\(\int \frac {1}{(d+e x)^4 (a^2+2 a b x+b^2 x^2)} \, dx\) [1514]

Optimal result

Integrand size = 26, antiderivative size = 133 \[ \int \frac {1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {b^3}{(b d-a e)^4 (a+b x)}-\frac {e}{3 (b d-a e)^2 (d+e x)^3}-\frac {b e}{(b d-a e)^3 (d+e x)^2}-\frac {3 b^2 e}{(b d-a e)^4 (d+e x)}-\frac {4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac {4 b^3 e \log (d+e x)}{(b d-a e)^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 46} \[ \int \frac {1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {b^3}{(a+b x) (b d-a e)^4}-\frac {4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac {4 b^3 e \log (d+e x)}{(b d-a e)^5}-\frac {3 b^2 e}{(d+e x) (b d-a e)^4}-\frac {b e}{(d+e x)^2 (b d-a e)^3}-\frac {e}{3 (d+e x)^3 (b d-a e)^2} \]

[In]

Int[1/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

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

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 46

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] && Lt
Q[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {-\frac {3 b^3 (b d-a e)}{a+b x}+\frac {e (-b d+a e)^3}{(d+e x)^3}-\frac {3 b e (b d-a e)^2}{(d+e x)^2}-\frac {9 b^2 e (b d-a e)}{d+e x}-12 b^3 e \log (a+b x)+12 b^3 e \log (d+e x)}{3 (b d-a e)^5} \]

[In]

Integrate[1/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

((-3*b^3*(b*d - a*e))/(a + b*x) + (e*(-(b*d) + a*e)^3)/(d + e*x)^3 - (3*b*e*(b*d - a*e)^2)/(d + e*x)^2 - (9*b^
2*e*(b*d - a*e))/(d + e*x) - 12*b^3*e*Log[a + b*x] + 12*b^3*e*Log[d + e*x])/(3*(b*d - a*e)^5)

Maple [A] (verified)

Time = 2.68 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {b^{3}}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {4 b^{3} e \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {e}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{3}}-\frac {4 b^{3} e \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}-\frac {3 e \,b^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}+\frac {e b}{\left (a e -b d \right )^{3} \left (e x +d \right )^{2}}\)	\(131\)
risch	\(\frac {-\frac {4 b^{3} e^{3} x^{3}}{e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {2 e^{2} \left (a e +5 b d \right ) b^{2} x^{2}}{e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {2 \left (a^{2} e^{2}-8 a b d e -11 b^{2} d^{2}\right ) b e x}{3 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {a^{3} e^{3}-5 a^{2} b d \,e^{2}+13 a \,b^{2} d^{2} e +3 b^{3} d^{3}}{3 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (e x +d \right )^{3} \left (b x +a \right )}+\frac {4 b^{3} e \ln \left (-b x -a \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {4 b^{3} e \ln \left (e x +d \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\)	\(476\)
norman	\(\frac {-\frac {4 b^{3} e^{3} x^{3}}{e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {\left (-2 a \,b^{3} e^{5}-10 b^{4} d \,e^{4}\right ) x^{2}}{b \,e^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {-a^{3} b \,e^{6}+5 a^{2} b^{2} d \,e^{5}-13 a \,b^{3} d^{2} e^{4}-3 b^{4} d^{3} e^{3}}{3 e^{3} b \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (2 a^{2} b^{2} e^{6}-16 a \,b^{3} d \,e^{5}-22 b^{4} d^{2} e^{4}\right ) x}{3 e^{3} b \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (e x +d \right )^{3} \left (b x +a \right )}+\frac {4 b^{3} e \ln \left (b x +a \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {4 b^{3} e \ln \left (e x +d \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\)	\(513\)
parallelrisch	\(\frac {-a^{4} b \,e^{7}-36 \ln \left (e x +d \right ) x^{2} a \,b^{4} d \,e^{6}+10 a \,b^{4} d^{3} e^{4}+12 \ln \left (b x +a \right ) x^{3} a \,b^{4} e^{7}+36 \ln \left (b x +a \right ) x^{3} b^{5} d \,e^{6}-12 \ln \left (e x +d \right ) x^{3} a \,b^{4} e^{7}-36 \ln \left (e x +d \right ) x^{3} b^{5} d \,e^{6}+36 \ln \left (b x +a \right ) x^{2} b^{5} d^{2} e^{5}-36 \ln \left (e x +d \right ) x^{2} b^{5} d^{2} e^{5}+12 \ln \left (b x +a \right ) x \,b^{5} d^{3} e^{4}-12 \ln \left (e x +d \right ) x \,b^{5} d^{3} e^{4}+12 \ln \left (b x +a \right ) a \,b^{4} d^{3} e^{4}-12 \ln \left (e x +d \right ) a \,b^{4} d^{3} e^{4}-24 x^{2} a \,b^{4} d \,e^{6}-18 x \,a^{2} b^{3} d \,e^{6}-6 x a \,b^{4} d^{2} e^{5}+36 \ln \left (b x +a \right ) x a \,b^{4} d^{2} e^{5}-36 \ln \left (e x +d \right ) x a \,b^{4} d^{2} e^{5}+36 \ln \left (b x +a \right ) x^{2} a \,b^{4} d \,e^{6}+3 b^{5} d^{4} e^{3}+6 a^{3} b^{2} d \,e^{6}-18 a^{2} b^{3} d^{2} e^{5}-12 x^{3} a \,b^{4} e^{7}+12 x^{3} b^{5} d \,e^{6}-6 x^{2} a^{2} b^{3} e^{7}+30 x^{2} b^{5} d^{2} e^{5}+2 x \,a^{3} b^{2} e^{7}+22 x \,b^{5} d^{3} e^{4}+12 \ln \left (b x +a \right ) x^{4} b^{5} e^{7}-12 \ln \left (e x +d \right ) x^{4} b^{5} e^{7}}{3 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) \left (e x +d \right )^{3} \left (b x +a \right ) b \,e^{3}}\)	\(557\)

[In]

int(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (131) = 262\).

Time = 0.32 (sec) , antiderivative size = 753, normalized size of antiderivative = 5.66 \[ \int \frac {1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {3 \, b^{4} d^{4} + 10 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 6 \, a^{3} b d e^{3} - a^{4} e^{4} + 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (5 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (11 \, b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x + 12 \, {\left (b^{4} e^{4} x^{4} + a b^{3} d^{3} e + {\left (3 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3}\right )} x^{2} + {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (b x + a\right ) - 12 \, {\left (b^{4} e^{4} x^{4} + a b^{3} d^{3} e + {\left (3 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3}\right )} x^{2} + {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (a b^{5} d^{8} - 5 \, a^{2} b^{4} d^{7} e + 10 \, a^{3} b^{3} d^{6} e^{2} - 10 \, a^{4} b^{2} d^{5} e^{3} + 5 \, a^{5} b d^{4} e^{4} - a^{6} d^{3} e^{5} + {\left (b^{6} d^{5} e^{3} - 5 \, a b^{5} d^{4} e^{4} + 10 \, a^{2} b^{4} d^{3} e^{5} - 10 \, a^{3} b^{3} d^{2} e^{6} + 5 \, a^{4} b^{2} d e^{7} - a^{5} b e^{8}\right )} x^{4} + {\left (3 \, b^{6} d^{6} e^{2} - 14 \, a b^{5} d^{5} e^{3} + 25 \, a^{2} b^{4} d^{4} e^{4} - 20 \, a^{3} b^{3} d^{3} e^{5} + 5 \, a^{4} b^{2} d^{2} e^{6} + 2 \, a^{5} b d e^{7} - a^{6} e^{8}\right )} x^{3} + 3 \, {\left (b^{6} d^{7} e - 4 \, a b^{5} d^{6} e^{2} + 5 \, a^{2} b^{4} d^{5} e^{3} - 5 \, a^{4} b^{2} d^{3} e^{5} + 4 \, a^{5} b d^{2} e^{6} - a^{6} d e^{7}\right )} x^{2} + {\left (b^{6} d^{8} - 2 \, a b^{5} d^{7} e - 5 \, a^{2} b^{4} d^{6} e^{2} + 20 \, a^{3} b^{3} d^{5} e^{3} - 25 \, a^{4} b^{2} d^{4} e^{4} + 14 \, a^{5} b d^{3} e^{5} - 3 \, a^{6} d^{2} e^{6}\right )} x\right )}} \]

[In]

integrate(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

-1/3*(3*b^4*d^4 + 10*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 6*a^3*b*d*e^3 - a^4*e^4 + 12*(b^4*d*e^3 - a*b^3*e^4)*x
^3 + 6*(5*b^4*d^2*e^2 - 4*a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 2*(11*b^4*d^3*e - 3*a*b^3*d^2*e^2 - 9*a^2*b^2*d*e^3
+ a^3*b*e^4)*x + 12*(b^4*e^4*x^4 + a*b^3*d^3*e + (3*b^4*d*e^3 + a*b^3*e^4)*x^3 + 3*(b^4*d^2*e^2 + a*b^3*d*e^3
)*x^2 + (b^4*d^3*e + 3*a*b^3*d^2*e^2)*x)*log(b*x + a) - 12*(b^4*e^4*x^4 + a*b^3*d^3*e + (3*b^4*d*e^3 + a*b^3*e
^4)*x^3 + 3*(b^4*d^2*e^2 + a*b^3*d*e^3)*x^2 + (b^4*d^3*e + 3*a*b^3*d^2*e^2)*x)*log(e*x + d))/(a*b^5*d^8 - 5*a^
2*b^4*d^7*e + 10*a^3*b^3*d^6*e^2 - 10*a^4*b^2*d^5*e^3 + 5*a^5*b*d^4*e^4 - a^6*d^3*e^5 + (b^6*d^5*e^3 - 5*a*b^5
*d^4*e^4 + 10*a^2*b^4*d^3*e^5 - 10*a^3*b^3*d^2*e^6 + 5*a^4*b^2*d*e^7 - a^5*b*e^8)*x^4 + (3*b^6*d^6*e^2 - 14*a*
b^5*d^5*e^3 + 25*a^2*b^4*d^4*e^4 - 20*a^3*b^3*d^3*e^5 + 5*a^4*b^2*d^2*e^6 + 2*a^5*b*d*e^7 - a^6*e^8)*x^3 + 3*(
b^6*d^7*e - 4*a*b^5*d^6*e^2 + 5*a^2*b^4*d^5*e^3 - 5*a^4*b^2*d^3*e^5 + 4*a^5*b*d^2*e^6 - a^6*d*e^7)*x^2 + (b^6*
d^8 - 2*a*b^5*d^7*e - 5*a^2*b^4*d^6*e^2 + 20*a^3*b^3*d^5*e^3 - 25*a^4*b^2*d^4*e^4 + 14*a^5*b*d^3*e^5 - 3*a^6*d
^2*e^6)*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (117) = 234\).

Time = 1.31 (sec) , antiderivative size = 882, normalized size of antiderivative = 6.63 \[ \int \frac {1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx=- \frac {4 b^{3} e \log {\left (x + \frac {- \frac {4 a^{6} b^{3} e^{7}}{\left (a e - b d\right )^{5}} + \frac {24 a^{5} b^{4} d e^{6}}{\left (a e - b d\right )^{5}} - \frac {60 a^{4} b^{5} d^{2} e^{5}}{\left (a e - b d\right )^{5}} + \frac {80 a^{3} b^{6} d^{3} e^{4}}{\left (a e - b d\right )^{5}} - \frac {60 a^{2} b^{7} d^{4} e^{3}}{\left (a e - b d\right )^{5}} + \frac {24 a b^{8} d^{5} e^{2}}{\left (a e - b d\right )^{5}} + 4 a b^{3} e^{2} - \frac {4 b^{9} d^{6} e}{\left (a e - b d\right )^{5}} + 4 b^{4} d e}{8 b^{4} e^{2}} \right )}}{\left (a e - b d\right )^{5}} + \frac {4 b^{3} e \log {\left (x + \frac {\frac {4 a^{6} b^{3} e^{7}}{\left (a e - b d\right )^{5}} - \frac {24 a^{5} b^{4} d e^{6}}{\left (a e - b d\right )^{5}} + \frac {60 a^{4} b^{5} d^{2} e^{5}}{\left (a e - b d\right )^{5}} - \frac {80 a^{3} b^{6} d^{3} e^{4}}{\left (a e - b d\right )^{5}} + \frac {60 a^{2} b^{7} d^{4} e^{3}}{\left (a e - b d\right )^{5}} - \frac {24 a b^{8} d^{5} e^{2}}{\left (a e - b d\right )^{5}} + 4 a b^{3} e^{2} + \frac {4 b^{9} d^{6} e}{\left (a e - b d\right )^{5}} + 4 b^{4} d e}{8 b^{4} e^{2}} \right )}}{\left (a e - b d\right )^{5}} + \frac {- a^{3} e^{3} + 5 a^{2} b d e^{2} - 13 a b^{2} d^{2} e - 3 b^{3} d^{3} - 12 b^{3} e^{3} x^{3} + x^{2} \left (- 6 a b^{2} e^{3} - 30 b^{3} d e^{2}\right ) + x \left (2 a^{2} b e^{3} - 16 a b^{2} d e^{2} - 22 b^{3} d^{2} e\right )}{3 a^{5} d^{3} e^{4} - 12 a^{4} b d^{4} e^{3} + 18 a^{3} b^{2} d^{5} e^{2} - 12 a^{2} b^{3} d^{6} e + 3 a b^{4} d^{7} + x^{4} \cdot \left (3 a^{4} b e^{7} - 12 a^{3} b^{2} d e^{6} + 18 a^{2} b^{3} d^{2} e^{5} - 12 a b^{4} d^{3} e^{4} + 3 b^{5} d^{4} e^{3}\right ) + x^{3} \cdot \left (3 a^{5} e^{7} - 3 a^{4} b d e^{6} - 18 a^{3} b^{2} d^{2} e^{5} + 42 a^{2} b^{3} d^{3} e^{4} - 33 a b^{4} d^{4} e^{3} + 9 b^{5} d^{5} e^{2}\right ) + x^{2} \cdot \left (9 a^{5} d e^{6} - 27 a^{4} b d^{2} e^{5} + 18 a^{3} b^{2} d^{3} e^{4} + 18 a^{2} b^{3} d^{4} e^{3} - 27 a b^{4} d^{5} e^{2} + 9 b^{5} d^{6} e\right ) + x \left (9 a^{5} d^{2} e^{5} - 33 a^{4} b d^{3} e^{4} + 42 a^{3} b^{2} d^{4} e^{3} - 18 a^{2} b^{3} d^{5} e^{2} - 3 a b^{4} d^{6} e + 3 b^{5} d^{7}\right )} \]

[In]

integrate(1/(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-4*b**3*e*log(x + (-4*a**6*b**3*e**7/(a*e - b*d)**5 + 24*a**5*b**4*d*e**6/(a*e - b*d)**5 - 60*a**4*b**5*d**2*e
**5/(a*e - b*d)**5 + 80*a**3*b**6*d**3*e**4/(a*e - b*d)**5 - 60*a**2*b**7*d**4*e**3/(a*e - b*d)**5 + 24*a*b**8
*d**5*e**2/(a*e - b*d)**5 + 4*a*b**3*e**2 - 4*b**9*d**6*e/(a*e - b*d)**5 + 4*b**4*d*e)/(8*b**4*e**2))/(a*e - b
*d)**5 + 4*b**3*e*log(x + (4*a**6*b**3*e**7/(a*e - b*d)**5 - 24*a**5*b**4*d*e**6/(a*e - b*d)**5 + 60*a**4*b**5
*d**2*e**5/(a*e - b*d)**5 - 80*a**3*b**6*d**3*e**4/(a*e - b*d)**5 + 60*a**2*b**7*d**4*e**3/(a*e - b*d)**5 - 24
*a*b**8*d**5*e**2/(a*e - b*d)**5 + 4*a*b**3*e**2 + 4*b**9*d**6*e/(a*e - b*d)**5 + 4*b**4*d*e)/(8*b**4*e**2))/(
a*e - b*d)**5 + (-a**3*e**3 + 5*a**2*b*d*e**2 - 13*a*b**2*d**2*e - 3*b**3*d**3 - 12*b**3*e**3*x**3 + x**2*(-6*
a*b**2*e**3 - 30*b**3*d*e**2) + x*(2*a**2*b*e**3 - 16*a*b**2*d*e**2 - 22*b**3*d**2*e))/(3*a**5*d**3*e**4 - 12*
a**4*b*d**4*e**3 + 18*a**3*b**2*d**5*e**2 - 12*a**2*b**3*d**6*e + 3*a*b**4*d**7 + x**4*(3*a**4*b*e**7 - 12*a**
3*b**2*d*e**6 + 18*a**2*b**3*d**2*e**5 - 12*a*b**4*d**3*e**4 + 3*b**5*d**4*e**3) + x**3*(3*a**5*e**7 - 3*a**4*
b*d*e**6 - 18*a**3*b**2*d**2*e**5 + 42*a**2*b**3*d**3*e**4 - 33*a*b**4*d**4*e**3 + 9*b**5*d**5*e**2) + x**2*(9
*a**5*d*e**6 - 27*a**4*b*d**2*e**5 + 18*a**3*b**2*d**3*e**4 + 18*a**2*b**3*d**4*e**3 - 27*a*b**4*d**5*e**2 + 9
*b**5*d**6*e) + x*(9*a**5*d**2*e**5 - 33*a**4*b*d**3*e**4 + 42*a**3*b**2*d**4*e**3 - 18*a**2*b**3*d**5*e**2 -
3*a*b**4*d**6*e + 3*b**5*d**7))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (131) = 262\).

Time = 0.24 (sec) , antiderivative size = 599, normalized size of antiderivative = 4.50 \[ \int \frac {1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {4 \, b^{3} e \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {4 \, b^{3} e \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {12 \, b^{3} e^{3} x^{3} + 3 \, b^{3} d^{3} + 13 \, a b^{2} d^{2} e - 5 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (5 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (11 \, b^{3} d^{2} e + 8 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{3 \, {\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} + {\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} + {\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \, {\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} + {\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \]

[In]

integrate(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

-4*b^3*e*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5
*e^5) + 4*b^3*e*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^
4 - a^5*e^5) - 1/3*(12*b^3*e^3*x^3 + 3*b^3*d^3 + 13*a*b^2*d^2*e - 5*a^2*b*d*e^2 + a^3*e^3 + 6*(5*b^3*d*e^2 + a
*b^2*e^3)*x^2 + 2*(11*b^3*d^2*e + 8*a*b^2*d*e^2 - a^2*b*e^3)*x)/(a*b^4*d^7 - 4*a^2*b^3*d^6*e + 6*a^3*b^2*d^5*e
^2 - 4*a^4*b*d^4*e^3 + a^5*d^3*e^4 + (b^5*d^4*e^3 - 4*a*b^4*d^3*e^4 + 6*a^2*b^3*d^2*e^5 - 4*a^3*b^2*d*e^6 + a^
4*b*e^7)*x^4 + (3*b^5*d^5*e^2 - 11*a*b^4*d^4*e^3 + 14*a^2*b^3*d^3*e^4 - 6*a^3*b^2*d^2*e^5 - a^4*b*d*e^6 + a^5*
e^7)*x^3 + 3*(b^5*d^6*e - 3*a*b^4*d^5*e^2 + 2*a^2*b^3*d^4*e^3 + 2*a^3*b^2*d^3*e^4 - 3*a^4*b*d^2*e^5 + a^5*d*e^
6)*x^2 + (b^5*d^7 - a*b^4*d^6*e - 6*a^2*b^3*d^5*e^2 + 14*a^3*b^2*d^4*e^3 - 11*a^4*b*d^3*e^4 + 3*a^5*d^2*e^5)*x
)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (131) = 262\).

Time = 0.26 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.63 \[ \int \frac {1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {4 \, b^{4} e \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac {4 \, b^{3} e^{2} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {3 \, b^{4} d^{4} + 10 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 6 \, a^{3} b d e^{3} - a^{4} e^{4} + 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (5 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (11 \, b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \, {\left (b d - a e\right )}^{5} {\left (b x + a\right )} {\left (e x + d\right )}^{3}} \]

[In]

integrate(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

-4*b^4*e*log(abs(b*x + a))/(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^
4 - a^5*b*e^5) + 4*b^3*e^2*log(abs(e*x + d))/(b^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^3*b^2*d^
2*e^4 + 5*a^4*b*d*e^5 - a^5*e^6) - 1/3*(3*b^4*d^4 + 10*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 6*a^3*b*d*e^3 - a^4*
e^4 + 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(5*b^4*d^2*e^2 - 4*a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 2*(11*b^4*d^3*e -
3*a*b^3*d^2*e^2 - 9*a^2*b^2*d*e^3 + a^3*b*e^4)*x)/((b*d - a*e)^5*(b*x + a)*(e*x + d)^3)

Mupad [B] (verification not implemented)

Time = 10.47 (sec) , antiderivative size = 534, normalized size of antiderivative = 4.02 \[ \int \frac {1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {8\,b^3\,e\,\mathrm {atanh}\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{{\left (a\,e-b\,d\right )}^5}+\frac {2\,b\,e\,x\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5}\right )}{{\left (a\,e-b\,d\right )}^5}-\frac {\frac {a^3\,e^3-5\,a^2\,b\,d\,e^2+13\,a\,b^2\,d^2\,e+3\,b^3\,d^3}{3\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {4\,b^3\,e^3\,x^3}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {2\,b^2\,x^2\,\left (a\,e^3+5\,b\,d\,e^2\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {2\,b\,x\,\left (-a^2\,e^3+8\,a\,b\,d\,e^2+11\,b^2\,d^2\,e\right )}{3\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}}{x^3\,\left (a\,e^3+3\,b\,d\,e^2\right )+x^2\,\left (3\,b\,d^2\,e+3\,a\,d\,e^2\right )+a\,d^3+x\,\left (b\,d^3+3\,a\,e\,d^2\right )+b\,e^3\,x^4} \]

[In]

int(1/((d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)),x)

[Out]

(8*b^3*e*atanh((a^5*e^5 + b^5*d^5 + 2*a^2*b^3*d^3*e^2 + 2*a^3*b^2*d^2*e^3 - 3*a*b^4*d^4*e - 3*a^4*b*d*e^4)/(a*
e - b*d)^5 + (2*b*e*x*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3))/(a*e - b*d)^5))
/(a*e - b*d)^5 - ((a^3*e^3 + 3*b^3*d^3 + 13*a*b^2*d^2*e - 5*a^2*b*d*e^2)/(3*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2
*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e^3)) + (4*b^3*e^3*x^3)/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*
e - 4*a^3*b*d*e^3) + (2*b^2*x^2*(a*e^3 + 5*b*d*e^2))/(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e -
4*a^3*b*d*e^3) + (2*b*x*(11*b^2*d^2*e - a^2*e^3 + 8*a*b*d*e^2))/(3*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*
a*b^3*d^3*e - 4*a^3*b*d*e^3)))/(x^3*(a*e^3 + 3*b*d*e^2) + x^2*(3*a*d*e^2 + 3*b*d^2*e) + a*d^3 + x*(b*d^3 + 3*a
*d^2*e) + b*e^3*x^4)