3.1535 \(\int \frac {1}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=181 \[ -\frac {e^5}{(d+e x) (b d-a e)^6}-\frac {6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac {6 b e^5 \log (d+e x)}{(b d-a e)^7}-\frac {5 b e^4}{(a+b x) (b d-a e)^6}+\frac {2 b e^3}{(a+b x)^2 (b d-a e)^5}-\frac {b e^2}{(a+b x)^3 (b d-a e)^4}+\frac {b e}{2 (a+b x)^4 (b d-a e)^3}-\frac {b}{5 (a+b x)^5 (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 181, 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} \[ -\frac {e^5}{(d+e x) (b d-a e)^6}-\frac {5 b e^4}{(a+b x) (b d-a e)^6}+\frac {2 b e^3}{(a+b x)^2 (b d-a e)^5}-\frac {b e^2}{(a+b x)^3 (b d-a e)^4}-\frac {6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac {6 b e^5 \log (d+e x)}{(b d-a e)^7}+\frac {b e}{2 (a+b x)^4 (b d-a e)^3}-\frac {b}{5 (a+b x)^5 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.13, size = 167, normalized size = 0.92 \[ \frac {\frac {10 e^5 (a e-b d)}{d+e x}-\frac {50 b e^4 (b d-a e)}{a+b x}+\frac {20 b e^3 (b d-a e)^2}{(a+b x)^2}-\frac {10 b e^2 (b d-a e)^3}{(a+b x)^3}+\frac {5 b e (b d-a e)^4}{(a+b x)^4}-\frac {2 b (b d-a e)^5}{(a+b x)^5}-60 b e^5 \log (a+b x)+60 b e^5 \log (d+e x)}{10 (b d-a e)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*b*(b*d - a*e)^5)/(a + b*x)^5 + (5*b*e*(b*d - a*e)^4)/(a + b*x)^4 - (10*b*e^2*(b*d - a*e)^3)/(a + b*x)^3 +
(20*b*e^3*(b*d - a*e)^2)/(a + b*x)^2 - (50*b*e^4*(b*d - a*e))/(a + b*x) + (10*e^5*(-(b*d) + a*e))/(d + e*x) -
60*b*e^5*Log[a + b*x] + 60*b*e^5*Log[d + e*x])/(10*(b*d - a*e)^7)

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fricas [B]  time = 0.72, size = 1420, normalized size = 7.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(2*b^6*d^6 - 15*a*b^5*d^5*e + 50*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 150*a^4*b^2*d^2*e^4 - 77*a^5*b*
d*e^5 - 10*a^6*e^6 + 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 30*(b^6*d^2*e^4 + 8*a*b^5*d*e^5 - 9*a^2*b^4*e^6)*x^4 - 1
0*(b^6*d^3*e^3 - 15*a*b^5*d^2*e^4 - 33*a^2*b^4*d*e^5 + 47*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 10*a*b^5*d^3*e^3
+ 60*a^2*b^4*d^2*e^4 + 26*a^3*b^3*d*e^5 - 77*a^4*b^2*e^6)*x^2 - (3*b^6*d^5*e - 25*a*b^5*d^4*e^2 + 100*a^2*b^4
*d^3*e^3 - 300*a^3*b^3*d^2*e^4 + 85*a^4*b^2*d*e^5 + 137*a^5*b*e^6)*x + 60*(b^6*e^6*x^6 + a^5*b*d*e^5 + (b^6*d*
e^5 + 5*a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 + 2*a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 5*(2*a^3
*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + (5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*log(b*x + a) - 60*(b^6*e^6*x^6 + a^5*b*d*e^5
+ (b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 + 2*a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 +
5*(2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + (5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*log(e*x + d))/(a^5*b^7*d^8 - 7*a^6*b
^6*d^7*e + 21*a^7*b^5*d^6*e^2 - 35*a^8*b^4*d^5*e^3 + 35*a^9*b^3*d^4*e^4 - 21*a^10*b^2*d^3*e^5 + 7*a^11*b*d^2*e
^6 - a^12*d*e^7 + (b^12*d^7*e - 7*a*b^11*d^6*e^2 + 21*a^2*b^10*d^5*e^3 - 35*a^3*b^9*d^4*e^4 + 35*a^4*b^8*d^3*e
^5 - 21*a^5*b^7*d^2*e^6 + 7*a^6*b^6*d*e^7 - a^7*b^5*e^8)*x^6 + (b^12*d^8 - 2*a*b^11*d^7*e - 14*a^2*b^10*d^6*e^
2 + 70*a^3*b^9*d^5*e^3 - 140*a^4*b^8*d^4*e^4 + 154*a^5*b^7*d^3*e^5 - 98*a^6*b^6*d^2*e^6 + 34*a^7*b^5*d*e^7 - 5
*a^8*b^4*e^8)*x^5 + 5*(a*b^11*d^8 - 5*a^2*b^10*d^7*e + 7*a^3*b^9*d^6*e^2 + 7*a^4*b^8*d^5*e^3 - 35*a^5*b^7*d^4*
e^4 + 49*a^6*b^6*d^3*e^5 - 35*a^7*b^5*d^2*e^6 + 13*a^8*b^4*d*e^7 - 2*a^9*b^3*e^8)*x^4 + 10*(a^2*b^10*d^8 - 6*a
^3*b^9*d^7*e + 14*a^4*b^8*d^6*e^2 - 14*a^5*b^7*d^5*e^3 + 14*a^7*b^5*d^3*e^5 - 14*a^8*b^4*d^2*e^6 + 6*a^9*b^3*d
*e^7 - a^10*b^2*e^8)*x^3 + 5*(2*a^3*b^9*d^8 - 13*a^4*b^8*d^7*e + 35*a^5*b^7*d^6*e^2 - 49*a^6*b^6*d^5*e^3 + 35*
a^7*b^5*d^4*e^4 - 7*a^8*b^4*d^3*e^5 - 7*a^9*b^3*d^2*e^6 + 5*a^10*b^2*d*e^7 - a^11*b*e^8)*x^2 + (5*a^4*b^8*d^8
- 34*a^5*b^7*d^7*e + 98*a^6*b^6*d^6*e^2 - 154*a^7*b^5*d^5*e^3 + 140*a^8*b^4*d^4*e^4 - 70*a^9*b^3*d^3*e^5 + 14*
a^10*b^2*d^2*e^6 + 2*a^11*b*d*e^7 - a^12*e^8)*x)

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giac [B]  time = 0.21, size = 451, normalized size = 2.49 \[ -\frac {6 \, b e^{6} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{7} d^{7} e - 7 \, a b^{6} d^{6} e^{2} + 21 \, a^{2} b^{5} d^{5} e^{3} - 35 \, a^{3} b^{4} d^{4} e^{4} + 35 \, a^{4} b^{3} d^{3} e^{5} - 21 \, a^{5} b^{2} d^{2} e^{6} + 7 \, a^{6} b d e^{7} - a^{7} e^{8}} - \frac {e^{11}}{{\left (b^{6} d^{6} e^{6} - 6 \, a b^{5} d^{5} e^{7} + 15 \, a^{2} b^{4} d^{4} e^{8} - 20 \, a^{3} b^{3} d^{3} e^{9} + 15 \, a^{4} b^{2} d^{2} e^{10} - 6 \, a^{5} b d e^{11} + a^{6} e^{12}\right )} {\left (x e + d\right )}} - \frac {87 \, b^{6} e^{5} - \frac {385 \, {\left (b^{6} d e^{6} - a b^{5} e^{7}\right )} e^{\left (-1\right )}}{x e + d} + \frac {650 \, {\left (b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {500 \, {\left (b^{6} d^{3} e^{8} - 3 \, a b^{5} d^{2} e^{9} + 3 \, a^{2} b^{4} d e^{10} - a^{3} b^{3} e^{11}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {150 \, {\left (b^{6} d^{4} e^{9} - 4 \, a b^{5} d^{3} e^{10} + 6 \, a^{2} b^{4} d^{2} e^{11} - 4 \, a^{3} b^{3} d e^{12} + a^{4} b^{2} e^{13}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}}{10 \, {\left (b d - a e\right )}^{7} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*b*e^6*log(abs(b - b*d/(x*e + d) + a*e/(x*e + d)))/(b^7*d^7*e - 7*a*b^6*d^6*e^2 + 21*a^2*b^5*d^5*e^3 - 35*a^
3*b^4*d^4*e^4 + 35*a^4*b^3*d^3*e^5 - 21*a^5*b^2*d^2*e^6 + 7*a^6*b*d*e^7 - a^7*e^8) - e^11/((b^6*d^6*e^6 - 6*a*
b^5*d^5*e^7 + 15*a^2*b^4*d^4*e^8 - 20*a^3*b^3*d^3*e^9 + 15*a^4*b^2*d^2*e^10 - 6*a^5*b*d*e^11 + a^6*e^12)*(x*e
+ d)) - 1/10*(87*b^6*e^5 - 385*(b^6*d*e^6 - a*b^5*e^7)*e^(-1)/(x*e + d) + 650*(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a
^2*b^4*e^9)*e^(-2)/(x*e + d)^2 - 500*(b^6*d^3*e^8 - 3*a*b^5*d^2*e^9 + 3*a^2*b^4*d*e^10 - a^3*b^3*e^11)*e^(-3)/
(x*e + d)^3 + 150*(b^6*d^4*e^9 - 4*a*b^5*d^3*e^10 + 6*a^2*b^4*d^2*e^11 - 4*a^3*b^3*d*e^12 + a^4*b^2*e^13)*e^(-
4)/(x*e + d)^4)/((b*d - a*e)^7*(b - b*d/(x*e + d) + a*e/(x*e + d))^5)

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maple [A]  time = 0.06, size = 178, normalized size = 0.98 \[ \frac {6 b \,e^{5} \ln \left (b x +a \right )}{\left (a e -b d \right )^{7}}-\frac {6 b \,e^{5} \ln \left (e x +d \right )}{\left (a e -b d \right )^{7}}-\frac {5 b \,e^{4}}{\left (a e -b d \right )^{6} \left (b x +a \right )}-\frac {e^{5}}{\left (a e -b d \right )^{6} \left (e x +d \right )}-\frac {2 b \,e^{3}}{\left (a e -b d \right )^{5} \left (b x +a \right )^{2}}-\frac {b \,e^{2}}{\left (a e -b d \right )^{4} \left (b x +a \right )^{3}}-\frac {b e}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{4}}-\frac {b}{5 \left (a e -b d \right )^{2} \left (b x +a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.56, size = 1154, normalized size = 6.38 \[ -\frac {6 \, b e^{5} \log \left (b x + a\right )}{b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 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^{6} b d e^{6} - a^{7} e^{7}} + \frac {6 \, b e^{5} \log \left (e x + d\right )}{b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 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^{6} b d e^{6} - a^{7} e^{7}} - \frac {60 \, b^{5} e^{5} x^{5} + 2 \, b^{5} d^{5} - 13 \, a b^{4} d^{4} e + 37 \, a^{2} b^{3} d^{3} e^{2} - 63 \, a^{3} b^{2} d^{2} e^{3} + 87 \, a^{4} b d e^{4} + 10 \, a^{5} e^{5} + 30 \, {\left (b^{5} d e^{4} + 9 \, a b^{4} e^{5}\right )} x^{4} - 10 \, {\left (b^{5} d^{2} e^{3} - 14 \, a b^{4} d e^{4} - 47 \, a^{2} b^{3} e^{5}\right )} x^{3} + 5 \, {\left (b^{5} d^{3} e^{2} - 9 \, a b^{4} d^{2} e^{3} + 51 \, a^{2} b^{3} d e^{4} + 77 \, a^{3} b^{2} e^{5}\right )} x^{2} - {\left (3 \, b^{5} d^{4} e - 22 \, a b^{4} d^{3} e^{2} + 78 \, a^{2} b^{3} d^{2} e^{3} - 222 \, a^{3} b^{2} d e^{4} - 137 \, a^{4} b e^{5}\right )} x}{10 \, {\left (a^{5} b^{6} d^{7} - 6 \, a^{6} b^{5} d^{6} e + 15 \, a^{7} b^{4} d^{5} e^{2} - 20 \, a^{8} b^{3} d^{4} e^{3} + 15 \, a^{9} b^{2} d^{3} e^{4} - 6 \, a^{10} b d^{2} e^{5} + a^{11} d e^{6} + {\left (b^{11} d^{6} e - 6 \, a b^{10} d^{5} e^{2} + 15 \, a^{2} b^{9} d^{4} e^{3} - 20 \, a^{3} b^{8} d^{3} e^{4} + 15 \, a^{4} b^{7} d^{2} e^{5} - 6 \, a^{5} b^{6} d e^{6} + a^{6} b^{5} e^{7}\right )} x^{6} + {\left (b^{11} d^{7} - a b^{10} d^{6} e - 15 \, a^{2} b^{9} d^{5} e^{2} + 55 \, a^{3} b^{8} d^{4} e^{3} - 85 \, a^{4} b^{7} d^{3} e^{4} + 69 \, a^{5} b^{6} d^{2} e^{5} - 29 \, a^{6} b^{5} d e^{6} + 5 \, a^{7} b^{4} e^{7}\right )} x^{5} + 5 \, {\left (a b^{10} d^{7} - 4 \, a^{2} b^{9} d^{6} e + 3 \, a^{3} b^{8} d^{5} e^{2} + 10 \, a^{4} b^{7} d^{4} e^{3} - 25 \, a^{5} b^{6} d^{3} e^{4} + 24 \, a^{6} b^{5} d^{2} e^{5} - 11 \, a^{7} b^{4} d e^{6} + 2 \, a^{8} b^{3} e^{7}\right )} x^{4} + 10 \, {\left (a^{2} b^{9} d^{7} - 5 \, a^{3} b^{8} d^{6} e + 9 \, a^{4} b^{7} d^{5} e^{2} - 5 \, a^{5} b^{6} d^{4} e^{3} - 5 \, a^{6} b^{5} d^{3} e^{4} + 9 \, a^{7} b^{4} d^{2} e^{5} - 5 \, a^{8} b^{3} d e^{6} + a^{9} b^{2} e^{7}\right )} x^{3} + 5 \, {\left (2 \, a^{3} b^{8} d^{7} - 11 \, a^{4} b^{7} d^{6} e + 24 \, a^{5} b^{6} d^{5} e^{2} - 25 \, a^{6} b^{5} d^{4} e^{3} + 10 \, a^{7} b^{4} d^{3} e^{4} + 3 \, a^{8} b^{3} d^{2} e^{5} - 4 \, a^{9} b^{2} d e^{6} + a^{10} b e^{7}\right )} x^{2} + {\left (5 \, a^{4} b^{7} d^{7} - 29 \, a^{5} b^{6} d^{6} e + 69 \, a^{6} b^{5} d^{5} e^{2} - 85 \, a^{7} b^{4} d^{4} e^{3} + 55 \, a^{8} b^{3} d^{3} e^{4} - 15 \, a^{9} b^{2} d^{2} e^{5} - a^{10} b d e^{6} + a^{11} e^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*b*e^5*log(b*x + a)/(b^7*d^7 - 7*a*b^6*d^6*e + 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^6*b*d*e^6 - a^7*e^7) + 6*b*e^5*log(e*x + d)/(b^7*d^7 - 7*a*b^6*d^6*e + 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^6*b*d*e^6 - a^7*e^7) - 1/10*(60*b^
5*e^5*x^5 + 2*b^5*d^5 - 13*a*b^4*d^4*e + 37*a^2*b^3*d^3*e^2 - 63*a^3*b^2*d^2*e^3 + 87*a^4*b*d*e^4 + 10*a^5*e^5
+ 30*(b^5*d*e^4 + 9*a*b^4*e^5)*x^4 - 10*(b^5*d^2*e^3 - 14*a*b^4*d*e^4 - 47*a^2*b^3*e^5)*x^3 + 5*(b^5*d^3*e^2
- 9*a*b^4*d^2*e^3 + 51*a^2*b^3*d*e^4 + 77*a^3*b^2*e^5)*x^2 - (3*b^5*d^4*e - 22*a*b^4*d^3*e^2 + 78*a^2*b^3*d^2*
e^3 - 222*a^3*b^2*d*e^4 - 137*a^4*b*e^5)*x)/(a^5*b^6*d^7 - 6*a^6*b^5*d^6*e + 15*a^7*b^4*d^5*e^2 - 20*a^8*b^3*d
^4*e^3 + 15*a^9*b^2*d^3*e^4 - 6*a^10*b*d^2*e^5 + a^11*d*e^6 + (b^11*d^6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*
e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4*b^7*d^2*e^5 - 6*a^5*b^6*d*e^6 + a^6*b^5*e^7)*x^6 + (b^11*d^7 - a*b^10*d^6*e
- 15*a^2*b^9*d^5*e^2 + 55*a^3*b^8*d^4*e^3 - 85*a^4*b^7*d^3*e^4 + 69*a^5*b^6*d^2*e^5 - 29*a^6*b^5*d*e^6 + 5*a^7
*b^4*e^7)*x^5 + 5*(a*b^10*d^7 - 4*a^2*b^9*d^6*e + 3*a^3*b^8*d^5*e^2 + 10*a^4*b^7*d^4*e^3 - 25*a^5*b^6*d^3*e^4
+ 24*a^6*b^5*d^2*e^5 - 11*a^7*b^4*d*e^6 + 2*a^8*b^3*e^7)*x^4 + 10*(a^2*b^9*d^7 - 5*a^3*b^8*d^6*e + 9*a^4*b^7*d
^5*e^2 - 5*a^5*b^6*d^4*e^3 - 5*a^6*b^5*d^3*e^4 + 9*a^7*b^4*d^2*e^5 - 5*a^8*b^3*d*e^6 + a^9*b^2*e^7)*x^3 + 5*(2
*a^3*b^8*d^7 - 11*a^4*b^7*d^6*e + 24*a^5*b^6*d^5*e^2 - 25*a^6*b^5*d^4*e^3 + 10*a^7*b^4*d^3*e^4 + 3*a^8*b^3*d^2
*e^5 - 4*a^9*b^2*d*e^6 + a^10*b*e^7)*x^2 + (5*a^4*b^7*d^7 - 29*a^5*b^6*d^6*e + 69*a^6*b^5*d^5*e^2 - 85*a^7*b^4
*d^4*e^3 + 55*a^8*b^3*d^3*e^4 - 15*a^9*b^2*d^2*e^5 - a^10*b*d*e^6 + a^11*e^7)*x)

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mupad [B]  time = 1.18, size = 1047, normalized size = 5.78 \[ \frac {12\,b\,e^5\,\mathrm {atanh}\left (\frac {a^7\,e^7-5\,a^6\,b\,d\,e^6+9\,a^5\,b^2\,d^2\,e^5-5\,a^4\,b^3\,d^3\,e^4-5\,a^3\,b^4\,d^4\,e^3+9\,a^2\,b^5\,d^5\,e^2-5\,a\,b^6\,d^6\,e+b^7\,d^7}{{\left (a\,e-b\,d\right )}^7}+\frac {2\,b\,e\,x\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^7}\right )}{{\left (a\,e-b\,d\right )}^7}-\frac {\frac {10\,a^5\,e^5+87\,a^4\,b\,d\,e^4-63\,a^3\,b^2\,d^2\,e^3+37\,a^2\,b^3\,d^3\,e^2-13\,a\,b^4\,d^4\,e+2\,b^5\,d^5}{10\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {e^3\,x^3\,\left (47\,a^2\,b^3\,e^2+14\,a\,b^4\,d\,e-b^5\,d^2\right )}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}+\frac {e^2\,x^2\,\left (77\,a^3\,b^2\,e^3+51\,a^2\,b^3\,d\,e^2-9\,a\,b^4\,d^2\,e+b^5\,d^3\right )}{2\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}+\frac {6\,b^5\,e^5\,x^5}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}+\frac {3\,e^4\,x^4\,\left (d\,b^5+9\,a\,e\,b^4\right )}{a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}+\frac {e\,x\,\left (137\,a^4\,b\,e^4+222\,a^3\,b^2\,d\,e^3-78\,a^2\,b^3\,d^2\,e^2+22\,a\,b^4\,d^3\,e-3\,b^5\,d^4\right )}{10\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}}{x^5\,\left (d\,b^5+5\,a\,e\,b^4\right )+x^3\,\left (10\,e\,a^3\,b^2+10\,d\,a^2\,b^3\right )+a^5\,d+x\,\left (e\,a^5+5\,b\,d\,a^4\right )+x^2\,\left (5\,e\,a^4\,b+10\,d\,a^3\,b^2\right )+x^4\,\left (10\,e\,a^2\,b^3+5\,d\,a\,b^4\right )+b^5\,e\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(12*b*e^5*atanh((a^7*e^7 + b^7*d^7 + 9*a^2*b^5*d^5*e^2 - 5*a^3*b^4*d^4*e^3 - 5*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^2
*e^5 - 5*a*b^6*d^6*e - 5*a^6*b*d*e^6)/(a*e - b*d)^7 + (2*b*e*x*(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5))/(a*e - b*d)^7))/(a*e - b*d)^7 - ((10*a^5*
e^5 + 2*b^5*d^5 + 37*a^2*b^3*d^3*e^2 - 63*a^3*b^2*d^2*e^3 - 13*a*b^4*d^4*e + 87*a^4*b*d*e^4)/(10*(a^6*e^6 + b^
6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5)) + (e^3*
x^3*(47*a^2*b^3*e^2 - b^5*d^2 + 14*a*b^4*d*e))/(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5) + (e^2*x^2*(b^5*d^3 + 77*a^3*b^2*e^3 + 51*a^2*b^3*d*e^2 -
9*a*b^4*d^2*e))/(2*(a^6*e^6 + b^6*d^6 + 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*b^5
*d^5*e - 6*a^5*b*d*e^5)) + (6*b^5*e^5*x^5)/(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5) + (3*e^4*x^4*(b^5*d + 9*a*b^4*e))/(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5) + (e*x*(137*a^4*b*e^4 -
3*b^5*d^4 + 222*a^3*b^2*d*e^3 - 78*a^2*b^3*d^2*e^2 + 22*a*b^4*d^3*e))/(10*(a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e - 6*a^5*b*d*e^5)))/(x^5*(b^5*d + 5*a*b^4*e) + x
^3*(10*a^2*b^3*d + 10*a^3*b^2*e) + a^5*d + x*(a^5*e + 5*a^4*b*d) + x^2*(10*a^3*b^2*d + 5*a^4*b*e) + x^4*(10*a^
2*b^3*e + 5*a*b^4*d) + b^5*e*x^6)

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sympy [B]  time = 5.18, size = 1518, normalized size = 8.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*b*e**5*log(x + (-6*a**8*b*e**13/(a*e - b*d)**7 + 48*a**7*b**2*d*e**12/(a*e - b*d)**7 - 168*a**6*b**3*d**2*e
**11/(a*e - b*d)**7 + 336*a**5*b**4*d**3*e**10/(a*e - b*d)**7 - 420*a**4*b**5*d**4*e**9/(a*e - b*d)**7 + 336*a
**3*b**6*d**5*e**8/(a*e - b*d)**7 - 168*a**2*b**7*d**6*e**7/(a*e - b*d)**7 + 48*a*b**8*d**7*e**6/(a*e - b*d)**
7 + 6*a*b*e**6 - 6*b**9*d**8*e**5/(a*e - b*d)**7 + 6*b**2*d*e**5)/(12*b**2*e**6))/(a*e - b*d)**7 + 6*b*e**5*lo
g(x + (6*a**8*b*e**13/(a*e - b*d)**7 - 48*a**7*b**2*d*e**12/(a*e - b*d)**7 + 168*a**6*b**3*d**2*e**11/(a*e - b
*d)**7 - 336*a**5*b**4*d**3*e**10/(a*e - b*d)**7 + 420*a**4*b**5*d**4*e**9/(a*e - b*d)**7 - 336*a**3*b**6*d**5
*e**8/(a*e - b*d)**7 + 168*a**2*b**7*d**6*e**7/(a*e - b*d)**7 - 48*a*b**8*d**7*e**6/(a*e - b*d)**7 + 6*a*b*e**
6 + 6*b**9*d**8*e**5/(a*e - b*d)**7 + 6*b**2*d*e**5)/(12*b**2*e**6))/(a*e - b*d)**7 + (-10*a**5*e**5 - 87*a**4
*b*d*e**4 + 63*a**3*b**2*d**2*e**3 - 37*a**2*b**3*d**3*e**2 + 13*a*b**4*d**4*e - 2*b**5*d**5 - 60*b**5*e**5*x*
*5 + x**4*(-270*a*b**4*e**5 - 30*b**5*d*e**4) + x**3*(-470*a**2*b**3*e**5 - 140*a*b**4*d*e**4 + 10*b**5*d**2*e
**3) + x**2*(-385*a**3*b**2*e**5 - 255*a**2*b**3*d*e**4 + 45*a*b**4*d**2*e**3 - 5*b**5*d**3*e**2) + x*(-137*a*
*4*b*e**5 - 222*a**3*b**2*d*e**4 + 78*a**2*b**3*d**2*e**3 - 22*a*b**4*d**3*e**2 + 3*b**5*d**4*e))/(10*a**11*d*
e**6 - 60*a**10*b*d**2*e**5 + 150*a**9*b**2*d**3*e**4 - 200*a**8*b**3*d**4*e**3 + 150*a**7*b**4*d**5*e**2 - 60
*a**6*b**5*d**6*e + 10*a**5*b**6*d**7 + x**6*(10*a**6*b**5*e**7 - 60*a**5*b**6*d*e**6 + 150*a**4*b**7*d**2*e**
5 - 200*a**3*b**8*d**3*e**4 + 150*a**2*b**9*d**4*e**3 - 60*a*b**10*d**5*e**2 + 10*b**11*d**6*e) + x**5*(50*a**
7*b**4*e**7 - 290*a**6*b**5*d*e**6 + 690*a**5*b**6*d**2*e**5 - 850*a**4*b**7*d**3*e**4 + 550*a**3*b**8*d**4*e*
*3 - 150*a**2*b**9*d**5*e**2 - 10*a*b**10*d**6*e + 10*b**11*d**7) + x**4*(100*a**8*b**3*e**7 - 550*a**7*b**4*d
*e**6 + 1200*a**6*b**5*d**2*e**5 - 1250*a**5*b**6*d**3*e**4 + 500*a**4*b**7*d**4*e**3 + 150*a**3*b**8*d**5*e**
2 - 200*a**2*b**9*d**6*e + 50*a*b**10*d**7) + x**3*(100*a**9*b**2*e**7 - 500*a**8*b**3*d*e**6 + 900*a**7*b**4*
d**2*e**5 - 500*a**6*b**5*d**3*e**4 - 500*a**5*b**6*d**4*e**3 + 900*a**4*b**7*d**5*e**2 - 500*a**3*b**8*d**6*e
+ 100*a**2*b**9*d**7) + x**2*(50*a**10*b*e**7 - 200*a**9*b**2*d*e**6 + 150*a**8*b**3*d**2*e**5 + 500*a**7*b**
4*d**3*e**4 - 1250*a**6*b**5*d**4*e**3 + 1200*a**5*b**6*d**5*e**2 - 550*a**4*b**7*d**6*e + 100*a**3*b**8*d**7)
+ x*(10*a**11*e**7 - 10*a**10*b*d*e**6 - 150*a**9*b**2*d**2*e**5 + 550*a**8*b**3*d**3*e**4 - 850*a**7*b**4*d*
*4*e**3 + 690*a**6*b**5*d**5*e**2 - 290*a**5*b**6*d**6*e + 50*a**4*b**7*d**7))

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