 + 4*b^6*d^6 - 31*a*b^5*d^5*e + 109*a^2*b^4*d^4*e^2
- 241*a^3*b^3*d^3*e^3 + 459*a^4*b^2*d^2*e^4 + 130*a^5*b*d*e^5 - 10*a^6*e^6 + 630*(b^6*d*e^5 + 3*a*b^5*e^6)*x^5
+ 70*(2*b^6*d^2*e^4 + 41*a*b^5*d*e^5 + 47*a^2*b^4*e^6)*x^4 - 35*(b^6*d^3*e^3 - 19*a*b^5*d^2*e^4 - 145*a^2*b^4
*d*e^5 - 77*a^3*b^3*e^6)*x^3 + 7*(2*b^6*d^4*e^2 - 23*a*b^5*d^3*e^3 + 177*a^2*b^4*d^2*e^4 + 607*a^3*b^3*d*e^5 +
137*a^4*b^2*e^6)*x^2 - 7*(b^6*d^5*e - 9*a*b^5*d^4*e^2 + 41*a^2*b^4*d^3*e^3 - 159*a^3*b^3*d^2*e^4 - 224*a^4*b^
2*d*e^5 - 10*a^5*b*e^6)*x)/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e^2 - 35*a^8*b^4*d^6*e^3 + 35*a^9*b
^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d^7*e^2 - 7*a*b^11*d^6*e^3 + 21*a^2
*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7 + 7*a^6*b^6*d*e^8 - a^7*b^5*e^9)*
x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5*e^4 - 105*a^4*b^8*d^4*e^5 + 133*a
^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6 + (b^12*d^9 + 3*a*b^11*d^8*e - 39*
a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^4*e^5 + 147*a^6*b^6*d^3*e^6 - 141*
a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3*a^2*b^10*d^8*e - 5*a^3*b^9*d^7*e^
2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5*d^3*e^6 - 15*a^8*b^4*d^2*e^7 + 10
*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e + 15*a^4*b^8*d^7*e^2 + 7*a^5*b^7*d
^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^9*b^3*d^2*e^7 + 3*a^10*b^2*d*e^8 -
a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2 - 147*a^6*b^6*d^6*e^3 + 21*a^7*b^5
*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 - 3*a^11*b*d*e^8 - a^12*e^9)*x^2 +
(5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3 + 105*a^8*b^4*d^5*e^4 - 35*a^9*b^
3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x)

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mupad [B]  time = 1.47, size = 1427, normalized size = 6.49 \[ \frac {\frac {-10\,a^6\,e^6+130\,a^5\,b\,d\,e^5+459\,a^4\,b^2\,d^2\,e^4-241\,a^3\,b^3\,d^3\,e^3+109\,a^2\,b^4\,d^4\,e^2-31\,a\,b^5\,d^5\,e+4\,b^6\,d^6}{20\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {7\,e^3\,x^3\,\left (77\,a^3\,b^3\,e^3+145\,a^2\,b^4\,d\,e^2+19\,a\,b^5\,d^2\,e-b^6\,d^3\right )}{4\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {21\,b^6\,e^6\,x^6}{a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7}+\frac {7\,e^2\,x^2\,\left (137\,a^4\,b^2\,e^4+607\,a^3\,b^3\,d\,e^3+177\,a^2\,b^4\,d^2\,e^2-23\,a\,b^5\,d^3\,e+2\,b^6\,d^4\right )}{20\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {7\,e^4\,x^4\,\left (47\,a^2\,b^4\,e^2+41\,a\,b^5\,d\,e+2\,b^6\,d^2\right )}{2\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {7\,e\,x\,\left (10\,a^5\,b\,e^5+224\,a^4\,b^2\,d\,e^4+159\,a^3\,b^3\,d^2\,e^3-41\,a^2\,b^4\,d^3\,e^2+9\,a\,b^5\,d^4\,e-b^6\,d^5\right )}{20\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}+\frac {63\,b\,e^4\,x^5\,\left (d\,b^5\,e+3\,a\,b^4\,e^2\right )}{2\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}}{x^3\,\left (5\,a^4\,b\,e^2+20\,a^3\,b^2\,d\,e+10\,a^2\,b^3\,d^2\right )+x^4\,\left (10\,a^3\,b^2\,e^2+20\,a^2\,b^3\,d\,e+5\,a\,b^4\,d^2\right )+x\,\left (2\,e\,a^5\,d+5\,b\,a^4\,d^2\right )+x^2\,\left (a^5\,e^2+10\,a^4\,b\,d\,e+10\,a^3\,b^2\,d^2\right )+x^5\,\left (10\,a^2\,b^3\,e^2+10\,a\,b^4\,d\,e+b^5\,d^2\right )+x^6\,\left (2\,d\,b^5\,e+5\,a\,b^4\,e^2\right )+a^5\,d^2+b^5\,e^2\,x^7}-\frac {42\,b^2\,e^5\,\mathrm {atanh}\left (\frac {a^8\,e^8-6\,a^7\,b\,d\,e^7+14\,a^6\,b^2\,d^2\,e^6-14\,a^5\,b^3\,d^3\,e^5+14\,a^3\,b^5\,d^5\,e^3-14\,a^2\,b^6\,d^6\,e^2+6\,a\,b^7\,d^7\,e-b^8\,d^8}{{\left (a\,e-b\,d\right )}^8}+\frac {2\,b\,e\,x\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{{\left (a\,e-b\,d\right )}^8}\right )}{{\left (a\,e-b\,d\right )}^8} \]

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)^3),x)

[Out]

((4*b^6*d^6 - 10*a^6*e^6 + 109*a^2*b^4*d^4*e^2 - 241*a^3*b^3*d^3*e^3 + 459*a^4*b^2*d^2*e^4 - 31*a*b^5*d^5*e +
130*a^5*b*d*e^5)/(20*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^
5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6)) + (7*e^3*x^3*(77*a^3*b^3*e^3 - b^6*d^3 + 145*a^2*b^4*d*e^2 + 1
9*a*b^5*d^2*e))/(4*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*
b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6)) + (21*b^6*e^6*x^6)/(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*
a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6) + (7*e^2*x^2*(2*b^6
*d^4 + 137*a^4*b^2*e^4 + 607*a^3*b^3*d*e^3 + 177*a^2*b^4*d^2*e^2 - 23*a*b^5*d^3*e))/(20*(a^7*e^7 - b^7*d^7 - 2
1*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e
^6)) + (7*e^4*x^4*(2*b^6*d^2 + 47*a^2*b^4*e^2 + 41*a*b^5*d*e))/(2*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35
*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6)) + (7*e*x*(10*a^5*
b*e^5 - b^6*d^5 + 224*a^4*b^2*d*e^4 - 41*a^2*b^4*d^3*e^2 + 159*a^3*b^3*d^2*e^3 + 9*a*b^5*d^4*e))/(20*(a^7*e^7
- b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e
- 7*a^6*b*d*e^6)) + (63*b*e^4*x^5*(3*a*b^4*e^2 + b^5*d*e))/(2*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3
*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6)))/(x^3*(5*a^4*b*e^2 +
10*a^2*b^3*d^2 + 20*a^3*b^2*d*e) + x^4*(5*a*b^4*d^2 + 10*a^3*b^2*e^2 + 20*a^2*b^3*d*e) + x*(5*a^4*b*d^2 + 2*a^
5*d*e) + x^2*(a^5*e^2 + 10*a^3*b^2*d^2 + 10*a^4*b*d*e) + x^5*(b^5*d^2 + 10*a^2*b^3*e^2 + 10*a*b^4*d*e) + x^6*(
5*a*b^4*e^2 + 2*b^5*d*e) + a^5*d^2 + b^5*e^2*x^7) - (42*b^2*e^5*atanh((a^8*e^8 - b^8*d^8 - 14*a^2*b^6*d^6*e^2
+ 14*a^3*b^5*d^5*e^3 - 14*a^5*b^3*d^3*e^5 + 14*a^6*b^2*d^2*e^6 + 6*a*b^7*d^7*e - 6*a^7*b*d*e^7)/(a*e - b*d)^8
+ (2*b*e*x*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*
e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6))/(a*e - b*d)^8))/(a*e - b*d)^8

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sympy [B]  time = 8.39, size = 1974, normalized size = 8.97 \[ \text {result too large to display} \]

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)**3,x)

[Out]

21*b**2*e**5*log(x + (-21*a**9*b**2*e**14/(a*e - b*d)**8 + 189*a**8*b**3*d*e**13/(a*e - b*d)**8 - 756*a**7*b**
4*d**2*e**12/(a*e - b*d)**8 + 1764*a**6*b**5*d**3*e**11/(a*e - b*d)**8 - 2646*a**5*b**6*d**4*e**10/(a*e - b*d)
**8 + 2646*a**4*b**7*d**5*e**9/(a*e - b*d)**8 - 1764*a**3*b**8*d**6*e**8/(a*e - b*d)**8 + 756*a**2*b**9*d**7*e
**7/(a*e - b*d)**8 - 189*a*b**10*d**8*e**6/(a*e - b*d)**8 + 21*a*b**2*e**6 + 21*b**11*d**9*e**5/(a*e - b*d)**8
+ 21*b**3*d*e**5)/(42*b**3*e**6))/(a*e - b*d)**8 - 21*b**2*e**5*log(x + (21*a**9*b**2*e**14/(a*e - b*d)**8 -
189*a**8*b**3*d*e**13/(a*e - b*d)**8 + 756*a**7*b**4*d**2*e**12/(a*e - b*d)**8 - 1764*a**6*b**5*d**3*e**11/(a*
e - b*d)**8 + 2646*a**5*b**6*d**4*e**10/(a*e - b*d)**8 - 2646*a**4*b**7*d**5*e**9/(a*e - b*d)**8 + 1764*a**3*b
**8*d**6*e**8/(a*e - b*d)**8 - 756*a**2*b**9*d**7*e**7/(a*e - b*d)**8 + 189*a*b**10*d**8*e**6/(a*e - b*d)**8 +
21*a*b**2*e**6 - 21*b**11*d**9*e**5/(a*e - b*d)**8 + 21*b**3*d*e**5)/(42*b**3*e**6))/(a*e - b*d)**8 + (-10*a*
*6*e**6 + 130*a**5*b*d*e**5 + 459*a**4*b**2*d**2*e**4 - 241*a**3*b**3*d**3*e**3 + 109*a**2*b**4*d**4*e**2 - 31
*a*b**5*d**5*e + 4*b**6*d**6 + 420*b**6*e**6*x**6 + x**5*(1890*a*b**5*e**6 + 630*b**6*d*e**5) + x**4*(3290*a**
2*b**4*e**6 + 2870*a*b**5*d*e**5 + 140*b**6*d**2*e**4) + x**3*(2695*a**3*b**3*e**6 + 5075*a**2*b**4*d*e**5 + 6
65*a*b**5*d**2*e**4 - 35*b**6*d**3*e**3) + x**2*(959*a**4*b**2*e**6 + 4249*a**3*b**3*d*e**5 + 1239*a**2*b**4*d
**2*e**4 - 161*a*b**5*d**3*e**3 + 14*b**6*d**4*e**2) + x*(70*a**5*b*e**6 + 1568*a**4*b**2*d*e**5 + 1113*a**3*b
**3*d**2*e**4 - 287*a**2*b**4*d**3*e**3 + 63*a*b**5*d**4*e**2 - 7*b**6*d**5*e))/(20*a**12*d**2*e**7 - 140*a**1
1*b*d**3*e**6 + 420*a**10*b**2*d**4*e**5 - 700*a**9*b**3*d**5*e**4 + 700*a**8*b**4*d**6*e**3 - 420*a**7*b**5*d
**7*e**2 + 140*a**6*b**6*d**8*e - 20*a**5*b**7*d**9 + x**7*(20*a**7*b**5*e**9 - 140*a**6*b**6*d*e**8 + 420*a**
5*b**7*d**2*e**7 - 700*a**4*b**8*d**3*e**6 + 700*a**3*b**9*d**4*e**5 - 420*a**2*b**10*d**5*e**4 + 140*a*b**11*
d**6*e**3 - 20*b**12*d**7*e**2) + x**6*(100*a**8*b**4*e**9 - 660*a**7*b**5*d*e**8 + 1820*a**6*b**6*d**2*e**7 -
2660*a**5*b**7*d**3*e**6 + 2100*a**4*b**8*d**4*e**5 - 700*a**3*b**9*d**5*e**4 - 140*a**2*b**10*d**6*e**3 + 18
0*a*b**11*d**7*e**2 - 40*b**12*d**8*e) + x**5*(200*a**9*b**3*e**9 - 1200*a**8*b**4*d*e**8 + 2820*a**7*b**5*d**
2*e**7 - 2940*a**6*b**6*d**3*e**6 + 420*a**5*b**7*d**4*e**5 + 2100*a**4*b**8*d**5*e**4 - 2100*a**3*b**9*d**6*e
**3 + 780*a**2*b**10*d**7*e**2 - 60*a*b**11*d**8*e - 20*b**12*d**9) + x**4*(200*a**10*b**2*e**9 - 1000*a**9*b*
*3*d*e**8 + 1500*a**8*b**4*d**2*e**7 + 700*a**7*b**5*d**3*e**6 - 4900*a**6*b**6*d**4*e**5 + 6300*a**5*b**7*d**
5*e**4 - 3500*a**4*b**8*d**6*e**3 + 500*a**3*b**9*d**7*e**2 + 300*a**2*b**10*d**8*e - 100*a*b**11*d**9) + x**3
*(100*a**11*b*e**9 - 300*a**10*b**2*d*e**8 - 500*a**9*b**3*d**2*e**7 + 3500*a**8*b**4*d**3*e**6 - 6300*a**7*b*
*5*d**4*e**5 + 4900*a**6*b**6*d**5*e**4 - 700*a**5*b**7*d**6*e**3 - 1500*a**4*b**8*d**7*e**2 + 1000*a**3*b**9*
d**8*e - 200*a**2*b**10*d**9) + x**2*(20*a**12*e**9 + 60*a**11*b*d*e**8 - 780*a**10*b**2*d**2*e**7 + 2100*a**9
*b**3*d**3*e**6 - 2100*a**8*b**4*d**4*e**5 - 420*a**7*b**5*d**5*e**4 + 2940*a**6*b**6*d**6*e**3 - 2820*a**5*b*
*7*d**7*e**2 + 1200*a**4*b**8*d**8*e - 200*a**3*b**9*d**9) + x*(40*a**12*d*e**8 - 180*a**11*b*d**2*e**7 + 140*
a**10*b**2*d**3*e**6 + 700*a**9*b**3*d**4*e**5 - 2100*a**8*b**4*d**5*e**4 + 2660*a**7*b**5*d**6*e**3 - 1820*a*
*6*b**6*d**7*e**2 + 660*a**5*b**7*d**8*e - 100*a**4*b**8*d**9))

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