\(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^2} \, dx\) [1491]

Optimal result

Integrand size = 26, antiderivative size = 156 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {15 b^2 (b d-a e)^4 x}{e^6}-\frac {(b d-a e)^6}{e^7 (d+e x)}-\frac {10 b^3 (b d-a e)^3 (d+e x)^2}{e^7}+\frac {5 b^4 (b d-a e)^2 (d+e x)^3}{e^7}-\frac {3 b^5 (b d-a e) (d+e x)^4}{2 e^7}+\frac {b^6 (d+e x)^5}{5 e^7}-\frac {6 b (b d-a e)^5 \log (d+e x)}{e^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=-\frac {3 b^5 (d+e x)^4 (b d-a e)}{2 e^7}+\frac {5 b^4 (d+e x)^3 (b d-a e)^2}{e^7}-\frac {10 b^3 (d+e x)^2 (b d-a e)^3}{e^7}+\frac {15 b^2 x (b d-a e)^4}{e^6}-\frac {(b d-a e)^6}{e^7 (d+e x)}-\frac {6 b (b d-a e)^5 \log (d+e x)}{e^7}+\frac {b^6 (d+e x)^5}{5 e^7} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {60 a^5 b d e^5-10 a^6 e^6+150 a^4 b^2 e^4 \left (-d^2+d e x+e^2 x^2\right )+100 a^3 b^3 e^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+50 a^2 b^4 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b^5 e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+b^6 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )-60 b (b d-a e)^5 (d+e x) \log (d+e x)}{10 e^7 (d+e x)} \]

[In]

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

[Out]

(60*a^5*b*d*e^5 - 10*a^6*e^6 + 150*a^4*b^2*e^4*(-d^2 + d*e*x + e^2*x^2) + 100*a^3*b^3*e^3*(2*d^3 - 4*d^2*e*x -
3*d*e^2*x^2 + e^3*x^3) + 50*a^2*b^4*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b^
5*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + b^6*(-10*d^6 + 50*d^5*
e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) - 60*b*(b*d - a*e)^5*(d + e*x
)*Log[d + e*x])/(10*e^7*(d + e*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(344\) vs. \(2(152)=304\).

Time = 2.21 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.21

method	result	size
norman	\(\frac {\frac {b^{3} \left (10 a^{3} e^{3}-10 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{3}}{e^{4}}+\frac {\left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+30 a^{4} b^{2} d^{2} e^{4}-60 a^{3} b^{3} d^{3} e^{3}+60 a^{2} b^{4} d^{4} e^{2}-30 a \,b^{5} d^{5} e +6 b^{6} d^{6}\right ) x}{d \,e^{6}}+\frac {b^{6} x^{6}}{5 e}+\frac {3 b^{2} \left (5 e^{4} a^{4}-10 b \,e^{3} d \,a^{3}+10 b^{2} e^{2} d^{2} a^{2}-5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{e^{5}}+\frac {b^{4} \left (10 a^{2} e^{2}-5 a b d e +b^{2} d^{2}\right ) x^{4}}{2 e^{3}}+\frac {3 b^{5} \left (5 a e -b d \right ) x^{5}}{10 e^{2}}}{e x +d}+\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(345\)
default	\(\frac {b^{2} \left (\frac {1}{5} b^{4} x^{5} e^{4}+\frac {3}{2} x^{4} a \,b^{3} e^{4}-\frac {1}{2} x^{4} b^{4} d \,e^{3}+5 x^{3} a^{2} b^{2} e^{4}-4 x^{3} a \,b^{3} d \,e^{3}+x^{3} b^{4} d^{2} e^{2}+10 x^{2} a^{3} b \,e^{4}-15 x^{2} a^{2} b^{2} d \,e^{3}+9 x^{2} a \,b^{3} d^{2} e^{2}-2 x^{2} b^{4} d^{3} e +15 e^{4} a^{4} x -40 b \,e^{3} d \,a^{3} x +45 b^{2} e^{2} d^{2} a^{2} x -24 a \,b^{3} d^{3} e x +5 b^{4} d^{4} x \right )}{e^{6}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{e^{7} \left (e x +d \right )}+\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(359\)
risch	\(-\frac {b^{6} d^{6}}{e^{7} \left (e x +d \right )}+\frac {6 b \ln \left (e x +d \right ) a^{5}}{e^{2}}-\frac {6 b^{6} \ln \left (e x +d \right ) d^{5}}{e^{7}}+\frac {b^{6} x^{5}}{5 e^{2}}-\frac {a^{6}}{e \left (e x +d \right )}-\frac {30 b^{2} \ln \left (e x +d \right ) a^{4} d}{e^{3}}+\frac {60 b^{3} \ln \left (e x +d \right ) a^{3} d^{2}}{e^{4}}-\frac {60 b^{4} \ln \left (e x +d \right ) a^{2} d^{3}}{e^{5}}+\frac {30 b^{5} \ln \left (e x +d \right ) a \,d^{4}}{e^{6}}+\frac {3 b^{5} x^{4} a}{2 e^{2}}-\frac {b^{6} x^{4} d}{2 e^{3}}+\frac {5 b^{4} x^{3} a^{2}}{e^{2}}+\frac {b^{6} x^{3} d^{2}}{e^{4}}+\frac {10 b^{3} x^{2} a^{3}}{e^{2}}-\frac {2 b^{6} x^{2} d^{3}}{e^{5}}+\frac {15 b^{2} a^{4} x}{e^{2}}+\frac {5 b^{6} d^{4} x}{e^{6}}-\frac {15 a^{2} b^{4} d^{4}}{e^{5} \left (e x +d \right )}+\frac {6 a \,b^{5} d^{5}}{e^{6} \left (e x +d \right )}+\frac {9 b^{5} x^{2} a \,d^{2}}{e^{4}}-\frac {40 b^{3} d \,a^{3} x}{e^{3}}+\frac {45 b^{4} d^{2} a^{2} x}{e^{4}}-\frac {24 b^{5} a \,d^{3} x}{e^{5}}+\frac {6 a^{5} b d}{e^{2} \left (e x +d \right )}-\frac {15 a^{4} b^{2} d^{2}}{e^{3} \left (e x +d \right )}+\frac {20 a^{3} b^{3} d^{3}}{e^{4} \left (e x +d \right )}-\frac {4 b^{5} x^{3} a d}{e^{3}}-\frac {15 b^{4} x^{2} a^{2} d}{e^{3}}\)	\(440\)
parallelrisch	\(\frac {15 x^{5} a \,b^{5} e^{6}-3 x^{5} b^{6} d \,e^{5}+50 x^{4} a^{2} b^{4} e^{6}+5 x^{4} b^{6} d^{2} e^{4}+100 x^{3} a^{3} b^{3} e^{6}-10 x^{3} b^{6} d^{3} e^{3}+150 x^{2} a^{4} b^{2} e^{6}-10 a^{6} e^{6}-60 b^{6} d^{6}+300 a \,b^{5} d^{5} e -300 a^{4} b^{2} d^{2} e^{4}+600 a^{3} b^{3} d^{3} e^{3}-600 a^{2} b^{4} d^{4} e^{2}+60 a^{5} b d \,e^{5}-300 \ln \left (e x +d \right ) x \,a^{4} b^{2} d \,e^{5}+600 \ln \left (e x +d \right ) x \,a^{3} b^{3} d^{2} e^{4}+300 \ln \left (e x +d \right ) x a \,b^{5} d^{4} e^{2}-600 \ln \left (e x +d \right ) x \,a^{2} b^{4} d^{3} e^{3}-60 \ln \left (e x +d \right ) b^{6} d^{6}+2 x^{6} b^{6} e^{6}+600 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}-600 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}+300 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +60 \ln \left (e x +d \right ) x \,a^{5} b \,e^{6}-60 \ln \left (e x +d \right ) x \,b^{6} d^{5} e +30 x^{2} b^{6} d^{4} e^{2}-25 x^{4} a \,b^{5} d \,e^{5}-100 x^{3} a^{2} b^{4} d \,e^{5}+50 x^{3} a \,b^{5} d^{2} e^{4}-300 x^{2} a^{3} b^{3} d \,e^{5}+300 x^{2} a^{2} b^{4} d^{2} e^{4}-150 x^{2} a \,b^{5} d^{3} e^{3}+60 \ln \left (e x +d \right ) a^{5} b d \,e^{5}-300 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}}{10 e^{7} \left (e x +d \right )}\)	\(519\)

[In]

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

[Out]

(b^3*(10*a^3*e^3-10*a^2*b*d*e^2+5*a*b^2*d^2*e-b^3*d^3)/e^4*x^3+(a^6*e^6-6*a^5*b*d*e^5+30*a^4*b^2*d^2*e^4-60*a^
3*b^3*d^3*e^3+60*a^2*b^4*d^4*e^2-30*a*b^5*d^5*e+6*b^6*d^6)/d/e^6*x+1/5*b^6/e*x^6+3*b^2*(5*a^4*e^4-10*a^3*b*d*e
^3+10*a^2*b^2*d^2*e^2-5*a*b^3*d^3*e+b^4*d^4)/e^5*x^2+1/2*b^4*(10*a^2*e^2-5*a*b*d*e+b^2*d^2)/e^3*x^4+3/10*b^5*(
5*a*e-b*d)/e^2*x^5)/(e*x+d)+6*b/e^7*(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)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (152) = 304\).

Time = 0.30 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.18 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {2 \, b^{6} e^{6} x^{6} - 10 \, b^{6} d^{6} + 60 \, a b^{5} d^{5} e - 150 \, a^{2} b^{4} d^{4} e^{2} + 200 \, a^{3} b^{3} d^{3} e^{3} - 150 \, a^{4} b^{2} d^{2} e^{4} + 60 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - 5 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} - 10 \, {\left (b^{6} d^{3} e^{3} - 5 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} - 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 5 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} - 10 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (5 \, b^{6} d^{5} e - 24 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} - 40 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5}\right )} x - 60 \, {\left (b^{6} d^{6} - 5 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} - a^{5} b d e^{5} + {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{8} x + d e^{7}\right )}} \]

[In]

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

[Out]

1/10*(2*b^6*e^6*x^6 - 10*b^6*d^6 + 60*a*b^5*d^5*e - 150*a^2*b^4*d^4*e^2 + 200*a^3*b^3*d^3*e^3 - 150*a^4*b^2*d^
2*e^4 + 60*a^5*b*d*e^5 - 10*a^6*e^6 - 3*(b^6*d*e^5 - 5*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 5*a*b^5*d*e^5 + 10*a^
2*b^4*e^6)*x^4 - 10*(b^6*d^3*e^3 - 5*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 - 10*a^3*b^3*e^6)*x^3 + 30*(b^6*d^4*e^2
- 5*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 - 10*a^3*b^3*d*e^5 + 5*a^4*b^2*e^6)*x^2 + 10*(5*b^6*d^5*e - 24*a*b^5*d^
4*e^2 + 45*a^2*b^4*d^3*e^3 - 40*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5)*x - 60*(b^6*d^6 - 5*a*b^5*d^5*e + 10*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 - a^5*b*d*e^5 + (b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*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 - a^5*b*e^6)*x)*log(e*x + d))/(e^8*x + d*e^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (143) = 286\).

Time = 0.64 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {b^{6} x^{5}}{5 e^{2}} + \frac {6 b \left (a e - b d\right )^{5} \log {\left (d + e x \right )}}{e^{7}} + x^{4} \cdot \left (\frac {3 a b^{5}}{2 e^{2}} - \frac {b^{6} d}{2 e^{3}}\right ) + x^{3} \cdot \left (\frac {5 a^{2} b^{4}}{e^{2}} - \frac {4 a b^{5} d}{e^{3}} + \frac {b^{6} d^{2}}{e^{4}}\right ) + x^{2} \cdot \left (\frac {10 a^{3} b^{3}}{e^{2}} - \frac {15 a^{2} b^{4} d}{e^{3}} + \frac {9 a b^{5} d^{2}}{e^{4}} - \frac {2 b^{6} d^{3}}{e^{5}}\right ) + x \left (\frac {15 a^{4} b^{2}}{e^{2}} - \frac {40 a^{3} b^{3} d}{e^{3}} + \frac {45 a^{2} b^{4} d^{2}}{e^{4}} - \frac {24 a b^{5} d^{3}}{e^{5}} + \frac {5 b^{6} d^{4}}{e^{6}}\right ) + \frac {- a^{6} e^{6} + 6 a^{5} b d e^{5} - 15 a^{4} b^{2} d^{2} e^{4} + 20 a^{3} b^{3} d^{3} e^{3} - 15 a^{2} b^{4} d^{4} e^{2} + 6 a b^{5} d^{5} e - b^{6} d^{6}}{d e^{7} + e^{8} x} \]

[In]

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

[Out]

b**6*x**5/(5*e**2) + 6*b*(a*e - b*d)**5*log(d + e*x)/e**7 + x**4*(3*a*b**5/(2*e**2) - b**6*d/(2*e**3)) + x**3*
(5*a**2*b**4/e**2 - 4*a*b**5*d/e**3 + b**6*d**2/e**4) + x**2*(10*a**3*b**3/e**2 - 15*a**2*b**4*d/e**3 + 9*a*b*
*5*d**2/e**4 - 2*b**6*d**3/e**5) + x*(15*a**4*b**2/e**2 - 40*a**3*b**3*d/e**3 + 45*a**2*b**4*d**2/e**4 - 24*a*
b**5*d**3/e**5 + 5*b**6*d**4/e**6) + (-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)/(d*e**7 + e**8*x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (152) = 304\).

Time = 0.20 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=-\frac {b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}}{e^{8} x + d e^{7}} + \frac {2 \, b^{6} e^{4} x^{5} - 5 \, {\left (b^{6} d e^{3} - 3 \, a b^{5} e^{4}\right )} x^{4} + 10 \, {\left (b^{6} d^{2} e^{2} - 4 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{3} - 10 \, {\left (2 \, b^{6} d^{3} e - 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} - 10 \, a^{3} b^{3} e^{4}\right )} x^{2} + 10 \, {\left (5 \, b^{6} d^{4} - 24 \, a b^{5} d^{3} e + 45 \, a^{2} b^{4} d^{2} e^{2} - 40 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x}{10 \, e^{6}} - \frac {6 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

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

[Out]

-(b^6*d^6 - 6*a*b^5*d^5*e + 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^5*b*d*e^5 + a^6
*e^6)/(e^8*x + d*e^7) + 1/10*(2*b^6*e^4*x^5 - 5*(b^6*d*e^3 - 3*a*b^5*e^4)*x^4 + 10*(b^6*d^2*e^2 - 4*a*b^5*d*e^
3 + 5*a^2*b^4*e^4)*x^3 - 10*(2*b^6*d^3*e - 9*a*b^5*d^2*e^2 + 15*a^2*b^4*d*e^3 - 10*a^3*b^3*e^4)*x^2 + 10*(5*b^
6*d^4 - 24*a*b^5*d^3*e + 45*a^2*b^4*d^2*e^2 - 40*a^3*b^3*d*e^3 + 15*a^4*b^2*e^4)*x)/e^6 - 6*(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)*log(e*x + d)/e^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (152) = 304\).

Time = 0.26 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.86 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {{\left (2 \, b^{6} - \frac {15 \, {\left (b^{6} d e - a b^{5} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {50 \, {\left (b^{6} d^{2} e^{2} - 2 \, a b^{5} d e^{3} + a^{2} b^{4} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {100 \, {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {150 \, {\left (b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}}\right )} {\left (e x + d\right )}^{5}}{10 \, e^{7}} + \frac {6 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{7}} - \frac {\frac {b^{6} d^{6} e^{5}}{e x + d} - \frac {6 \, a b^{5} d^{5} e^{6}}{e x + d} + \frac {15 \, a^{2} b^{4} d^{4} e^{7}}{e x + d} - \frac {20 \, a^{3} b^{3} d^{3} e^{8}}{e x + d} + \frac {15 \, a^{4} b^{2} d^{2} e^{9}}{e x + d} - \frac {6 \, a^{5} b d e^{10}}{e x + d} + \frac {a^{6} e^{11}}{e x + d}}{e^{12}} \]

[In]

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

[Out]

1/10*(2*b^6 - 15*(b^6*d*e - a*b^5*e^2)/((e*x + d)*e) + 50*(b^6*d^2*e^2 - 2*a*b^5*d*e^3 + a^2*b^4*e^4)/((e*x +
d)^2*e^2) - 100*(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)/((e*x + d)^3*e^3) + 150*(b^6*d
^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4*b^2*e^8)/((e*x + d)^4*e^4))*(e*x + d)^5/e
^7 + 6*(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)*log(a
bs(e*x + d)/((e*x + d)^2*abs(e)))/e^7 - (b^6*d^6*e^5/(e*x + d) - 6*a*b^5*d^5*e^6/(e*x + d) + 15*a^2*b^4*d^4*e^
7/(e*x + d) - 20*a^3*b^3*d^3*e^8/(e*x + d) + 15*a^4*b^2*d^2*e^9/(e*x + d) - 6*a^5*b*d*e^10/(e*x + d) + a^6*e^1
1/(e*x + d))/e^12

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 523, normalized size of antiderivative = 3.35 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=x^4\,\left (\frac {3\,a\,b^5}{2\,e^2}-\frac {b^6\,d}{2\,e^3}\right )-x^3\,\left (\frac {2\,d\,\left (\frac {6\,a\,b^5}{e^2}-\frac {2\,b^6\,d}{e^3}\right )}{3\,e}-\frac {5\,a^2\,b^4}{e^2}+\frac {b^6\,d^2}{3\,e^4}\right )+x^2\,\left (\frac {10\,a^3\,b^3}{e^2}+\frac {d\,\left (\frac {2\,d\,\left (\frac {6\,a\,b^5}{e^2}-\frac {2\,b^6\,d}{e^3}\right )}{e}-\frac {15\,a^2\,b^4}{e^2}+\frac {b^6\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {6\,a\,b^5}{e^2}-\frac {2\,b^6\,d}{e^3}\right )}{2\,e^2}\right )+x\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {6\,a\,b^5}{e^2}-\frac {2\,b^6\,d}{e^3}\right )}{e}-\frac {15\,a^2\,b^4}{e^2}+\frac {b^6\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {20\,a^3\,b^3}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {6\,a\,b^5}{e^2}-\frac {2\,b^6\,d}{e^3}\right )}{e}-\frac {15\,a^2\,b^4}{e^2}+\frac {b^6\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {6\,a\,b^5}{e^2}-\frac {2\,b^6\,d}{e^3}\right )}{e^2}\right )}{e}+\frac {15\,a^4\,b^2}{e^2}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-6\,a^5\,b\,e^5+30\,a^4\,b^2\,d\,e^4-60\,a^3\,b^3\,d^2\,e^3+60\,a^2\,b^4\,d^3\,e^2-30\,a\,b^5\,d^4\,e+6\,b^6\,d^5\right )}{e^7}+\frac {b^6\,x^5}{5\,e^2}-\frac {a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6}{e\,\left (x\,e^7+d\,e^6\right )} \]

[In]

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

[Out]

x^4*((3*a*b^5)/(2*e^2) - (b^6*d)/(2*e^3)) - x^3*((2*d*((6*a*b^5)/e^2 - (2*b^6*d)/e^3))/(3*e) - (5*a^2*b^4)/e^2
+ (b^6*d^2)/(3*e^4)) + x^2*((10*a^3*b^3)/e^2 + (d*((2*d*((6*a*b^5)/e^2 - (2*b^6*d)/e^3))/e - (15*a^2*b^4)/e^2
+ (b^6*d^2)/e^4))/e - (d^2*((6*a*b^5)/e^2 - (2*b^6*d)/e^3))/(2*e^2)) + x*((d^2*((2*d*((6*a*b^5)/e^2 - (2*b^6*
d)/e^3))/e - (15*a^2*b^4)/e^2 + (b^6*d^2)/e^4))/e^2 - (2*d*((20*a^3*b^3)/e^2 + (2*d*((2*d*((6*a*b^5)/e^2 - (2*
b^6*d)/e^3))/e - (15*a^2*b^4)/e^2 + (b^6*d^2)/e^4))/e - (d^2*((6*a*b^5)/e^2 - (2*b^6*d)/e^3))/e^2))/e + (15*a^
4*b^2)/e^2) - (log(d + e*x)*(6*b^6*d^5 - 6*a^5*b*e^5 + 30*a^4*b^2*d*e^4 + 60*a^2*b^4*d^3*e^2 - 60*a^3*b^3*d^2*
e^3 - 30*a*b^5*d^4*e))/e^7 + (b^6*x^5)/(5*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)/(e*(d*e^6 + e^7*x))