Integrand size = 31, antiderivative size = 186 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=-\frac {21 b^2 (b d-a e)^5 x}{e^7}+\frac {(b d-a e)^7}{e^8 (d+e x)}+\frac {35 b^3 (b d-a e)^4 (d+e x)^2}{2 e^8}-\frac {35 b^4 (b d-a e)^3 (d+e x)^3}{3 e^8}+\frac {21 b^5 (b d-a e)^2 (d+e x)^4}{4 e^8}-\frac {7 b^6 (b d-a e) (d+e x)^5}{5 e^8}+\frac {b^7 (d+e x)^6}{6 e^8}+\frac {7 b (b d-a e)^6 \log (d+e x)}{e^8} \] Output:
-21*b^2*(-a*e+b*d)^5*x/e^7+(-a*e+b*d)^7/e^8/(e*x+d)+35/2*b^3*(-a*e+b*d)^4* (e*x+d)^2/e^8-35/3*b^4*(-a*e+b*d)^3*(e*x+d)^3/e^8+21/4*b^5*(-a*e+b*d)^2*(e *x+d)^4/e^8-7/5*b^6*(-a*e+b*d)*(e*x+d)^5/e^8+1/6*b^7*(e*x+d)^6/e^8+7*b*(-a *e+b*d)^6*ln(e*x+d)/e^8
Leaf count is larger than twice the leaf count of optimal. \(387\) vs. \(2(186)=372\).
Time = 0.14 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.08 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {420 a^6 b d e^6-60 a^7 e^7+1260 a^5 b^2 e^5 \left (-d^2+d e x+e^2 x^2\right )+1050 a^4 b^3 e^4 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+700 a^3 b^4 e^3 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+105 a^2 b^5 e^2 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+42 a b^6 e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+b^7 \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )+420 b (b d-a e)^6 (d+e x) \log (d+e x)}{60 e^8 (d+e x)} \] Input:
Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^2,x]
Output:
(420*a^6*b*d*e^6 - 60*a^7*e^7 + 1260*a^5*b^2*e^5*(-d^2 + d*e*x + e^2*x^2) + 1050*a^4*b^3*e^4*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 700*a^3*b ^4*e^3*(-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) + 105* a^2*b^5*e^2*(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) + 42*a*b^6*e*(-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) + b^7*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e ^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7) + 420*b*(b*d - a*e)^6*(d + e*x)*Log[d + e*x])/(60*e^8*(d + e*x))
Time = 0.82 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^7}{(d+e x)^2}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^7}{(d+e x)^2}dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {7 b^6 (d+e x)^4 (b d-a e)}{e^7}+\frac {21 b^5 (d+e x)^3 (b d-a e)^2}{e^7}-\frac {35 b^4 (d+e x)^2 (b d-a e)^3}{e^7}+\frac {35 b^3 (d+e x) (b d-a e)^4}{e^7}-\frac {21 b^2 (b d-a e)^5}{e^7}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)}+\frac {(a e-b d)^7}{e^7 (d+e x)^2}+\frac {b^7 (d+e x)^5}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7 b^6 (d+e x)^5 (b d-a e)}{5 e^8}+\frac {21 b^5 (d+e x)^4 (b d-a e)^2}{4 e^8}-\frac {35 b^4 (d+e x)^3 (b d-a e)^3}{3 e^8}+\frac {35 b^3 (d+e x)^2 (b d-a e)^4}{2 e^8}-\frac {21 b^2 x (b d-a e)^5}{e^7}+\frac {(b d-a e)^7}{e^8 (d+e x)}+\frac {7 b (b d-a e)^6 \log (d+e x)}{e^8}+\frac {b^7 (d+e x)^6}{6 e^8}\) |
Input:
Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^2,x]
Output:
(-21*b^2*(b*d - a*e)^5*x)/e^7 + (b*d - a*e)^7/(e^8*(d + e*x)) + (35*b^3*(b *d - a*e)^4*(d + e*x)^2)/(2*e^8) - (35*b^4*(b*d - a*e)^3*(d + e*x)^3)/(3*e ^8) + (21*b^5*(b*d - a*e)^2*(d + e*x)^4)/(4*e^8) - (7*b^6*(b*d - a*e)*(d + e*x)^5)/(5*e^8) + (b^7*(d + e*x)^6)/(6*e^8) + (7*b*(b*d - a*e)^6*Log[d + e*x])/e^8
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(176)=352\).
Time = 1.35 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.41
| method | result | size |
| norman | \(\frac {\frac {\left (a^{7} e^{7}-7 a^{6} b d \,e^{6}+42 a^{5} b^{2} d^{2} e^{5}-105 a^{4} b^{3} d^{3} e^{4}+140 a^{3} b^{4} d^{4} e^{3}-105 a^{2} b^{5} d^{5} e^{2}+42 a \,b^{6} d^{6} e -7 b^{7} d^{7}\right ) x}{d \,e^{7}}+\frac {b^{7} x^{7}}{6 e}+\frac {7 b^{2} \left (6 e^{5} a^{5}-15 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}-15 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) x^{2}}{2 e^{6}}+\frac {7 b^{3} \left (15 a^{4} e^{4}-20 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}-6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{3}}{6 e^{5}}+\frac {7 b^{4} \left (20 e^{3} a^{3}-15 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{4}}{12 e^{4}}+\frac {7 b^{5} \left (15 e^{2} a^{2}-6 a b d e +b^{2} d^{2}\right ) x^{5}}{20 e^{3}}+\frac {7 b^{6} \left (6 a e -b d \right ) x^{6}}{30 e^{2}}}{e x +d}+\frac {7 b \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(449\) |
| default | \(\frac {b^{2} \left (\frac {1}{6} b^{5} x^{6} e^{5}+\frac {7}{5} x^{5} a \,b^{4} e^{5}-\frac {2}{5} x^{5} b^{5} d \,e^{4}+\frac {21}{4} x^{4} a^{2} b^{3} e^{5}-\frac {7}{2} x^{4} a \,b^{4} d \,e^{4}+\frac {3}{4} x^{4} b^{5} d^{2} e^{3}+\frac {35}{3} x^{3} a^{3} b^{2} e^{5}-14 x^{3} a^{2} b^{3} d \,e^{4}+7 x^{3} a \,b^{4} d^{2} e^{3}-\frac {4}{3} x^{3} b^{5} d^{3} e^{2}+\frac {35}{2} a^{4} b \,e^{5} x^{2}-35 a^{3} b^{2} d \,e^{4} x^{2}+\frac {63}{2} x^{2} a^{2} b^{3} d^{2} e^{3}-14 x^{2} a \,b^{4} d^{3} e^{2}+\frac {5}{2} b^{5} d^{4} e \,x^{2}+21 e^{5} a^{5} x -70 a^{4} b d \,e^{4} x +105 a^{3} b^{2} d^{2} e^{3} x -84 a^{2} b^{3} d^{3} e^{2} x +35 a \,b^{4} d^{4} e x -6 b^{5} d^{5} x \right )}{e^{7}}+\frac {7 b \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {a^{7} e^{7}-7 a^{6} b d \,e^{6}+21 a^{5} b^{2} d^{2} e^{5}-35 a^{4} b^{3} d^{3} e^{4}+35 a^{3} b^{4} d^{4} e^{3}-21 a^{2} b^{5} d^{5} e^{2}+7 a \,b^{6} d^{6} e -b^{7} d^{7}}{e^{8} \left (e x +d \right )}\) | \(478\) |
| risch | \(\frac {7 b \ln \left (e x +d \right ) a^{6}}{e^{2}}+\frac {7 b^{7} \ln \left (e x +d \right ) d^{6}}{e^{8}}+\frac {b^{7} d^{7}}{e^{8} \left (e x +d \right )}+\frac {105 b^{3} \ln \left (e x +d \right ) a^{4} d^{2}}{e^{4}}-\frac {140 b^{4} \ln \left (e x +d \right ) a^{3} d^{3}}{e^{5}}+\frac {105 b^{5} \ln \left (e x +d \right ) a^{2} d^{4}}{e^{6}}-\frac {42 b^{6} \ln \left (e x +d \right ) a \,d^{5}}{e^{7}}+\frac {7 a^{6} b d}{e^{2} \left (e x +d \right )}-\frac {21 a^{5} b^{2} d^{2}}{e^{3} \left (e x +d \right )}+\frac {35 a^{4} b^{3} d^{3}}{e^{4} \left (e x +d \right )}-\frac {35 a^{3} b^{4} d^{4}}{e^{5} \left (e x +d \right )}+\frac {21 a^{2} b^{5} d^{5}}{e^{6} \left (e x +d \right )}-\frac {7 a \,b^{6} d^{6}}{e^{7} \left (e x +d \right )}+\frac {7 b^{6} x^{5} a}{5 e^{2}}-\frac {2 b^{7} x^{5} d}{5 e^{3}}+\frac {21 b^{5} x^{4} a^{2}}{4 e^{2}}+\frac {3 b^{7} x^{4} d^{2}}{4 e^{4}}+\frac {35 b^{4} x^{3} a^{3}}{3 e^{2}}-\frac {4 b^{7} x^{3} d^{3}}{3 e^{5}}+\frac {35 b^{3} a^{4} x^{2}}{2 e^{2}}+\frac {5 b^{7} d^{4} x^{2}}{2 e^{6}}+\frac {21 b^{2} a^{5} x}{e^{2}}-\frac {6 b^{7} d^{5} x}{e^{7}}+\frac {105 b^{4} a^{3} d^{2} x}{e^{4}}-\frac {84 b^{5} a^{2} d^{3} x}{e^{5}}+\frac {35 b^{6} a \,d^{4} x}{e^{6}}-\frac {42 b^{2} \ln \left (e x +d \right ) a^{5} d}{e^{3}}+\frac {7 b^{6} x^{3} a \,d^{2}}{e^{4}}-\frac {35 b^{4} a^{3} d \,x^{2}}{e^{3}}+\frac {63 b^{5} x^{2} a^{2} d^{2}}{2 e^{4}}-\frac {14 b^{6} x^{2} a \,d^{3}}{e^{5}}-\frac {70 b^{3} a^{4} d x}{e^{3}}-\frac {7 b^{6} x^{4} a d}{2 e^{3}}-\frac {14 b^{5} x^{3} a^{2} d}{e^{3}}+\frac {b^{7} x^{6}}{6 e^{2}}-\frac {a^{7}}{e \left (e x +d \right )}\) | \(571\) |
| parallelrisch | \(\frac {420 b^{7} d^{7}+420 a^{6} b d \,e^{6}-2520 a^{5} b^{2} d^{2} e^{5}+6300 a^{4} b^{3} d^{3} e^{4}-2520 \ln \left (e x +d \right ) x \,a^{5} b^{2} d \,e^{6}+6300 \ln \left (e x +d \right ) x \,a^{4} b^{3} d^{2} e^{5}-2520 \ln \left (e x +d \right ) x a \,b^{6} d^{5} e^{2}-8400 a^{3} b^{4} d^{4} e^{3}+6300 a^{2} b^{5} d^{5} e^{2}-2520 a \,b^{6} d^{6} e -8400 \ln \left (e x +d \right ) x \,a^{3} b^{4} d^{3} e^{4}+6300 \ln \left (e x +d \right ) x \,a^{2} b^{5} d^{4} e^{3}+420 \ln \left (e x +d \right ) x \,a^{6} b \,e^{7}+420 \ln \left (e x +d \right ) x \,b^{7} d^{6} e -525 x^{4} a^{2} b^{5} d \,e^{6}+210 x^{4} a \,b^{6} d^{2} e^{5}-126 x^{5} a \,b^{6} d \,e^{6}+420 \ln \left (e x +d \right ) a^{6} b d \,e^{6}-2520 \ln \left (e x +d \right ) a^{5} b^{2} d^{2} e^{5}+6300 \ln \left (e x +d \right ) a^{4} b^{3} d^{3} e^{4}-3150 x^{2} a^{2} b^{5} d^{3} e^{4}+1260 x^{2} a \,b^{6} d^{4} e^{3}-1400 x^{3} a^{3} b^{4} d \,e^{6}-8400 \ln \left (e x +d \right ) a^{3} b^{4} d^{4} e^{3}+6300 \ln \left (e x +d \right ) a^{2} b^{5} d^{5} e^{2}-2520 \ln \left (e x +d \right ) a \,b^{6} d^{6} e -3150 x^{2} a^{4} b^{3} d \,e^{6}+4200 x^{2} a^{3} b^{4} d^{2} e^{5}+1050 x^{3} a^{2} b^{5} d^{2} e^{5}-420 x^{3} a \,b^{6} d^{3} e^{4}+1260 x^{2} a^{5} b^{2} e^{7}-210 x^{2} b^{7} d^{5} e^{2}+1050 x^{3} a^{4} b^{3} e^{7}+70 x^{3} b^{7} d^{4} e^{3}+700 x^{4} a^{3} b^{4} e^{7}-35 x^{4} b^{7} d^{3} e^{4}+315 x^{5} a^{2} b^{5} e^{7}+21 x^{5} b^{7} d^{2} e^{5}+84 x^{6} a \,b^{6} e^{7}-14 x^{6} b^{7} d \,e^{6}-60 a^{7} e^{7}+10 x^{7} b^{7} e^{7}+420 \ln \left (e x +d \right ) b^{7} d^{7}}{60 e^{8} \left (e x +d \right )}\) | \(666\) |
Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
((a^7*e^7-7*a^6*b*d*e^6+42*a^5*b^2*d^2*e^5-105*a^4*b^3*d^3*e^4+140*a^3*b^4 *d^4*e^3-105*a^2*b^5*d^5*e^2+42*a*b^6*d^6*e-7*b^7*d^7)/d/e^7*x+1/6/e*b^7*x ^7+7/2*b^2*(6*a^5*e^5-15*a^4*b*d*e^4+20*a^3*b^2*d^2*e^3-15*a^2*b^3*d^3*e^2 +6*a*b^4*d^4*e-b^5*d^5)/e^6*x^2+7/6*b^3*(15*a^4*e^4-20*a^3*b*d*e^3+15*a^2* b^2*d^2*e^2-6*a*b^3*d^3*e+b^4*d^4)/e^5*x^3+7/12*b^4*(20*a^3*e^3-15*a^2*b*d *e^2+6*a*b^2*d^2*e-b^3*d^3)/e^4*x^4+7/20*b^5*(15*a^2*e^2-6*a*b*d*e+b^2*d^2 )/e^3*x^5+7/30*b^6*(6*a*e-b*d)/e^2*x^6)/(e*x+d)+7*b/e^8*(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)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (176) = 352\).
Time = 0.08 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.38 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {10 \, b^{7} e^{7} x^{7} + 60 \, b^{7} d^{7} - 420 \, a b^{6} d^{6} e + 1260 \, a^{2} b^{5} d^{5} e^{2} - 2100 \, a^{3} b^{4} d^{4} e^{3} + 2100 \, a^{4} b^{3} d^{3} e^{4} - 1260 \, a^{5} b^{2} d^{2} e^{5} + 420 \, a^{6} b d e^{6} - 60 \, a^{7} e^{7} - 14 \, {\left (b^{7} d e^{6} - 6 \, a b^{6} e^{7}\right )} x^{6} + 21 \, {\left (b^{7} d^{2} e^{5} - 6 \, a b^{6} d e^{6} + 15 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \, {\left (b^{7} d^{3} e^{4} - 6 \, a b^{6} d^{2} e^{5} + 15 \, a^{2} b^{5} d e^{6} - 20 \, a^{3} b^{4} e^{7}\right )} x^{4} + 70 \, {\left (b^{7} d^{4} e^{3} - 6 \, a b^{6} d^{3} e^{4} + 15 \, a^{2} b^{5} d^{2} e^{5} - 20 \, a^{3} b^{4} d e^{6} + 15 \, a^{4} b^{3} e^{7}\right )} x^{3} - 210 \, {\left (b^{7} d^{5} e^{2} - 6 \, a b^{6} d^{4} e^{3} + 15 \, a^{2} b^{5} d^{3} e^{4} - 20 \, a^{3} b^{4} d^{2} e^{5} + 15 \, a^{4} b^{3} d e^{6} - 6 \, a^{5} b^{2} e^{7}\right )} x^{2} - 60 \, {\left (6 \, b^{7} d^{6} e - 35 \, a b^{6} d^{5} e^{2} + 84 \, a^{2} b^{5} d^{4} e^{3} - 105 \, a^{3} b^{4} d^{3} e^{4} + 70 \, a^{4} b^{3} d^{2} e^{5} - 21 \, a^{5} b^{2} d e^{6}\right )} x + 420 \, {\left (b^{7} d^{7} - 6 \, a b^{6} d^{6} e + 15 \, a^{2} b^{5} d^{5} e^{2} - 20 \, a^{3} b^{4} d^{4} e^{3} + 15 \, a^{4} b^{3} d^{3} e^{4} - 6 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} + {\left (b^{7} d^{6} e - 6 \, a b^{6} d^{5} e^{2} + 15 \, a^{2} b^{5} d^{4} e^{3} - 20 \, a^{3} b^{4} d^{3} e^{4} + 15 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} + a^{6} b e^{7}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{9} x + d e^{8}\right )}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^2,x, algorithm="fricas")
Output:
1/60*(10*b^7*e^7*x^7 + 60*b^7*d^7 - 420*a*b^6*d^6*e + 1260*a^2*b^5*d^5*e^2 - 2100*a^3*b^4*d^4*e^3 + 2100*a^4*b^3*d^3*e^4 - 1260*a^5*b^2*d^2*e^5 + 42 0*a^6*b*d*e^6 - 60*a^7*e^7 - 14*(b^7*d*e^6 - 6*a*b^6*e^7)*x^6 + 21*(b^7*d^ 2*e^5 - 6*a*b^6*d*e^6 + 15*a^2*b^5*e^7)*x^5 - 35*(b^7*d^3*e^4 - 6*a*b^6*d^ 2*e^5 + 15*a^2*b^5*d*e^6 - 20*a^3*b^4*e^7)*x^4 + 70*(b^7*d^4*e^3 - 6*a*b^6 *d^3*e^4 + 15*a^2*b^5*d^2*e^5 - 20*a^3*b^4*d*e^6 + 15*a^4*b^3*e^7)*x^3 - 2 10*(b^7*d^5*e^2 - 6*a*b^6*d^4*e^3 + 15*a^2*b^5*d^3*e^4 - 20*a^3*b^4*d^2*e^ 5 + 15*a^4*b^3*d*e^6 - 6*a^5*b^2*e^7)*x^2 - 60*(6*b^7*d^6*e - 35*a*b^6*d^5 *e^2 + 84*a^2*b^5*d^4*e^3 - 105*a^3*b^4*d^3*e^4 + 70*a^4*b^3*d^2*e^5 - 21* a^5*b^2*d*e^6)*x + 420*(b^7*d^7 - 6*a*b^6*d^6*e + 15*a^2*b^5*d^5*e^2 - 20* a^3*b^4*d^4*e^3 + 15*a^4*b^3*d^3*e^4 - 6*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 + ( b^7*d^6*e - 6*a*b^6*d^5*e^2 + 15*a^2*b^5*d^4*e^3 - 20*a^3*b^4*d^3*e^4 + 15 *a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^6 + a^6*b*e^7)*x)*log(e*x + d))/(e^9*x + d*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (172) = 344\).
Time = 1.14 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.30 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {b^{7} x^{6}}{6 e^{2}} + \frac {7 b \left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{8}} + x^{5} \cdot \left (\frac {7 a b^{6}}{5 e^{2}} - \frac {2 b^{7} d}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {21 a^{2} b^{5}}{4 e^{2}} - \frac {7 a b^{6} d}{2 e^{3}} + \frac {3 b^{7} d^{2}}{4 e^{4}}\right ) + x^{3} \cdot \left (\frac {35 a^{3} b^{4}}{3 e^{2}} - \frac {14 a^{2} b^{5} d}{e^{3}} + \frac {7 a b^{6} d^{2}}{e^{4}} - \frac {4 b^{7} d^{3}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {35 a^{4} b^{3}}{2 e^{2}} - \frac {35 a^{3} b^{4} d}{e^{3}} + \frac {63 a^{2} b^{5} d^{2}}{2 e^{4}} - \frac {14 a b^{6} d^{3}}{e^{5}} + \frac {5 b^{7} d^{4}}{2 e^{6}}\right ) + x \left (\frac {21 a^{5} b^{2}}{e^{2}} - \frac {70 a^{4} b^{3} d}{e^{3}} + \frac {105 a^{3} b^{4} d^{2}}{e^{4}} - \frac {84 a^{2} b^{5} d^{3}}{e^{5}} + \frac {35 a b^{6} d^{4}}{e^{6}} - \frac {6 b^{7} d^{5}}{e^{7}}\right ) + \frac {- a^{7} e^{7} + 7 a^{6} b d e^{6} - 21 a^{5} b^{2} d^{2} e^{5} + 35 a^{4} b^{3} d^{3} e^{4} - 35 a^{3} b^{4} d^{4} e^{3} + 21 a^{2} b^{5} d^{5} e^{2} - 7 a b^{6} d^{6} e + b^{7} d^{7}}{d e^{8} + e^{9} x} \] Input:
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**2,x)
Output:
b**7*x**6/(6*e**2) + 7*b*(a*e - b*d)**6*log(d + e*x)/e**8 + x**5*(7*a*b**6 /(5*e**2) - 2*b**7*d/(5*e**3)) + x**4*(21*a**2*b**5/(4*e**2) - 7*a*b**6*d/ (2*e**3) + 3*b**7*d**2/(4*e**4)) + x**3*(35*a**3*b**4/(3*e**2) - 14*a**2*b **5*d/e**3 + 7*a*b**6*d**2/e**4 - 4*b**7*d**3/(3*e**5)) + x**2*(35*a**4*b* *3/(2*e**2) - 35*a**3*b**4*d/e**3 + 63*a**2*b**5*d**2/(2*e**4) - 14*a*b**6 *d**3/e**5 + 5*b**7*d**4/(2*e**6)) + x*(21*a**5*b**2/e**2 - 70*a**4*b**3*d /e**3 + 105*a**3*b**4*d**2/e**4 - 84*a**2*b**5*d**3/e**5 + 35*a*b**6*d**4/ e**6 - 6*b**7*d**5/e**7) + (-a**7*e**7 + 7*a**6*b*d*e**6 - 21*a**5*b**2*d* *2*e**5 + 35*a**4*b**3*d**3*e**4 - 35*a**3*b**4*d**4*e**3 + 21*a**2*b**5*d **5*e**2 - 7*a*b**6*d**6*e + b**7*d**7)/(d*e**8 + e**9*x)
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (176) = 352\).
Time = 0.03 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.51 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}}{e^{9} x + d e^{8}} + \frac {10 \, b^{7} e^{5} x^{6} - 12 \, {\left (2 \, b^{7} d e^{4} - 7 \, a b^{6} e^{5}\right )} x^{5} + 15 \, {\left (3 \, b^{7} d^{2} e^{3} - 14 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{4} - 20 \, {\left (4 \, b^{7} d^{3} e^{2} - 21 \, a b^{6} d^{2} e^{3} + 42 \, a^{2} b^{5} d e^{4} - 35 \, a^{3} b^{4} e^{5}\right )} x^{3} + 30 \, {\left (5 \, b^{7} d^{4} e - 28 \, a b^{6} d^{3} e^{2} + 63 \, a^{2} b^{5} d^{2} e^{3} - 70 \, a^{3} b^{4} d e^{4} + 35 \, a^{4} b^{3} e^{5}\right )} x^{2} - 60 \, {\left (6 \, b^{7} d^{5} - 35 \, a b^{6} d^{4} e + 84 \, a^{2} b^{5} d^{3} e^{2} - 105 \, a^{3} b^{4} d^{2} e^{3} + 70 \, a^{4} b^{3} d e^{4} - 21 \, a^{5} b^{2} e^{5}\right )} x}{60 \, e^{7}} + \frac {7 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^2,x, algorithm="maxima")
Output:
(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^ 4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)/(e^9*x + d*e ^8) + 1/60*(10*b^7*e^5*x^6 - 12*(2*b^7*d*e^4 - 7*a*b^6*e^5)*x^5 + 15*(3*b^ 7*d^2*e^3 - 14*a*b^6*d*e^4 + 21*a^2*b^5*e^5)*x^4 - 20*(4*b^7*d^3*e^2 - 21* a*b^6*d^2*e^3 + 42*a^2*b^5*d*e^4 - 35*a^3*b^4*e^5)*x^3 + 30*(5*b^7*d^4*e - 28*a*b^6*d^3*e^2 + 63*a^2*b^5*d^2*e^3 - 70*a^3*b^4*d*e^4 + 35*a^4*b^3*e^5 )*x^2 - 60*(6*b^7*d^5 - 35*a*b^6*d^4*e + 84*a^2*b^5*d^3*e^2 - 105*a^3*b^4* d^2*e^3 + 70*a^4*b^3*d*e^4 - 21*a^5*b^2*e^5)*x)/e^7 + 7*(b^7*d^6 - 6*a*b^6 *d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2*e^4 - 6* a^5*b^2*d*e^5 + a^6*b*e^6)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (176) = 352\).
Time = 0.20 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.04 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {{\left (10 \, b^{7} - \frac {84 \, {\left (b^{7} d e - a b^{6} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {315 \, {\left (b^{7} d^{2} e^{2} - 2 \, a b^{6} d e^{3} + a^{2} b^{5} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {700 \, {\left (b^{7} d^{3} e^{3} - 3 \, a b^{6} d^{2} e^{4} + 3 \, a^{2} b^{5} d e^{5} - a^{3} b^{4} e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {1050 \, {\left (b^{7} d^{4} e^{4} - 4 \, a b^{6} d^{3} e^{5} + 6 \, a^{2} b^{5} d^{2} e^{6} - 4 \, a^{3} b^{4} d e^{7} + a^{4} b^{3} e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}} - \frac {1260 \, {\left (b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}\right )}}{{\left (e x + d\right )}^{5} e^{5}}\right )} {\left (e x + d\right )}^{6}}{60 \, e^{8}} - \frac {7 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{8}} + \frac {\frac {b^{7} d^{7} e^{6}}{e x + d} - \frac {7 \, a b^{6} d^{6} e^{7}}{e x + d} + \frac {21 \, a^{2} b^{5} d^{5} e^{8}}{e x + d} - \frac {35 \, a^{3} b^{4} d^{4} e^{9}}{e x + d} + \frac {35 \, a^{4} b^{3} d^{3} e^{10}}{e x + d} - \frac {21 \, a^{5} b^{2} d^{2} e^{11}}{e x + d} + \frac {7 \, a^{6} b d e^{12}}{e x + d} - \frac {a^{7} e^{13}}{e x + d}}{e^{14}} \] Input:
integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^2,x, algorithm="giac")
Output:
1/60*(10*b^7 - 84*(b^7*d*e - a*b^6*e^2)/((e*x + d)*e) + 315*(b^7*d^2*e^2 - 2*a*b^6*d*e^3 + a^2*b^5*e^4)/((e*x + d)^2*e^2) - 700*(b^7*d^3*e^3 - 3*a*b ^6*d^2*e^4 + 3*a^2*b^5*d*e^5 - a^3*b^4*e^6)/((e*x + d)^3*e^3) + 1050*(b^7* d^4*e^4 - 4*a*b^6*d^3*e^5 + 6*a^2*b^5*d^2*e^6 - 4*a^3*b^4*d*e^7 + a^4*b^3* e^8)/((e*x + d)^4*e^4) - 1260*(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5* d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10)/((e*x + d)^ 5*e^5))*(e*x + d)^6/e^8 - 7*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*l og(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^8 + (b^7*d^7*e^6/(e*x + d) - 7*a*b ^6*d^6*e^7/(e*x + d) + 21*a^2*b^5*d^5*e^8/(e*x + d) - 35*a^3*b^4*d^4*e^9/( e*x + d) + 35*a^4*b^3*d^3*e^10/(e*x + d) - 21*a^5*b^2*d^2*e^11/(e*x + d) + 7*a^6*b*d*e^12/(e*x + d) - a^7*e^13/(e*x + d))/e^14
Time = 11.10 (sec) , antiderivative size = 839, normalized size of antiderivative = 4.51 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/(d + e*x)^2,x)
Output:
x^5*((7*a*b^6)/(5*e^2) - (2*b^7*d)/(5*e^3)) + x^2*((d^2*((2*d*((7*a*b^6)/e ^2 - (2*b^7*d)/e^3))/e - (21*a^2*b^5)/e^2 + (b^7*d^2)/e^4))/(2*e^2) - (d*( (35*a^3*b^4)/e^2 + (2*d*((2*d*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/e - (21*a^2 *b^5)/e^2 + (b^7*d^2)/e^4))/e - (d^2*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/e^2) )/e + (35*a^4*b^3)/(2*e^2)) - x^4*((d*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/(2* e) - (21*a^2*b^5)/(4*e^2) + (b^7*d^2)/(4*e^4)) + x^3*((35*a^3*b^4)/(3*e^2) + (2*d*((2*d*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/e - (21*a^2*b^5)/e^2 + (b^7 *d^2)/e^4))/(3*e) - (d^2*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/(3*e^2)) - x*((d ^2*((35*a^3*b^4)/e^2 + (2*d*((2*d*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/e - (21 *a^2*b^5)/e^2 + (b^7*d^2)/e^4))/e - (d^2*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/ e^2))/e^2 - (21*a^5*b^2)/e^2 + (2*d*((d^2*((2*d*((7*a*b^6)/e^2 - (2*b^7*d) /e^3))/e - (21*a^2*b^5)/e^2 + (b^7*d^2)/e^4))/e^2 - (2*d*((35*a^3*b^4)/e^2 + (2*d*((2*d*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/e - (21*a^2*b^5)/e^2 + (b^7 *d^2)/e^4))/e - (d^2*((7*a*b^6)/e^2 - (2*b^7*d)/e^3))/e^2))/e + (35*a^4*b^ 3)/e^2))/e) + (log(d + e*x)*(7*b^7*d^6 + 7*a^6*b*e^6 - 42*a^5*b^2*d*e^5 + 105*a^2*b^5*d^4*e^2 - 140*a^3*b^4*d^3*e^3 + 105*a^4*b^3*d^2*e^4 - 42*a*b^6 *d^5*e))/e^8 - (a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^ 3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^ 6)/(e*(d*e^7 + e^8*x)) + (b^7*x^6)/(6*e^2)
Time = 0.21 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.76 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^2} \, dx=\frac {420 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{8}+60 a^{7} e^{8} x +8400 a^{3} b^{4} d^{4} e^{4} x +4200 a^{3} b^{4} d^{3} e^{5} x^{2}-1400 a^{3} b^{4} d^{2} e^{6} x^{3}+700 a^{3} b^{4} d \,e^{7} x^{4}-6300 a^{2} b^{5} d^{5} e^{3} x -3150 a^{2} b^{5} d^{4} e^{4} x^{2}+1050 a^{2} b^{5} d^{3} e^{5} x^{3}-525 a^{2} b^{5} d^{2} e^{6} x^{4}+315 a^{2} b^{5} d \,e^{7} x^{5}+2520 a \,b^{6} d^{6} e^{2} x +1260 a \,b^{6} d^{5} e^{3} x^{2}-420 a \,b^{6} d^{4} e^{4} x^{3}+210 a \,b^{6} d^{3} e^{5} x^{4}-126 a \,b^{6} d^{2} e^{6} x^{5}+84 a \,b^{6} d \,e^{7} x^{6}+420 \,\mathrm {log}\left (e x +d \right ) a^{6} b \,d^{2} e^{6}-2520 \,\mathrm {log}\left (e x +d \right ) a^{5} b^{2} d^{3} e^{5}+6300 \,\mathrm {log}\left (e x +d \right ) a^{4} b^{3} d^{4} e^{4}-8400 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{4} d^{5} e^{3}+6300 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{5} d^{6} e^{2}-2520 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{7} e +420 \,\mathrm {log}\left (e x +d \right ) b^{7} d^{7} e x -420 a^{6} b d \,e^{7} x +2520 a^{5} b^{2} d^{2} e^{6} x +1260 a^{5} b^{2} d \,e^{7} x^{2}-6300 a^{4} b^{3} d^{3} e^{5} x -3150 a^{4} b^{3} d^{2} e^{6} x^{2}+1050 a^{4} b^{3} d \,e^{7} x^{3}+420 \,\mathrm {log}\left (e x +d \right ) a^{6} b d \,e^{7} x -2520 \,\mathrm {log}\left (e x +d \right ) a^{5} b^{2} d^{2} e^{6} x +6300 \,\mathrm {log}\left (e x +d \right ) a^{4} b^{3} d^{3} e^{5} x -8400 \,\mathrm {log}\left (e x +d \right ) a^{3} b^{4} d^{4} e^{4} x +6300 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{5} d^{5} e^{3} x -2520 \,\mathrm {log}\left (e x +d \right ) a \,b^{6} d^{6} e^{2} x -420 b^{7} d^{7} e x -210 b^{7} d^{6} e^{2} x^{2}+70 b^{7} d^{5} e^{3} x^{3}-35 b^{7} d^{4} e^{4} x^{4}+21 b^{7} d^{3} e^{5} x^{5}-14 b^{7} d^{2} e^{6} x^{6}+10 b^{7} d \,e^{7} x^{7}}{60 d \,e^{8} \left (e x +d \right )} \] Input:
int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^2,x)
Output:
(420*log(d + e*x)*a**6*b*d**2*e**6 + 420*log(d + e*x)*a**6*b*d*e**7*x - 25 20*log(d + e*x)*a**5*b**2*d**3*e**5 - 2520*log(d + e*x)*a**5*b**2*d**2*e** 6*x + 6300*log(d + e*x)*a**4*b**3*d**4*e**4 + 6300*log(d + e*x)*a**4*b**3* d**3*e**5*x - 8400*log(d + e*x)*a**3*b**4*d**5*e**3 - 8400*log(d + e*x)*a* *3*b**4*d**4*e**4*x + 6300*log(d + e*x)*a**2*b**5*d**6*e**2 + 6300*log(d + e*x)*a**2*b**5*d**5*e**3*x - 2520*log(d + e*x)*a*b**6*d**7*e - 2520*log(d + e*x)*a*b**6*d**6*e**2*x + 420*log(d + e*x)*b**7*d**8 + 420*log(d + e*x) *b**7*d**7*e*x + 60*a**7*e**8*x - 420*a**6*b*d*e**7*x + 2520*a**5*b**2*d** 2*e**6*x + 1260*a**5*b**2*d*e**7*x**2 - 6300*a**4*b**3*d**3*e**5*x - 3150* a**4*b**3*d**2*e**6*x**2 + 1050*a**4*b**3*d*e**7*x**3 + 8400*a**3*b**4*d** 4*e**4*x + 4200*a**3*b**4*d**3*e**5*x**2 - 1400*a**3*b**4*d**2*e**6*x**3 + 700*a**3*b**4*d*e**7*x**4 - 6300*a**2*b**5*d**5*e**3*x - 3150*a**2*b**5*d **4*e**4*x**2 + 1050*a**2*b**5*d**3*e**5*x**3 - 525*a**2*b**5*d**2*e**6*x* *4 + 315*a**2*b**5*d*e**7*x**5 + 2520*a*b**6*d**6*e**2*x + 1260*a*b**6*d** 5*e**3*x**2 - 420*a*b**6*d**4*e**4*x**3 + 210*a*b**6*d**3*e**5*x**4 - 126* a*b**6*d**2*e**6*x**5 + 84*a*b**6*d*e**7*x**6 - 420*b**7*d**7*e*x - 210*b* *7*d**6*e**2*x**2 + 70*b**7*d**5*e**3*x**3 - 35*b**7*d**4*e**4*x**4 + 21*b **7*d**3*e**5*x**5 - 14*b**7*d**2*e**6*x**6 + 10*b**7*d*e**7*x**7)/(60*d*e **8*(d + e*x))