\(\int \frac {(d+e x)^5}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [1516]

Optimal result

Integrand size = 26, antiderivative size = 129 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^4 (5 b d-4 a e) x}{b^5}+\frac {e^5 x^2}{2 b^4}-\frac {(b d-a e)^5}{3 b^6 (a+b x)^3}-\frac {5 e (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac {10 e^2 (b d-a e)^3}{b^6 (a+b x)}+\frac {10 e^3 (b d-a e)^2 \log (a+b x)}{b^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 129, 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, 45} \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {10 e^3 (b d-a e)^2 \log (a+b x)}{b^6}-\frac {10 e^2 (b d-a e)^3}{b^6 (a+b x)}-\frac {5 e (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac {(b d-a e)^5}{3 b^6 (a+b x)^3}+\frac {e^4 x (5 b d-4 a e)}{b^5}+\frac {e^5 x^2}{2 b^4} \]

[In]

Int[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(e^4*(5*b*d - 4*a*e)*x)/b^5 + (e^5*x^2)/(2*b^4) - (b*d - a*e)^5/(3*b^6*(a + b*x)^3) - (5*e*(b*d - a*e)^4)/(2*b
^6*(a + b*x)^2) - (10*e^2*(b*d - a*e)^3)/(b^6*(a + b*x)) + (10*e^3*(b*d - a*e)^2*Log[a + b*x])/b^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 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.77 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {47 a^5 e^5+a^4 b e^4 (-130 d+81 e x)+a^3 b^2 e^3 \left (110 d^2-270 d e x-9 e^2 x^2\right )-a^2 b^3 e^2 \left (20 d^3-270 d^2 e x+90 d e^2 x^2+63 e^3 x^3\right )-5 a b^4 e \left (d^4+12 d^3 e x-36 d^2 e^2 x^2-18 d e^3 x^3+3 e^4 x^4\right )+b^5 \left (-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right )+60 e^3 (b d-a e)^2 (a+b x)^3 \log (a+b x)}{6 b^6 (a+b x)^3} \]

[In]

Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(47*a^5*e^5 + a^4*b*e^4*(-130*d + 81*e*x) + a^3*b^2*e^3*(110*d^2 - 270*d*e*x - 9*e^2*x^2) - a^2*b^3*e^2*(20*d^
3 - 270*d^2*e*x + 90*d*e^2*x^2 + 63*e^3*x^3) - 5*a*b^4*e*(d^4 + 12*d^3*e*x - 36*d^2*e^2*x^2 - 18*d*e^3*x^3 + 3
*e^4*x^4) + b^5*(-2*d^5 - 15*d^4*e*x - 60*d^3*e^2*x^2 + 30*d*e^4*x^4 + 3*e^5*x^5) + 60*e^3*(b*d - a*e)^2*(a +
b*x)^3*Log[a + b*x])/(6*b^6*(a + b*x)^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(251\) vs. \(2(123)=246\).

Time = 2.30 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.95

method	result	size
default	\(-\frac {e^{4} \left (-\frac {1}{2} b e \,x^{2}+4 a e x -5 b d x \right )}{b^{5}}-\frac {-a^{5} e^{5}+5 a^{4} b d \,e^{4}-10 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e +b^{5} d^{5}}{3 b^{6} \left (b x +a \right )^{3}}+\frac {10 e^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{6}}-\frac {5 e \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 b^{6} \left (b x +a \right )^{2}}+\frac {10 e^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{6} \left (b x +a \right )}\)	\(252\)
norman	\(\frac {\frac {\left (30 a^{3} e^{5}-60 a^{2} b d \,e^{4}+30 a \,b^{2} d^{2} e^{3}-10 b^{3} d^{3} e^{2}\right ) x^{2}}{b^{4}}+\frac {110 a^{5} e^{5}-220 a^{4} b d \,e^{4}+110 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e -2 b^{5} d^{5}}{6 b^{6}}+\frac {e^{5} x^{5}}{2 b}+\frac {\left (90 e^{5} a^{4}-180 a^{3} b d \,e^{4}+90 a^{2} b^{2} d^{2} e^{3}-20 a \,b^{3} d^{3} e^{2}-5 b^{4} d^{4} e \right ) x}{2 b^{5}}-\frac {5 e^{4} \left (a e -2 b d \right ) x^{4}}{2 b^{2}}}{\left (b x +a \right )^{3}}+\frac {10 e^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(255\)
risch	\(\frac {e^{5} x^{2}}{2 b^{4}}-\frac {4 e^{5} a x}{b^{5}}+\frac {5 e^{4} d x}{b^{4}}+\frac {\left (10 a^{3} b \,e^{5}-30 a^{2} b^{2} d \,e^{4}+30 a \,b^{3} d^{2} e^{3}-10 b^{4} d^{3} e^{2}\right ) x^{2}+\frac {5 e \left (7 e^{4} a^{4}-20 b \,e^{3} d \,a^{3}+18 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e -b^{4} d^{4}\right ) x}{2}+\frac {47 a^{5} e^{5}-130 a^{4} b d \,e^{4}+110 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e -2 b^{5} d^{5}}{6 b}}{b^{5} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}+\frac {10 e^{5} \ln \left (b x +a \right ) a^{2}}{b^{6}}-\frac {20 e^{4} \ln \left (b x +a \right ) a d}{b^{5}}+\frac {10 e^{3} \ln \left (b x +a \right ) d^{2}}{b^{4}}\)	\(287\)
parallelrisch	\(\frac {-2 b^{5} d^{5}+110 a^{5} e^{5}-220 a^{4} b d \,e^{4}+110 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e -540 a^{3} b^{2} d \,e^{4} x -360 \ln \left (b x +a \right ) x^{2} a^{2} b^{3} d \,e^{4}+180 \ln \left (b x +a \right ) x^{2} a \,b^{4} d^{2} e^{3}+270 a^{4} b \,e^{5} x -360 x^{2} a^{2} b^{3} d \,e^{4}+180 x^{2} a \,b^{4} d^{2} e^{3}+180 \ln \left (b x +a \right ) x \,a^{4} b \,e^{5}-120 \ln \left (b x +a \right ) a^{4} b d \,e^{4}-120 \ln \left (b x +a \right ) x^{3} a \,b^{4} d \,e^{4}+180 \ln \left (b x +a \right ) x \,a^{2} b^{3} d^{2} e^{3}-360 \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{4}+60 \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{3}+30 x^{4} b^{5} d \,e^{4}-15 b^{5} d^{4} e x +3 x^{5} e^{5} b^{5}+60 \ln \left (b x +a \right ) a^{5} e^{5}-15 x^{4} a \,b^{4} e^{5}+60 \ln \left (b x +a \right ) x^{3} a^{2} b^{3} e^{5}+60 \ln \left (b x +a \right ) x^{3} b^{5} d^{2} e^{3}+180 \ln \left (b x +a \right ) x^{2} a^{3} b^{2} e^{5}+180 x^{2} a^{3} b^{2} e^{5}-60 x^{2} b^{5} d^{3} e^{2}+270 x \,a^{2} b^{3} d^{2} e^{3}-60 x a \,b^{4} d^{3} e^{2}}{6 b^{6} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}\)	\(477\)

[In]

int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (123) = 246\).

Time = 0.33 (sec) , antiderivative size = 426, normalized size of antiderivative = 3.30 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {3 \, b^{5} e^{5} x^{5} - 2 \, b^{5} d^{5} - 5 \, a b^{4} d^{4} e - 20 \, a^{2} b^{3} d^{3} e^{2} + 110 \, a^{3} b^{2} d^{2} e^{3} - 130 \, a^{4} b d e^{4} + 47 \, a^{5} e^{5} + 15 \, {\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 9 \, {\left (10 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} - 3 \, {\left (20 \, b^{5} d^{3} e^{2} - 60 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 3 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (5 \, b^{5} d^{4} e + 20 \, a b^{4} d^{3} e^{2} - 90 \, a^{2} b^{3} d^{2} e^{3} + 90 \, a^{3} b^{2} d e^{4} - 27 \, a^{4} b e^{5}\right )} x + 60 \, {\left (a^{3} b^{2} d^{2} e^{3} - 2 \, a^{4} b d e^{4} + a^{5} e^{5} + {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (a^{2} b^{3} d^{2} e^{3} - 2 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \]

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/6*(3*b^5*e^5*x^5 - 2*b^5*d^5 - 5*a*b^4*d^4*e - 20*a^2*b^3*d^3*e^2 + 110*a^3*b^2*d^2*e^3 - 130*a^4*b*d*e^4 +
47*a^5*e^5 + 15*(2*b^5*d*e^4 - a*b^4*e^5)*x^4 + 9*(10*a*b^4*d*e^4 - 7*a^2*b^3*e^5)*x^3 - 3*(20*b^5*d^3*e^2 - 6
0*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 + 3*a^3*b^2*e^5)*x^2 - 3*(5*b^5*d^4*e + 20*a*b^4*d^3*e^2 - 90*a^2*b^3*d^2*e
^3 + 90*a^3*b^2*d*e^4 - 27*a^4*b*e^5)*x + 60*(a^3*b^2*d^2*e^3 - 2*a^4*b*d*e^4 + a^5*e^5 + (b^5*d^2*e^3 - 2*a*b
^4*d*e^4 + a^2*b^3*e^5)*x^3 + 3*(a*b^4*d^2*e^3 - 2*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 + a^4*b*e^5)*x)*log(b*x + a))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (119) = 238\).

Time = 1.73 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.20 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=x \left (- \frac {4 a e^{5}}{b^{5}} + \frac {5 d e^{4}}{b^{4}}\right ) + \frac {47 a^{5} e^{5} - 130 a^{4} b d e^{4} + 110 a^{3} b^{2} d^{2} e^{3} - 20 a^{2} b^{3} d^{3} e^{2} - 5 a b^{4} d^{4} e - 2 b^{5} d^{5} + x^{2} \cdot \left (60 a^{3} b^{2} e^{5} - 180 a^{2} b^{3} d e^{4} + 180 a b^{4} d^{2} e^{3} - 60 b^{5} d^{3} e^{2}\right ) + x \left (105 a^{4} b e^{5} - 300 a^{3} b^{2} d e^{4} + 270 a^{2} b^{3} d^{2} e^{3} - 60 a b^{4} d^{3} e^{2} - 15 b^{5} d^{4} e\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} + \frac {e^{5} x^{2}}{2 b^{4}} + \frac {10 e^{3} \left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{6}} \]

[In]

integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

x*(-4*a*e**5/b**5 + 5*d*e**4/b**4) + (47*a**5*e**5 - 130*a**4*b*d*e**4 + 110*a**3*b**2*d**2*e**3 - 20*a**2*b**
3*d**3*e**2 - 5*a*b**4*d**4*e - 2*b**5*d**5 + x**2*(60*a**3*b**2*e**5 - 180*a**2*b**3*d*e**4 + 180*a*b**4*d**2
*e**3 - 60*b**5*d**3*e**2) + x*(105*a**4*b*e**5 - 300*a**3*b**2*d*e**4 + 270*a**2*b**3*d**2*e**3 - 60*a*b**4*d
**3*e**2 - 15*b**5*d**4*e))/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + e**5*x**2/(2*b**4)
+ 10*e**3*(a*e - b*d)**2*log(a + b*x)/b**6

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (123) = 246\).

Time = 0.22 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.18 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {2 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} - 110 \, a^{3} b^{2} d^{2} e^{3} + 130 \, a^{4} b d e^{4} - 47 \, a^{5} e^{5} + 60 \, {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \, {\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} - 18 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 7 \, a^{4} b e^{5}\right )} x}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac {b e^{5} x^{2} + 2 \, {\left (5 \, b d e^{4} - 4 \, a e^{5}\right )} x}{2 \, b^{5}} + \frac {10 \, {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \log \left (b x + a\right )}{b^{6}} \]

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

-1/6*(2*b^5*d^5 + 5*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 - 110*a^3*b^2*d^2*e^3 + 130*a^4*b*d*e^4 - 47*a^5*e^5 + 60
*(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 15*(b^5*d^4*e + 4*a*b^4*d^3*e^2 - 18*a^
2*b^3*d^2*e^3 + 20*a^3*b^2*d*e^4 - 7*a^4*b*e^5)*x)/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 1/2*(b*e^
5*x^2 + 2*(5*b*d*e^4 - 4*a*e^5)*x)/b^5 + 10*(b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*log(b*x + a)/b^6

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (123) = 246\).

Time = 0.25 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.03 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {10 \, {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {b^{4} e^{5} x^{2} + 10 \, b^{4} d e^{4} x - 8 \, a b^{3} e^{5} x}{2 \, b^{8}} - \frac {2 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} - 110 \, a^{3} b^{2} d^{2} e^{3} + 130 \, a^{4} b d e^{4} - 47 \, a^{5} e^{5} + 60 \, {\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \, {\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} - 18 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 7 \, a^{4} b e^{5}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{6}} \]

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

10*(b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*log(abs(b*x + a))/b^6 + 1/2*(b^4*e^5*x^2 + 10*b^4*d*e^4*x - 8*a*b^3*e
^5*x)/b^8 - 1/6*(2*b^5*d^5 + 5*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 - 110*a^3*b^2*d^2*e^3 + 130*a^4*b*d*e^4 - 47*a
^5*e^5 + 60*(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 15*(b^5*d^4*e + 4*a*b^4*d^3*
e^2 - 18*a^2*b^3*d^2*e^3 + 20*a^3*b^2*d*e^4 - 7*a^4*b*e^5)*x)/((b*x + a)^3*b^6)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.22 \[ \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^5\,x^2}{2\,b^4}-x\,\left (\frac {4\,a\,e^5}{b^5}-\frac {5\,d\,e^4}{b^4}\right )-\frac {\frac {-47\,a^5\,e^5+130\,a^4\,b\,d\,e^4-110\,a^3\,b^2\,d^2\,e^3+20\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e+2\,b^5\,d^5}{6\,b}+x\,\left (-\frac {35\,a^4\,e^5}{2}+50\,a^3\,b\,d\,e^4-45\,a^2\,b^2\,d^2\,e^3+10\,a\,b^3\,d^3\,e^2+\frac {5\,b^4\,d^4\,e}{2}\right )-x^2\,\left (10\,a^3\,b\,e^5-30\,a^2\,b^2\,d\,e^4+30\,a\,b^3\,d^2\,e^3-10\,b^4\,d^3\,e^2\right )}{a^3\,b^5+3\,a^2\,b^6\,x+3\,a\,b^7\,x^2+b^8\,x^3}+\frac {\ln \left (a+b\,x\right )\,\left (10\,a^2\,e^5-20\,a\,b\,d\,e^4+10\,b^2\,d^2\,e^3\right )}{b^6} \]

[In]

int((d + e*x)^5/(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

(e^5*x^2)/(2*b^4) - x*((4*a*e^5)/b^5 - (5*d*e^4)/b^4) - ((2*b^5*d^5 - 47*a^5*e^5 + 20*a^2*b^3*d^3*e^2 - 110*a^
3*b^2*d^2*e^3 + 5*a*b^4*d^4*e + 130*a^4*b*d*e^4)/(6*b) + x*((5*b^4*d^4*e)/2 - (35*a^4*e^5)/2 + 10*a*b^3*d^3*e^
2 - 45*a^2*b^2*d^2*e^3 + 50*a^3*b*d*e^4) - x^2*(10*a^3*b*e^5 - 10*b^4*d^3*e^2 + 30*a*b^3*d^2*e^3 - 30*a^2*b^2*
d*e^4))/(a^3*b^5 + b^8*x^3 + 3*a^2*b^6*x + 3*a*b^7*x^2) + (log(a + b*x)*(10*a^2*e^5 + 10*b^2*d^2*e^3 - 20*a*b*
d*e^4))/b^6