\(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^6} \, dx\) [1495]

Optimal result

Integrand size = 26, antiderivative size = 155 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx=\frac {b^6 x}{e^6}-\frac {(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac {3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac {5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac {10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac {15 b^4 (b d-a e)^2}{e^7 (d+e x)}-\frac {6 b^5 (b d-a e) \log (d+e x)}{e^7} \]

[Out]

b^6*x/e^6-1/5*(-a*e+b*d)^6/e^7/(e*x+d)^5+3/2*b*(-a*e+b*d)^5/e^7/(e*x+d)^4-5*b^2*(-a*e+b*d)^4/e^7/(e*x+d)^3+10*
b^3*(-a*e+b*d)^3/e^7/(e*x+d)^2-15*b^4*(-a*e+b*d)^2/e^7/(e*x+d)-6*b^5*(-a*e+b*d)*ln(e*x+d)/e^7

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 155, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx=-\frac {6 b^5 (b d-a e) \log (d+e x)}{e^7}-\frac {15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac {10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac {5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac {3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac {(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac {b^6 x}{e^6} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^6,x]

[Out]

(b^6*x)/e^6 - (b*d - a*e)^6/(5*e^7*(d + e*x)^5) + (3*b*(b*d - a*e)^5)/(2*e^7*(d + e*x)^4) - (5*b^2*(b*d - a*e)
^4)/(e^7*(d + e*x)^3) + (10*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)^2) - (15*b^4*(b*d - a*e)^2)/(e^7*(d + e*x)) - (6
*b^5*(b*d - a*e)*Log[d + e*x])/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 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 {(a+b x)^6}{(d+e x)^6} \, dx \\ & = \int \left (\frac {b^6}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^6}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^5}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^4}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^3}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^2}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)}\right ) \, dx \\ & = \frac {b^6 x}{e^6}-\frac {(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac {3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac {5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac {10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac {15 b^4 (b d-a e)^2}{e^7 (d+e x)}-\frac {6 b^5 (b d-a e) \log (d+e x)}{e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx=-\frac {2 a^6 e^6+3 a^5 b e^5 (d+5 e x)+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+30 a^2 b^4 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-a b^5 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+b^6 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+60 b^5 (b d-a e) (d+e x)^5 \log (d+e x)}{10 e^7 (d+e x)^5} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^6,x]

[Out]

-1/10*(2*a^6*e^6 + 3*a^5*b*e^5*(d + 5*e*x) + 5*a^4*b^2*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 10*a^3*b^3*e^3*(d^3
+ 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 30*a^2*b^4*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5
*e^4*x^4) - a*b^5*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) + b^6*(87*d^6 +
375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + 60*b^5*(b*d -
a*e)*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(347\) vs. \(2(151)=302\).

Time = 2.27 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.25

method	result	size
risch	\(\frac {b^{6} x}{e^{6}}+\frac {\left (-15 e^{5} a^{2} b^{4}+30 d \,e^{4} a \,b^{5}-15 d^{2} e^{3} b^{6}\right ) x^{4}-10 b^{3} e^{2} \left (a^{3} e^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}\right ) x^{3}-5 b^{2} e \left (e^{4} a^{4}+2 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-22 a \,b^{3} d^{3} e +13 b^{4} d^{4}\right ) x^{2}-\frac {b \left (3 a^{5} e^{5}+5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}+30 a^{2} b^{3} d^{3} e^{2}-125 a \,b^{4} d^{4} e +77 b^{5} d^{5}\right ) x}{2}-\frac {2 a^{6} e^{6}+3 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+30 a^{2} b^{4} d^{4} e^{2}-137 a \,b^{5} d^{5} e +87 b^{6} d^{6}}{10 e}}{e^{6} \left (e x +d \right )^{5}}+\frac {6 b^{5} \ln \left (e x +d \right ) a}{e^{6}}-\frac {6 b^{6} \ln \left (e x +d \right ) d}{e^{7}}\)	\(348\)
default	\(\frac {b^{6} x}{e^{6}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{7} \left (e x +d \right )}-\frac {5 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{e^{7} \left (e x +d \right )^{3}}-\frac {3 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{2 e^{7} \left (e x +d \right )^{4}}-\frac {10 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{7} \left (e x +d \right )^{2}}+\frac {6 b^{5} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{7}}\)	\(349\)
norman	\(\frac {\frac {b^{6} x^{6}}{e}-\frac {2 a^{6} e^{6}+3 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+30 a^{2} b^{4} d^{4} e^{2}-137 a \,b^{5} d^{5} e +137 b^{6} d^{6}}{10 e^{7}}-\frac {5 \left (3 e^{2} a^{2} b^{4}-6 d e a \,b^{5}+6 b^{6} d^{2}\right ) x^{4}}{e^{3}}-\frac {10 \left (e^{3} a^{3} b^{3}+3 d \,e^{2} a^{2} b^{4}-9 d^{2} e a \,b^{5}+9 d^{3} b^{6}\right ) x^{3}}{e^{4}}-\frac {5 \left (e^{4} a^{4} b^{2}+2 d \,e^{3} a^{3} b^{3}+6 d^{2} e^{2} a^{2} b^{4}-22 d^{3} e a \,b^{5}+22 d^{4} b^{6}\right ) x^{2}}{e^{5}}-\frac {\left (3 a^{5} b \,e^{5}+5 d \,e^{4} a^{4} b^{2}+10 d^{2} e^{3} a^{3} b^{3}+30 d^{3} e^{2} a^{2} b^{4}-125 d^{4} e a \,b^{5}+125 d^{5} b^{6}\right ) x}{2 e^{6}}}{\left (e x +d \right )^{5}}+\frac {6 b^{5} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{7}}\)	\(349\)
parallelrisch	\(\frac {-15 x \,a^{5} b \,e^{6}-625 x \,b^{6} d^{5} e -150 x^{4} a^{2} b^{4} e^{6}-300 x^{4} b^{6} d^{2} e^{4}-100 x^{3} a^{3} b^{3} e^{6}-900 x^{3} b^{6} d^{3} e^{3}-50 x^{2} a^{4} b^{2} e^{6}-2 a^{6} e^{6}-137 b^{6} d^{6}+137 a \,b^{5} d^{5} e -5 a^{4} b^{2} d^{2} e^{4}-10 a^{3} b^{3} d^{3} e^{3}-30 a^{2} b^{4} d^{4} e^{2}-3 a^{5} b d \,e^{5}-600 \ln \left (e x +d \right ) x^{3} b^{6} d^{3} e^{3}+300 \ln \left (e x +d \right ) x^{4} a \,b^{5} d \,e^{5}+600 \ln \left (e x +d \right ) x^{2} a \,b^{5} d^{3} e^{3}+300 \ln \left (e x +d \right ) x a \,b^{5} d^{4} e^{2}+600 \ln \left (e x +d \right ) x^{3} a \,b^{5} d^{2} e^{4}-60 \ln \left (e x +d \right ) b^{6} d^{6}+10 x^{6} b^{6} e^{6}-300 \ln \left (e x +d \right ) x^{4} b^{6} d^{2} e^{4}+60 \ln \left (e x +d \right ) x^{5} a \,b^{5} e^{6}-60 \ln \left (e x +d \right ) x^{5} b^{6} d \,e^{5}+60 \ln \left (e x +d \right ) a \,b^{5} d^{5} e -600 \ln \left (e x +d \right ) x^{2} b^{6} d^{4} e^{2}-300 \ln \left (e x +d \right ) x \,b^{6} d^{5} e -1100 x^{2} b^{6} d^{4} e^{2}+300 x^{4} a \,b^{5} d \,e^{5}-300 x^{3} a^{2} b^{4} d \,e^{5}+900 x^{3} a \,b^{5} d^{2} e^{4}-100 x^{2} a^{3} b^{3} d \,e^{5}-300 x^{2} a^{2} b^{4} d^{2} e^{4}+1100 x^{2} a \,b^{5} d^{3} e^{3}-25 x \,a^{4} b^{2} d \,e^{5}-50 x \,a^{3} b^{3} d^{2} e^{4}-150 x \,a^{2} b^{4} d^{3} e^{3}+625 x a \,b^{5} d^{4} e^{2}}{10 e^{7} \left (e x +d \right )^{5}}\)	\(575\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

b^6*x/e^6+((-15*a^2*b^4*e^5+30*a*b^5*d*e^4-15*b^6*d^2*e^3)*x^4-10*b^3*e^2*(a^3*e^3+3*a^2*b*d*e^2-9*a*b^2*d^2*e
+5*b^3*d^3)*x^3-5*b^2*e*(a^4*e^4+2*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-22*a*b^3*d^3*e+13*b^4*d^4)*x^2-1/2*b*(3*a^5*e
^5+5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3+30*a^2*b^3*d^3*e^2-125*a*b^4*d^4*e+77*b^5*d^5)*x-1/10/e*(2*a^6*e^6+3*a^5*b
*d*e^5+5*a^4*b^2*d^2*e^4+10*a^3*b^3*d^3*e^3+30*a^2*b^4*d^4*e^2-137*a*b^5*d^5*e+87*b^6*d^6))/e^6/(e*x+d)^5+6*b^
5/e^6*ln(e*x+d)*a-6*b^6/e^7*ln(e*x+d)*d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (151) = 302\).

Time = 0.31 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.50 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx=\frac {10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \, {\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \, {\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \, {\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - a b^{5} d^{5} e + {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \, {\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/10*(10*b^6*e^6*x^6 + 50*b^6*d*e^5*x^5 - 87*b^6*d^6 + 137*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 10*a^3*b^3*d^3*e
^3 - 5*a^4*b^2*d^2*e^4 - 3*a^5*b*d*e^5 - 2*a^6*e^6 - 50*(b^6*d^2*e^4 - 6*a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 - 10
0*(4*b^6*d^3*e^3 - 9*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 - 50*(12*b^6*d^4*e^2 - 22*a*b^5*d^3*e^
3 + 6*a^2*b^4*d^2*e^4 + 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 - 5*(75*b^6*d^5*e - 125*a*b^5*d^4*e^2 + 30*a^2*b^4*
d^3*e^3 + 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - a*b^5*d^5*e + (b^6*d*e^5 - a*b
^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - a*b^5*d*e^5)*x^4 + 10*(b^6*d^3*e^3 - a*b^5*d^2*e^4)*x^3 + 10*(b^6*d^4*e^2 - a*b
^5*d^3*e^3)*x^2 + 5*(b^6*d^5*e - a*b^5*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 +
10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx=\text {Timed out} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**6,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (151) = 302\).

Time = 0.25 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.56 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx=\frac {b^{6} x}{e^{6}} - \frac {87 \, b^{6} d^{6} - 137 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + 2 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (5 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (13 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (77 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} - \frac {6 \, {\left (b^{6} d - a b^{5} e\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^6,x, algorithm="maxima")

[Out]

b^6*x/e^6 - 1/10*(87*b^6*d^6 - 137*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 +
3*a^5*b*d*e^5 + 2*a^6*e^6 + 150*(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 100*(5*b^6*d^3*e^3 - 9*a*b^
5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 50*(13*b^6*d^4*e^2 - 22*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 2
*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 5*(77*b^6*d^5*e - 125*a*b^5*d^4*e^2 + 30*a^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*
e^4 + 5*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^
8*x + d^5*e^7) - 6*(b^6*d - a*b^5*e)*log(e*x + d)/e^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (151) = 302\).

Time = 0.27 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx=\frac {b^{6} x}{e^{6}} - \frac {6 \, {\left (b^{6} d - a b^{5} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {87 \, b^{6} d^{6} - 137 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + 2 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (5 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (13 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (77 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x}{10 \, {\left (e x + d\right )}^{5} e^{7}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^6,x, algorithm="giac")

[Out]

b^6*x/e^6 - 6*(b^6*d - a*b^5*e)*log(abs(e*x + d))/e^7 - 1/10*(87*b^6*d^6 - 137*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^
2 + 10*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 3*a^5*b*d*e^5 + 2*a^6*e^6 + 150*(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^
2*b^4*e^6)*x^4 + 100*(5*b^6*d^3*e^3 - 9*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 50*(13*b^6*d^4*e^
2 - 22*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 5*(77*b^6*d^5*e - 125*a*b^5*d^
4*e^2 + 30*a^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x)/((e*x + d)^5*e^7)

Mupad [B] (verification not implemented)

Time = 10.23 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.57 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^6} \, dx=\frac {b^6\,x}{e^6}-\frac {\ln \left (d+e\,x\right )\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )}{e^7}-\frac {x^2\,\left (5\,a^4\,b^2\,e^5+10\,a^3\,b^3\,d\,e^4+30\,a^2\,b^4\,d^2\,e^3-110\,a\,b^5\,d^3\,e^2+65\,b^6\,d^4\,e\right )+x^4\,\left (15\,a^2\,b^4\,e^5-30\,a\,b^5\,d\,e^4+15\,b^6\,d^2\,e^3\right )+\frac {2\,a^6\,e^6+3\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4+10\,a^3\,b^3\,d^3\,e^3+30\,a^2\,b^4\,d^4\,e^2-137\,a\,b^5\,d^5\,e+87\,b^6\,d^6}{10\,e}+x\,\left (\frac {3\,a^5\,b\,e^5}{2}+\frac {5\,a^4\,b^2\,d\,e^4}{2}+5\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-\frac {125\,a\,b^5\,d^4\,e}{2}+\frac {77\,b^6\,d^5}{2}\right )+x^3\,\left (10\,a^3\,b^3\,e^5+30\,a^2\,b^4\,d\,e^4-90\,a\,b^5\,d^2\,e^3+50\,b^6\,d^3\,e^2\right )}{d^5\,e^6+5\,d^4\,e^7\,x+10\,d^3\,e^8\,x^2+10\,d^2\,e^9\,x^3+5\,d\,e^{10}\,x^4+e^{11}\,x^5} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^6,x)

[Out]

(b^6*x)/e^6 - (log(d + e*x)*(6*b^6*d - 6*a*b^5*e))/e^7 - (x^2*(65*b^6*d^4*e + 5*a^4*b^2*e^5 - 110*a*b^5*d^3*e^
2 + 10*a^3*b^3*d*e^4 + 30*a^2*b^4*d^2*e^3) + x^4*(15*a^2*b^4*e^5 + 15*b^6*d^2*e^3 - 30*a*b^5*d*e^4) + (2*a^6*e
^6 + 87*b^6*d^6 + 30*a^2*b^4*d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 - 137*a*b^5*d^5*e + 3*a^5*b*d*e^
5)/(10*e) + x*((77*b^6*d^5)/2 + (3*a^5*b*e^5)/2 + (5*a^4*b^2*d*e^4)/2 + 15*a^2*b^4*d^3*e^2 + 5*a^3*b^3*d^2*e^3
- (125*a*b^5*d^4*e)/2) + x^3*(10*a^3*b^3*e^5 + 50*b^6*d^3*e^2 - 90*a*b^5*d^2*e^3 + 30*a^2*b^4*d*e^4))/(d^5*e^
6 + e^11*x^5 + 5*d^4*e^7*x + 5*d*e^10*x^4 + 10*d^3*e^8*x^2 + 10*d^2*e^9*x^3)