\(\int \frac {(d+e x)^6}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [1515]

Optimal result

Integrand size = 26, antiderivative size = 156 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {15 e^4 (b d-a e)^2 x}{b^6}-\frac {(b d-a e)^6}{3 b^7 (a+b x)^3}-\frac {3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac {15 e^2 (b d-a e)^4}{b^7 (a+b x)}+\frac {3 e^5 (b d-a e) (a+b x)^2}{b^7}+\frac {e^6 (a+b x)^3}{3 b^7}+\frac {20 e^3 (b d-a e)^3 \log (a+b x)}{b^7} \]

[Out]

15*e^4*(-a*e+b*d)^2*x/b^6-1/3*(-a*e+b*d)^6/b^7/(b*x+a)^3-3*e*(-a*e+b*d)^5/b^7/(b*x+a)^2-15*e^2*(-a*e+b*d)^4/b^
7/(b*x+a)+3*e^5*(-a*e+b*d)*(b*x+a)^2/b^7+1/3*e^6*(b*x+a)^3/b^7+20*e^3*(-a*e+b*d)^3*ln(b*x+a)/b^7

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 156, 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} \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {3 e^5 (a+b x)^2 (b d-a e)}{b^7}+\frac {20 e^3 (b d-a e)^3 \log (a+b x)}{b^7}-\frac {15 e^2 (b d-a e)^4}{b^7 (a+b x)}-\frac {3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac {(b d-a e)^6}{3 b^7 (a+b x)^3}+\frac {e^6 (a+b x)^3}{3 b^7}+\frac {15 e^4 x (b d-a e)^2}{b^6} \]

[In]

Int[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(15*e^4*(b*d - a*e)^2*x)/b^6 - (b*d - a*e)^6/(3*b^7*(a + b*x)^3) - (3*e*(b*d - a*e)^5)/(b^7*(a + b*x)^2) - (15
*e^2*(b*d - a*e)^4)/(b^7*(a + b*x)) + (3*e^5*(b*d - a*e)*(a + b*x)^2)/b^7 + (e^6*(a + b*x)^3)/(3*b^7) + (20*e^
3*(b*d - a*e)^3*Log[a + b*x])/b^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 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)^6}{(a+b x)^4} \, dx \\ & = \int \left (\frac {15 e^4 (b d-a e)^2}{b^6}+\frac {(b d-a e)^6}{b^6 (a+b x)^4}+\frac {6 e (b d-a e)^5}{b^6 (a+b x)^3}+\frac {15 e^2 (b d-a e)^4}{b^6 (a+b x)^2}+\frac {20 e^3 (b d-a e)^3}{b^6 (a+b x)}+\frac {6 e^5 (b d-a e) (a+b x)}{b^6}+\frac {e^6 (a+b x)^2}{b^6}\right ) \, dx \\ & = \frac {15 e^4 (b d-a e)^2 x}{b^6}-\frac {(b d-a e)^6}{3 b^7 (a+b x)^3}-\frac {3 e (b d-a e)^5}{b^7 (a+b x)^2}-\frac {15 e^2 (b d-a e)^4}{b^7 (a+b x)}+\frac {3 e^5 (b d-a e) (a+b x)^2}{b^7}+\frac {e^6 (a+b x)^3}{3 b^7}+\frac {20 e^3 (b d-a e)^3 \log (a+b x)}{b^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.93 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-37 a^6 e^6+3 a^5 b e^5 (47 d-17 e x)+3 a^4 b^2 e^4 \left (-65 d^2+81 d e x+13 e^2 x^2\right )+a^3 b^3 e^3 \left (110 d^3-405 d^2 e x-27 d e^2 x^2+73 e^3 x^3\right )+3 a^2 b^4 e^2 \left (-5 d^4+90 d^3 e x-45 d^2 e^2 x^2-63 d e^3 x^3+5 e^4 x^4\right )-3 a b^5 e \left (d^5+15 d^4 e x-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d e^4 x^4+e^5 x^5\right )+b^6 \left (-d^6-9 d^5 e x-45 d^4 e^2 x^2+45 d^2 e^4 x^4+9 d e^5 x^5+e^6 x^6\right )-60 e^3 (-b d+a e)^3 (a+b x)^3 \log (a+b x)}{3 b^7 (a+b x)^3} \]

[In]

Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(-37*a^6*e^6 + 3*a^5*b*e^5*(47*d - 17*e*x) + 3*a^4*b^2*e^4*(-65*d^2 + 81*d*e*x + 13*e^2*x^2) + a^3*b^3*e^3*(11
0*d^3 - 405*d^2*e*x - 27*d*e^2*x^2 + 73*e^3*x^3) + 3*a^2*b^4*e^2*(-5*d^4 + 90*d^3*e*x - 45*d^2*e^2*x^2 - 63*d*
e^3*x^3 + 5*e^4*x^4) - 3*a*b^5*e*(d^5 + 15*d^4*e*x - 60*d^3*e^2*x^2 - 45*d^2*e^3*x^3 + 15*d*e^4*x^4 + e^5*x^5)
+ b^6*(-d^6 - 9*d^5*e*x - 45*d^4*e^2*x^2 + 45*d^2*e^4*x^4 + 9*d*e^5*x^5 + e^6*x^6) - 60*e^3*(-(b*d) + a*e)^3*
(a + b*x)^3*Log[a + b*x])/(3*b^7*(a + b*x)^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(343\) vs. \(2(152)=304\).

Time = 2.46 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.21

method	result	size
norman	\(\frac {-\frac {110 a^{6} e^{6}-330 a^{5} b d \,e^{5}+330 a^{4} b^{2} d^{2} e^{4}-110 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +b^{6} d^{6}}{3 b^{7}}+\frac {e^{6} x^{6}}{3 b}-\frac {3 \left (20 a^{4} e^{6}-60 a^{3} b d \,e^{5}+60 a^{2} b^{2} d^{2} e^{4}-20 a \,b^{3} d^{3} e^{3}+5 b^{4} d^{4} e^{2}\right ) x^{2}}{b^{5}}-\frac {3 \left (30 a^{5} e^{6}-90 a^{4} b d \,e^{5}+90 a^{3} b^{2} d^{2} e^{4}-30 a^{2} b^{3} d^{3} e^{3}+5 a \,b^{4} d^{4} e^{2}+b^{5} d^{5} e \right ) x}{b^{6}}+\frac {5 e^{4} \left (a^{2} e^{2}-3 a b d e +3 b^{2} d^{2}\right ) x^{4}}{b^{3}}-\frac {e^{5} \left (a e -3 b d \right ) x^{5}}{b^{2}}}{\left (b x +a \right )^{3}}-\frac {20 e^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{7}}\)	\(344\)
default	\(\frac {e^{4} \left (\frac {1}{3} b^{2} e^{2} x^{3}-2 x^{2} a b \,e^{2}+3 b^{2} d e \,x^{2}+10 a^{2} e^{2} x -24 a b d e x +15 b^{2} d^{2} x \right )}{b^{6}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{3 b^{7} \left (b x +a \right )^{3}}-\frac {20 e^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{7}}+\frac {3 e \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{b^{7} \left (b x +a \right )^{2}}-\frac {15 e^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{b^{7} \left (b x +a \right )}\)	\(346\)
risch	\(\frac {e^{6} x^{3}}{3 b^{4}}-\frac {2 e^{6} x^{2} a}{b^{5}}+\frac {3 e^{5} d \,x^{2}}{b^{4}}+\frac {10 e^{6} a^{2} x}{b^{6}}-\frac {24 e^{5} a d x}{b^{5}}+\frac {15 e^{4} d^{2} x}{b^{4}}+\frac {\left (-15 a^{4} b \,e^{6}+60 a^{3} d \,e^{5} b^{2}-90 a^{2} d^{2} e^{4} b^{3}+60 a \,b^{4} d^{3} e^{3}-15 d^{4} e^{2} b^{5}\right ) x^{2}-3 e \left (9 a^{5} e^{5}-35 a^{4} b d \,e^{4}+50 a^{3} b^{2} d^{2} e^{3}-30 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x -\frac {37 a^{6} e^{6}-141 a^{5} b d \,e^{5}+195 a^{4} b^{2} d^{2} e^{4}-110 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +b^{6} d^{6}}{3 b}}{b^{6} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}-\frac {20 e^{6} \ln \left (b x +a \right ) a^{3}}{b^{7}}+\frac {60 e^{5} \ln \left (b x +a \right ) a^{2} d}{b^{6}}-\frac {60 e^{4} \ln \left (b x +a \right ) a \,d^{2}}{b^{5}}+\frac {20 e^{3} \ln \left (b x +a \right ) d^{3}}{b^{4}}\)	\(386\)
parallelrisch	\(-\frac {270 x \,a^{5} b \,e^{6}+9 x \,b^{6} d^{5} e +3 x^{5} a \,b^{5} e^{6}-9 x^{5} b^{6} d \,e^{5}-15 x^{4} a^{2} b^{4} e^{6}-45 x^{4} b^{6} d^{2} e^{4}+180 x^{2} a^{4} b^{2} e^{6}+110 a^{6} e^{6}+b^{6} d^{6}+3 a \,b^{5} d^{5} e +330 a^{4} b^{2} d^{2} e^{4}-110 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-330 a^{5} b d \,e^{5}+180 \ln \left (b x +a \right ) x^{3} a \,b^{5} d^{2} e^{4}-540 \ln \left (b x +a \right ) x^{2} a^{3} b^{3} d \,e^{5}+540 \ln \left (b x +a \right ) x \,a^{3} b^{3} d^{2} e^{4}+60 \ln \left (b x +a \right ) a^{6} e^{6}-540 \ln \left (b x +a \right ) x \,a^{4} b^{2} d \,e^{5}-180 \ln \left (b x +a \right ) x^{3} a^{2} b^{4} d \,e^{5}+540 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} d^{2} e^{4}-180 \ln \left (b x +a \right ) x^{2} a \,b^{5} d^{3} e^{3}-180 \ln \left (b x +a \right ) x \,a^{2} b^{4} d^{3} e^{3}-x^{6} b^{6} e^{6}+60 \ln \left (b x +a \right ) x^{3} a^{3} b^{3} e^{6}-60 \ln \left (b x +a \right ) x^{3} b^{6} d^{3} e^{3}+180 \ln \left (b x +a \right ) x^{2} a^{4} b^{2} e^{6}+180 \ln \left (b x +a \right ) x \,a^{5} b \,e^{6}-180 \ln \left (b x +a \right ) a^{5} b d \,e^{5}+180 \ln \left (b x +a \right ) a^{4} b^{2} d^{2} e^{4}-60 \ln \left (b x +a \right ) a^{3} b^{3} d^{3} e^{3}+45 x^{2} b^{6} d^{4} e^{2}+45 x^{4} a \,b^{5} d \,e^{5}-540 x^{2} a^{3} b^{3} d \,e^{5}+540 x^{2} a^{2} b^{4} d^{2} e^{4}-180 x^{2} a \,b^{5} d^{3} e^{3}-810 x \,a^{4} b^{2} d \,e^{5}+810 x \,a^{3} b^{3} d^{2} e^{4}-270 x \,a^{2} b^{4} d^{3} e^{3}+45 x a \,b^{5} d^{4} e^{2}}{3 b^{7} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}\)	\(650\)

[In]

int((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

(-1/3*(110*a^6*e^6-330*a^5*b*d*e^5+330*a^4*b^2*d^2*e^4-110*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2+3*a*b^5*d^5*e+b^
6*d^6)/b^7+1/3*e^6/b*x^6-3*(20*a^4*e^6-60*a^3*b*d*e^5+60*a^2*b^2*d^2*e^4-20*a*b^3*d^3*e^3+5*b^4*d^4*e^2)/b^5*x
^2-3*(30*a^5*e^6-90*a^4*b*d*e^5+90*a^3*b^2*d^2*e^4-30*a^2*b^3*d^3*e^3+5*a*b^4*d^4*e^2+b^5*d^5*e)/b^6*x+5*e^4*(
a^2*e^2-3*a*b*d*e+3*b^2*d^2)/b^3*x^4-e^5*(a*e-3*b*d)/b^2*x^5)/(b*x+a)^3-20/b^7*e^3*(a^3*e^3-3*a^2*b*d*e^2+3*a*
b^2*d^2*e-b^3*d^3)*ln(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (152) = 304\).

Time = 0.32 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.70 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {b^{6} e^{6} x^{6} - b^{6} d^{6} - 3 \, a b^{5} d^{5} e - 15 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 195 \, a^{4} b^{2} d^{2} e^{4} + 141 \, a^{5} b d e^{5} - 37 \, a^{6} e^{6} + 3 \, {\left (3 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (3 \, b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + {\left (135 \, a b^{5} d^{2} e^{4} - 189 \, a^{2} b^{4} d e^{5} + 73 \, a^{3} b^{3} e^{6}\right )} x^{3} - 3 \, {\left (15 \, b^{6} d^{4} e^{2} - 60 \, a b^{5} d^{3} e^{3} + 45 \, a^{2} b^{4} d^{2} e^{4} + 9 \, a^{3} b^{3} d e^{5} - 13 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \, {\left (3 \, b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} - 90 \, a^{2} b^{4} d^{3} e^{3} + 135 \, a^{3} b^{3} d^{2} e^{4} - 81 \, a^{4} b^{2} d e^{5} + 17 \, a^{5} b e^{6}\right )} x + 60 \, {\left (a^{3} b^{3} d^{3} e^{3} - 3 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} - a^{6} e^{6} + {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (a b^{5} d^{3} e^{3} - 3 \, a^{2} b^{4} d^{2} e^{4} + 3 \, a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} d^{3} e^{3} - 3 \, a^{3} b^{3} d^{2} e^{4} + 3 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{3 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} \]

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/3*(b^6*e^6*x^6 - b^6*d^6 - 3*a*b^5*d^5*e - 15*a^2*b^4*d^4*e^2 + 110*a^3*b^3*d^3*e^3 - 195*a^4*b^2*d^2*e^4 +
141*a^5*b*d*e^5 - 37*a^6*e^6 + 3*(3*b^6*d*e^5 - a*b^5*e^6)*x^5 + 15*(3*b^6*d^2*e^4 - 3*a*b^5*d*e^5 + a^2*b^4*e
^6)*x^4 + (135*a*b^5*d^2*e^4 - 189*a^2*b^4*d*e^5 + 73*a^3*b^3*e^6)*x^3 - 3*(15*b^6*d^4*e^2 - 60*a*b^5*d^3*e^3
+ 45*a^2*b^4*d^2*e^4 + 9*a^3*b^3*d*e^5 - 13*a^4*b^2*e^6)*x^2 - 3*(3*b^6*d^5*e + 15*a*b^5*d^4*e^2 - 90*a^2*b^4*
d^3*e^3 + 135*a^3*b^3*d^2*e^4 - 81*a^4*b^2*d*e^5 + 17*a^5*b*e^6)*x + 60*(a^3*b^3*d^3*e^3 - 3*a^4*b^2*d^2*e^4 +
3*a^5*b*d*e^5 - a^6*e^6 + (b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 3*(a*b^5*d^3*
e^3 - 3*a^2*b^4*d^2*e^4 + 3*a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 3*(a^2*b^4*d^3*e^3 - 3*a^3*b^3*d^2*e^4 + 3*a^4*
b^2*d*e^5 - a^5*b*e^6)*x)*log(b*x + a))/(b^10*x^3 + 3*a*b^9*x^2 + 3*a^2*b^8*x + a^3*b^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (143) = 286\).

Time = 4.76 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.35 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=x^{2} \left (- \frac {2 a e^{6}}{b^{5}} + \frac {3 d e^{5}}{b^{4}}\right ) + x \left (\frac {10 a^{2} e^{6}}{b^{6}} - \frac {24 a d e^{5}}{b^{5}} + \frac {15 d^{2} e^{4}}{b^{4}}\right ) + \frac {- 37 a^{6} e^{6} + 141 a^{5} b d e^{5} - 195 a^{4} b^{2} d^{2} e^{4} + 110 a^{3} b^{3} d^{3} e^{3} - 15 a^{2} b^{4} d^{4} e^{2} - 3 a b^{5} d^{5} e - b^{6} d^{6} + x^{2} \left (- 45 a^{4} b^{2} e^{6} + 180 a^{3} b^{3} d e^{5} - 270 a^{2} b^{4} d^{2} e^{4} + 180 a b^{5} d^{3} e^{3} - 45 b^{6} d^{4} e^{2}\right ) + x \left (- 81 a^{5} b e^{6} + 315 a^{4} b^{2} d e^{5} - 450 a^{3} b^{3} d^{2} e^{4} + 270 a^{2} b^{4} d^{3} e^{3} - 45 a b^{5} d^{4} e^{2} - 9 b^{6} d^{5} e\right )}{3 a^{3} b^{7} + 9 a^{2} b^{8} x + 9 a b^{9} x^{2} + 3 b^{10} x^{3}} + \frac {e^{6} x^{3}}{3 b^{4}} - \frac {20 e^{3} \left (a e - b d\right )^{3} \log {\left (a + b x \right )}}{b^{7}} \]

[In]

integrate((e*x+d)**6/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

x**2*(-2*a*e**6/b**5 + 3*d*e**5/b**4) + x*(10*a**2*e**6/b**6 - 24*a*d*e**5/b**5 + 15*d**2*e**4/b**4) + (-37*a*
*6*e**6 + 141*a**5*b*d*e**5 - 195*a**4*b**2*d**2*e**4 + 110*a**3*b**3*d**3*e**3 - 15*a**2*b**4*d**4*e**2 - 3*a
*b**5*d**5*e - b**6*d**6 + x**2*(-45*a**4*b**2*e**6 + 180*a**3*b**3*d*e**5 - 270*a**2*b**4*d**2*e**4 + 180*a*b
**5*d**3*e**3 - 45*b**6*d**4*e**2) + x*(-81*a**5*b*e**6 + 315*a**4*b**2*d*e**5 - 450*a**3*b**3*d**2*e**4 + 270
*a**2*b**4*d**3*e**3 - 45*a*b**5*d**4*e**2 - 9*b**6*d**5*e))/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*
b**10*x**3) + e**6*x**3/(3*b**4) - 20*e**3*(a*e - b*d)**3*log(a + b*x)/b**7

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (152) = 304\).

Time = 0.21 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.40 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 195 \, a^{4} b^{2} d^{2} e^{4} - 141 \, a^{5} b d e^{5} + 37 \, a^{6} e^{6} + 45 \, {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{2} d e^{5} + 9 \, a^{5} b e^{6}\right )} x}{3 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} + \frac {b^{2} e^{6} x^{3} + 3 \, {\left (3 \, b^{2} d e^{5} - 2 \, a b e^{6}\right )} x^{2} + 3 \, {\left (15 \, b^{2} d^{2} e^{4} - 24 \, a b d e^{5} + 10 \, a^{2} e^{6}\right )} x}{3 \, b^{6}} + \frac {20 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \log \left (b x + a\right )}{b^{7}} \]

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

-1/3*(b^6*d^6 + 3*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 195*a^4*b^2*d^2*e^4 - 141*a^5*b*d*e
^5 + 37*a^6*e^6 + 45*(b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 +
9*(b^6*d^5*e + 5*a*b^5*d^4*e^2 - 30*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 - 35*a^4*b^2*d*e^5 + 9*a^5*b*e^6)*x)
/(b^10*x^3 + 3*a*b^9*x^2 + 3*a^2*b^8*x + a^3*b^7) + 1/3*(b^2*e^6*x^3 + 3*(3*b^2*d*e^5 - 2*a*b*e^6)*x^2 + 3*(15
*b^2*d^2*e^4 - 24*a*b*d*e^5 + 10*a^2*e^6)*x)/b^6 + 20*(b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6
)*log(b*x + a)/b^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (152) = 304\).

Time = 0.25 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.28 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {20 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 195 \, a^{4} b^{2} d^{2} e^{4} - 141 \, a^{5} b d e^{5} + 37 \, a^{6} e^{6} + 45 \, {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{2} d e^{5} + 9 \, a^{5} b e^{6}\right )} x}{3 \, {\left (b x + a\right )}^{3} b^{7}} + \frac {b^{8} e^{6} x^{3} + 9 \, b^{8} d e^{5} x^{2} - 6 \, a b^{7} e^{6} x^{2} + 45 \, b^{8} d^{2} e^{4} x - 72 \, a b^{7} d e^{5} x + 30 \, a^{2} b^{6} e^{6} x}{3 \, b^{12}} \]

[In]

integrate((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

20*(b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*log(abs(b*x + a))/b^7 - 1/3*(b^6*d^6 + 3*a*b^5*d^
5*e + 15*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 195*a^4*b^2*d^2*e^4 - 141*a^5*b*d*e^5 + 37*a^6*e^6 + 45*(b^6*
d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 9*(b^6*d^5*e + 5*a*b^5*d^
4*e^2 - 30*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 - 35*a^4*b^2*d*e^5 + 9*a^5*b*e^6)*x)/((b*x + a)^3*b^7) + 1/3*(
b^8*e^6*x^3 + 9*b^8*d*e^5*x^2 - 6*a*b^7*e^6*x^2 + 45*b^8*d^2*e^4*x - 72*a*b^7*d*e^5*x + 30*a^2*b^6*e^6*x)/b^12

Mupad [B] (verification not implemented)

Time = 9.95 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.52 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=x\,\left (\frac {4\,a\,\left (\frac {4\,a\,e^6}{b^5}-\frac {6\,d\,e^5}{b^4}\right )}{b}-\frac {6\,a^2\,e^6}{b^6}+\frac {15\,d^2\,e^4}{b^4}\right )-\frac {x^2\,\left (15\,a^4\,b\,e^6-60\,a^3\,b^2\,d\,e^5+90\,a^2\,b^3\,d^2\,e^4-60\,a\,b^4\,d^3\,e^3+15\,b^5\,d^4\,e^2\right )+\frac {37\,a^6\,e^6-141\,a^5\,b\,d\,e^5+195\,a^4\,b^2\,d^2\,e^4-110\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2+3\,a\,b^5\,d^5\,e+b^6\,d^6}{3\,b}+x\,\left (27\,a^5\,e^6-105\,a^4\,b\,d\,e^5+150\,a^3\,b^2\,d^2\,e^4-90\,a^2\,b^3\,d^3\,e^3+15\,a\,b^4\,d^4\,e^2+3\,b^5\,d^5\,e\right )}{a^3\,b^6+3\,a^2\,b^7\,x+3\,a\,b^8\,x^2+b^9\,x^3}-x^2\,\left (\frac {2\,a\,e^6}{b^5}-\frac {3\,d\,e^5}{b^4}\right )-\frac {\ln \left (a+b\,x\right )\,\left (20\,a^3\,e^6-60\,a^2\,b\,d\,e^5+60\,a\,b^2\,d^2\,e^4-20\,b^3\,d^3\,e^3\right )}{b^7}+\frac {e^6\,x^3}{3\,b^4} \]

[In]

int((d + e*x)^6/(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x*((4*a*((4*a*e^6)/b^5 - (6*d*e^5)/b^4))/b - (6*a^2*e^6)/b^6 + (15*d^2*e^4)/b^4) - (x^2*(15*a^4*b*e^6 + 15*b^5
*d^4*e^2 - 60*a*b^4*d^3*e^3 - 60*a^3*b^2*d*e^5 + 90*a^2*b^3*d^2*e^4) + (37*a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*
e^2 - 110*a^3*b^3*d^3*e^3 + 195*a^4*b^2*d^2*e^4 + 3*a*b^5*d^5*e - 141*a^5*b*d*e^5)/(3*b) + x*(27*a^5*e^6 + 3*b
^5*d^5*e + 15*a*b^4*d^4*e^2 - 90*a^2*b^3*d^3*e^3 + 150*a^3*b^2*d^2*e^4 - 105*a^4*b*d*e^5))/(a^3*b^6 + b^9*x^3
+ 3*a^2*b^7*x + 3*a*b^8*x^2) - x^2*((2*a*e^6)/b^5 - (3*d*e^5)/b^4) - (log(a + b*x)*(20*a^3*e^6 - 20*b^3*d^3*e^
3 + 60*a*b^2*d^2*e^4 - 60*a^2*b*d*e^5))/b^7 + (e^6*x^3)/(3*b^4)