Integrand size = 22, antiderivative size = 310 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {c^2 \left (20 B c d^3-10 A c d^2 e+12 a B d e^2-3 a A e^3\right ) x}{e^7}+\frac {c^2 \left (10 B c d^2-4 A c d e+3 a B e^2\right ) x^2}{2 e^6}-\frac {c^3 (4 B d-A e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{3 e^8 (d+e x)^3}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{2 e^8 (d+e x)^2}+\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^8 (d+e x)}-\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) \log (d+e x)}{e^8} \] Output:
-c^2*(-3*A*a*e^3-10*A*c*d^2*e+12*B*a*d*e^2+20*B*c*d^3)*x/e^7+1/2*c^2*(-4*A *c*d*e+3*B*a*e^2+10*B*c*d^2)*x^2/e^6-1/3*c^3*(-A*e+4*B*d)*x^3/e^5+1/4*B*c^ 3*x^4/e^4+1/3*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^3-1/2*(a*e^2+c*d^2)^2 *(-6*A*c*d*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^2+3*c*(a*e^2+c*d^2)*(-A*a*e^3- 5*A*c*d^2*e+3*B*a*d*e^2+7*B*c*d^3)/e^8/(e*x+d)-c*(4*A*c*d*e*(3*a*e^2+5*c*d ^2)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))*ln(e*x+d)/e^8
Time = 0.13 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {12 c^2 e \left (A e \left (10 c d^2+3 a e^2\right )-4 B \left (5 c d^3+3 a d e^2\right )\right ) x+6 c^2 e^2 \left (10 B c d^2-4 A c d e+3 a B e^2\right ) x^2+4 c^3 e^3 (-4 B d+A e) x^3+3 B c^3 e^4 x^4+\frac {4 (B d-A e) \left (c d^2+a e^2\right )^3}{(d+e x)^3}-\frac {6 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{(d+e x)^2}+\frac {36 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{d+e x}+12 c \left (-4 A c d e \left (5 c d^2+3 a e^2\right )+B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) \log (d+e x)}{12 e^8} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^4,x]
Output:
(12*c^2*e*(A*e*(10*c*d^2 + 3*a*e^2) - 4*B*(5*c*d^3 + 3*a*d*e^2))*x + 6*c^2 *e^2*(10*B*c*d^2 - 4*A*c*d*e + 3*a*B*e^2)*x^2 + 4*c^3*e^3*(-4*B*d + A*e)*x ^3 + 3*B*c^3*e^4*x^4 + (4*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(d + e*x)^3 - (6* (c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(d + e*x)^2 + (36*c*( c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(d + e*x ) + 12*c*(-4*A*c*d*e*(5*c*d^2 + 3*a*e^2) + B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*Log[d + e*x])/(12*e^8)
Time = 0.66 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)}-\frac {c^2 x \left (-3 a B e^2+4 A c d e-10 B c d^2\right )}{e^6}+\frac {c^2 \left (3 a A e^3-12 a B d e^2+10 A c d^2 e-20 B c d^3\right )}{e^7}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^4}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^2}+\frac {c^3 x^2 (A e-4 B d)}{e^5}+\frac {B c^3 x^3}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c \log (d+e x) \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8}+\frac {c^2 x^2 \left (3 a B e^2-4 A c d e+10 B c d^2\right )}{2 e^6}-\frac {c^2 x \left (-3 a A e^3+12 a B d e^2-10 A c d^2 e+20 B c d^3\right )}{e^7}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8 (d+e x)^3}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)}-\frac {c^3 x^3 (4 B d-A e)}{3 e^5}+\frac {B c^3 x^4}{4 e^4}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^4,x]
Output:
-((c^2*(20*B*c*d^3 - 10*A*c*d^2*e + 12*a*B*d*e^2 - 3*a*A*e^3)*x)/e^7) + (c ^2*(10*B*c*d^2 - 4*A*c*d*e + 3*a*B*e^2)*x^2)/(2*e^6) - (c^3*(4*B*d - A*e)* x^3)/(3*e^5) + (B*c^3*x^4)/(4*e^4) + ((B*d - A*e)*(c*d^2 + a*e^2)^3)/(3*e^ 8*(d + e*x)^3) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(2* e^8*(d + e*x)^2) + (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d *e^2 - a*A*e^3))/(e^8*(d + e*x)) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*( 35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*Log[d + e*x])/e^8
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 0.71 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.39
| method | result | size |
| norman | \(\frac {-\frac {2 A \,a^{3} e^{7}+6 A \,a^{2} c \,d^{2} e^{5}+132 A a \,c^{2} d^{4} e^{3}+220 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}-33 B \,a^{2} c \,d^{3} e^{4}-330 B a \,c^{2} d^{5} e^{2}-385 B \,c^{3} d^{7}}{6 e^{8}}-\frac {3 \left (A \,a^{2} c \,e^{5}+12 A a \,c^{2} d^{2} e^{3}+20 A \,c^{3} d^{4} e -3 B \,a^{2} c d \,e^{4}-30 B a \,c^{2} d^{3} e^{2}-35 B \,c^{3} d^{5}\right ) x^{2}}{e^{6}}-\frac {\left (6 A \,a^{2} c d \,e^{5}+108 A a \,c^{2} d^{3} e^{3}+180 A \,c^{3} d^{5} e +B \,a^{3} e^{6}-27 B \,a^{2} c \,d^{2} e^{4}-270 B a \,c^{2} d^{4} e^{2}-315 B \,c^{3} d^{6}\right ) x}{2 e^{7}}+\frac {B \,c^{3} x^{7}}{4 e}-\frac {c^{2} \left (4 A c d e -6 B a \,e^{2}-7 B c \,d^{2}\right ) x^{5}}{4 e^{3}}+\frac {c^{2} \left (12 A a \,e^{3}+20 A c \,d^{2} e -30 B a d \,e^{2}-35 B c \,d^{3}\right ) x^{4}}{4 e^{4}}+\frac {c^{3} \left (4 A e -7 B d \right ) x^{6}}{12 e^{2}}}{\left (e x +d \right )^{3}}-\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(431\) |
| default | \(\frac {c^{2} \left (\frac {1}{4} B c \,x^{4} e^{3}+\frac {1}{3} A \,x^{3} c \,e^{3}-\frac {4}{3} B \,x^{3} c d \,e^{2}-2 A \,x^{2} c d \,e^{2}+\frac {3}{2} B \,x^{2} a \,e^{3}+5 B \,x^{2} c \,d^{2} e +3 A a \,e^{3} x +10 A c \,d^{2} e x -12 B a d \,e^{2} x -20 B c \,d^{3} x \right )}{e^{7}}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {3 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{e^{8} \left (e x +d \right )}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{2 e^{8} \left (e x +d \right )^{2}}\) | \(432\) |
| risch | \(\frac {B \,c^{3} x^{4}}{4 e^{4}}+\frac {c^{3} A \,x^{3}}{3 e^{4}}-\frac {4 c^{3} B \,x^{3} d}{3 e^{5}}-\frac {2 c^{3} A \,x^{2} d}{e^{5}}+\frac {3 c^{2} B \,x^{2} a}{2 e^{4}}+\frac {5 c^{3} B \,x^{2} d^{2}}{e^{6}}+\frac {3 c^{2} A a x}{e^{4}}+\frac {10 c^{3} A \,d^{2} x}{e^{6}}-\frac {12 c^{2} B a d x}{e^{5}}-\frac {20 c^{3} B \,d^{3} x}{e^{7}}+\frac {\left (-3 a^{2} A c \,e^{6}-18 A \,d^{2} a \,c^{2} e^{4}-15 A \,d^{4} c^{3} e^{2}+9 B \,a^{2} c d \,e^{5}+30 B a \,c^{2} d^{3} e^{3}+21 B \,c^{3} d^{5} e \right ) x^{2}+\left (-3 A \,a^{2} c d \,e^{5}-30 A a \,c^{2} d^{3} e^{3}-27 A \,c^{3} d^{5} e -\frac {1}{2} B \,a^{3} e^{6}+\frac {27}{2} B \,a^{2} c \,d^{2} e^{4}+\frac {105}{2} B a \,c^{2} d^{4} e^{2}+\frac {77}{2} B \,c^{3} d^{6}\right ) x -\frac {2 A \,a^{3} e^{7}+6 A \,a^{2} c \,d^{2} e^{5}+78 A a \,c^{2} d^{4} e^{3}+74 A \,c^{3} d^{6} e +B \,a^{3} d \,e^{6}-33 B \,a^{2} c \,d^{3} e^{4}-141 B a \,c^{2} d^{5} e^{2}-107 B \,c^{3} d^{7}}{6 e}}{e^{7} \left (e x +d \right )^{3}}-\frac {12 c^{2} \ln \left (e x +d \right ) A a d}{e^{5}}-\frac {20 c^{3} \ln \left (e x +d \right ) A \,d^{3}}{e^{7}}+\frac {3 c \ln \left (e x +d \right ) B \,a^{2}}{e^{4}}+\frac {30 c^{2} \ln \left (e x +d \right ) B a \,d^{2}}{e^{6}}+\frac {35 c^{3} \ln \left (e x +d \right ) B \,d^{4}}{e^{8}}\) | \(479\) |
| parallelrisch | \(-\frac {432 A \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{4}-66 B \,a^{2} c \,d^{3} e^{4}+12 A \,a^{2} c \,d^{2} e^{5}+264 A a \,c^{2} d^{4} e^{3}-4 A \,x^{6} c^{3} e^{7}-3 B \,x^{7} c^{3} e^{7}-420 B \ln \left (e x +d \right ) c^{3} d^{7}+6 B x \,a^{3} e^{7}-1620 B x a \,c^{2} d^{4} e^{3}-108 B \ln \left (e x +d \right ) x \,a^{2} c \,d^{2} e^{5}-660 B a \,c^{2} d^{5} e^{2}+720 A \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{3}-1260 B \ln \left (e x +d \right ) x^{2} c^{3} d^{5} e^{2}-60 A \,x^{4} c^{3} d^{2} e^{5}+105 B \,x^{4} c^{3} d^{3} e^{4}-770 B \,c^{3} d^{7}-1080 B \ln \left (e x +d \right ) x a \,c^{2} d^{4} e^{3}-108 B \ln \left (e x +d \right ) x^{2} a^{2} c d \,e^{6}-1080 B \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{3} e^{4}+432 A \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{5}-360 B \ln \left (e x +d \right ) x^{3} a \,c^{2} d^{2} e^{5}+144 A \ln \left (e x +d \right ) x^{3} a \,c^{2} d \,e^{6}+240 A \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{4}-36 B \ln \left (e x +d \right ) x^{3} a^{2} c \,e^{7}-420 B \ln \left (e x +d \right ) x^{3} c^{3} d^{4} e^{3}+12 A \,x^{5} c^{3} d \,e^{6}+7 B \,x^{6} c^{3} d \,e^{6}+240 A \ln \left (e x +d \right ) c^{3} d^{6} e +720 A \ln \left (e x +d \right ) x \,c^{3} d^{5} e^{2}-1260 B \ln \left (e x +d \right ) x \,c^{3} d^{6} e +36 A \,x^{2} a^{2} c \,e^{7}+720 A \,x^{2} c^{3} d^{4} e^{3}-1260 B \,x^{2} c^{3} d^{5} e^{2}+1080 A x \,c^{3} d^{5} e^{2}-1890 B x \,c^{3} d^{6} e -18 B \,x^{5} a \,c^{2} e^{7}-21 B \,x^{5} c^{3} d^{2} e^{5}-36 A \,x^{4} a \,c^{2} e^{7}+440 A \,c^{3} d^{6} e +2 B \,a^{3} d \,e^{6}+144 A \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{3}-36 B \ln \left (e x +d \right ) a^{2} c \,d^{3} e^{4}-360 B \ln \left (e x +d \right ) a \,c^{2} d^{5} e^{2}+432 A \,x^{2} a \,c^{2} d^{2} e^{5}-108 B \,x^{2} a^{2} c d \,e^{6}-1080 B \,x^{2} a \,c^{2} d^{3} e^{4}+36 A x \,a^{2} c d \,e^{6}+648 A x a \,c^{2} d^{3} e^{4}-162 B x \,a^{2} c \,d^{2} e^{5}+90 B \,x^{4} a \,c^{2} d \,e^{6}+4 A \,a^{3} e^{7}}{12 e^{8} \left (e x +d \right )^{3}}\) | \(809\) |
Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^4,x,method=_RETURNVERBOSE)
Output:
(-1/6*(2*A*a^3*e^7+6*A*a^2*c*d^2*e^5+132*A*a*c^2*d^4*e^3+220*A*c^3*d^6*e+B *a^3*d*e^6-33*B*a^2*c*d^3*e^4-330*B*a*c^2*d^5*e^2-385*B*c^3*d^7)/e^8-3*(A* a^2*c*e^5+12*A*a*c^2*d^2*e^3+20*A*c^3*d^4*e-3*B*a^2*c*d*e^4-30*B*a*c^2*d^3 *e^2-35*B*c^3*d^5)/e^6*x^2-1/2*(6*A*a^2*c*d*e^5+108*A*a*c^2*d^3*e^3+180*A* c^3*d^5*e+B*a^3*e^6-27*B*a^2*c*d^2*e^4-270*B*a*c^2*d^4*e^2-315*B*c^3*d^6)/ e^7*x+1/4*B*c^3*x^7/e-1/4*c^2*(4*A*c*d*e-6*B*a*e^2-7*B*c*d^2)/e^3*x^5+1/4* c^2*(12*A*a*e^3+20*A*c*d^2*e-30*B*a*d*e^2-35*B*c*d^3)/e^4*x^4+1/12*c^3*(4* A*e-7*B*d)/e^2*x^6)/(e*x+d)^3-1/e^8*c*(12*A*a*c*d*e^3+20*A*c^2*d^3*e-3*B*a ^2*e^4-30*B*a*c*d^2*e^2-35*B*c^2*d^4)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 732 vs. \(2 (300) = 600\).
Time = 0.08 (sec) , antiderivative size = 732, normalized size of antiderivative = 2.36 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {3 \, B c^{3} e^{7} x^{7} + 214 \, B c^{3} d^{7} - 148 \, A c^{3} d^{6} e + 282 \, B a c^{2} d^{5} e^{2} - 156 \, A a c^{2} d^{4} e^{3} + 66 \, B a^{2} c d^{3} e^{4} - 12 \, A a^{2} c d^{2} e^{5} - 2 \, B a^{3} d e^{6} - 4 \, A a^{3} e^{7} - {\left (7 \, B c^{3} d e^{6} - 4 \, A c^{3} e^{7}\right )} x^{6} + 3 \, {\left (7 \, B c^{3} d^{2} e^{5} - 4 \, A c^{3} d e^{6} + 6 \, B a c^{2} e^{7}\right )} x^{5} - 3 \, {\left (35 \, B c^{3} d^{3} e^{4} - 20 \, A c^{3} d^{2} e^{5} + 30 \, B a c^{2} d e^{6} - 12 \, A a c^{2} e^{7}\right )} x^{4} - 2 \, {\left (278 \, B c^{3} d^{4} e^{3} - 146 \, A c^{3} d^{3} e^{4} + 189 \, B a c^{2} d^{2} e^{5} - 54 \, A a c^{2} d e^{6}\right )} x^{3} - 6 \, {\left (68 \, B c^{3} d^{5} e^{2} - 26 \, A c^{3} d^{4} e^{3} + 9 \, B a c^{2} d^{3} e^{4} + 18 \, A a c^{2} d^{2} e^{5} - 18 \, B a^{2} c d e^{6} + 6 \, A a^{2} c e^{7}\right )} x^{2} + 6 \, {\left (37 \, B c^{3} d^{6} e - 34 \, A c^{3} d^{5} e^{2} + 81 \, B a c^{2} d^{4} e^{3} - 54 \, A a c^{2} d^{3} e^{4} + 27 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x + 12 \, {\left (35 \, B c^{3} d^{7} - 20 \, A c^{3} d^{6} e + 30 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + {\left (35 \, B c^{3} d^{4} e^{3} - 20 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 3 \, {\left (35 \, B c^{3} d^{5} e^{2} - 20 \, A c^{3} d^{4} e^{3} + 30 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6}\right )} x^{2} + 3 \, {\left (35 \, B c^{3} d^{6} e - 20 \, A c^{3} d^{5} e^{2} + 30 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^4,x, algorithm="fricas")
Output:
1/12*(3*B*c^3*e^7*x^7 + 214*B*c^3*d^7 - 148*A*c^3*d^6*e + 282*B*a*c^2*d^5* e^2 - 156*A*a*c^2*d^4*e^3 + 66*B*a^2*c*d^3*e^4 - 12*A*a^2*c*d^2*e^5 - 2*B* a^3*d*e^6 - 4*A*a^3*e^7 - (7*B*c^3*d*e^6 - 4*A*c^3*e^7)*x^6 + 3*(7*B*c^3*d ^2*e^5 - 4*A*c^3*d*e^6 + 6*B*a*c^2*e^7)*x^5 - 3*(35*B*c^3*d^3*e^4 - 20*A*c ^3*d^2*e^5 + 30*B*a*c^2*d*e^6 - 12*A*a*c^2*e^7)*x^4 - 2*(278*B*c^3*d^4*e^3 - 146*A*c^3*d^3*e^4 + 189*B*a*c^2*d^2*e^5 - 54*A*a*c^2*d*e^6)*x^3 - 6*(68 *B*c^3*d^5*e^2 - 26*A*c^3*d^4*e^3 + 9*B*a*c^2*d^3*e^4 + 18*A*a*c^2*d^2*e^5 - 18*B*a^2*c*d*e^6 + 6*A*a^2*c*e^7)*x^2 + 6*(37*B*c^3*d^6*e - 34*A*c^3*d^ 5*e^2 + 81*B*a*c^2*d^4*e^3 - 54*A*a*c^2*d^3*e^4 + 27*B*a^2*c*d^2*e^5 - 6*A *a^2*c*d*e^6 - B*a^3*e^7)*x + 12*(35*B*c^3*d^7 - 20*A*c^3*d^6*e + 30*B*a*c ^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 + (35*B*c^3*d^4*e^3 - 20*A*c^3*d^3*e^4 + 30*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)* x^3 + 3*(35*B*c^3*d^5*e^2 - 20*A*c^3*d^4*e^3 + 30*B*a*c^2*d^3*e^4 - 12*A*a *c^2*d^2*e^5 + 3*B*a^2*c*d*e^6)*x^2 + 3*(35*B*c^3*d^6*e - 20*A*c^3*d^5*e^2 + 30*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)
Time = 8.24 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.71 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {B c^{3} x^{4}}{4 e^{4}} + \frac {c \left (- 12 A a c d e^{3} - 20 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 30 B a c d^{2} e^{2} + 35 B c^{2} d^{4}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{3} \left (\frac {A c^{3}}{3 e^{4}} - \frac {4 B c^{3} d}{3 e^{5}}\right ) + x^{2} \left (- \frac {2 A c^{3} d}{e^{5}} + \frac {3 B a c^{2}}{2 e^{4}} + \frac {5 B c^{3} d^{2}}{e^{6}}\right ) + x \left (\frac {3 A a c^{2}}{e^{4}} + \frac {10 A c^{3} d^{2}}{e^{6}} - \frac {12 B a c^{2} d}{e^{5}} - \frac {20 B c^{3} d^{3}}{e^{7}}\right ) + \frac {- 2 A a^{3} e^{7} - 6 A a^{2} c d^{2} e^{5} - 78 A a c^{2} d^{4} e^{3} - 74 A c^{3} d^{6} e - B a^{3} d e^{6} + 33 B a^{2} c d^{3} e^{4} + 141 B a c^{2} d^{5} e^{2} + 107 B c^{3} d^{7} + x^{2} \left (- 18 A a^{2} c e^{7} - 108 A a c^{2} d^{2} e^{5} - 90 A c^{3} d^{4} e^{3} + 54 B a^{2} c d e^{6} + 180 B a c^{2} d^{3} e^{4} + 126 B c^{3} d^{5} e^{2}\right ) + x \left (- 18 A a^{2} c d e^{6} - 180 A a c^{2} d^{3} e^{4} - 162 A c^{3} d^{5} e^{2} - 3 B a^{3} e^{7} + 81 B a^{2} c d^{2} e^{5} + 315 B a c^{2} d^{4} e^{3} + 231 B c^{3} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} \] Input:
integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**4,x)
Output:
B*c**3*x**4/(4*e**4) + c*(-12*A*a*c*d*e**3 - 20*A*c**2*d**3*e + 3*B*a**2*e **4 + 30*B*a*c*d**2*e**2 + 35*B*c**2*d**4)*log(d + e*x)/e**8 + x**3*(A*c** 3/(3*e**4) - 4*B*c**3*d/(3*e**5)) + x**2*(-2*A*c**3*d/e**5 + 3*B*a*c**2/(2 *e**4) + 5*B*c**3*d**2/e**6) + x*(3*A*a*c**2/e**4 + 10*A*c**3*d**2/e**6 - 12*B*a*c**2*d/e**5 - 20*B*c**3*d**3/e**7) + (-2*A*a**3*e**7 - 6*A*a**2*c*d **2*e**5 - 78*A*a*c**2*d**4*e**3 - 74*A*c**3*d**6*e - B*a**3*d*e**6 + 33*B *a**2*c*d**3*e**4 + 141*B*a*c**2*d**5*e**2 + 107*B*c**3*d**7 + x**2*(-18*A *a**2*c*e**7 - 108*A*a*c**2*d**2*e**5 - 90*A*c**3*d**4*e**3 + 54*B*a**2*c* d*e**6 + 180*B*a*c**2*d**3*e**4 + 126*B*c**3*d**5*e**2) + x*(-18*A*a**2*c* d*e**6 - 180*A*a*c**2*d**3*e**4 - 162*A*c**3*d**5*e**2 - 3*B*a**3*e**7 + 8 1*B*a**2*c*d**2*e**5 + 315*B*a*c**2*d**4*e**3 + 231*B*c**3*d**6*e))/(6*d** 3*e**8 + 18*d**2*e**9*x + 18*d*e**10*x**2 + 6*e**11*x**3)
Time = 0.06 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {107 \, B c^{3} d^{7} - 74 \, A c^{3} d^{6} e + 141 \, B a c^{2} d^{5} e^{2} - 78 \, A a c^{2} d^{4} e^{3} + 33 \, B a^{2} c d^{3} e^{4} - 6 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 2 \, A a^{3} e^{7} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B a c^{2} d^{3} e^{4} - 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} - A a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e - 54 \, A c^{3} d^{5} e^{2} + 105 \, B a c^{2} d^{4} e^{3} - 60 \, A a c^{2} d^{3} e^{4} + 27 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x}{6 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac {3 \, B c^{3} e^{3} x^{4} - 4 \, {\left (4 \, B c^{3} d e^{2} - A c^{3} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B c^{3} d^{2} e - 4 \, A c^{3} d e^{2} + 3 \, B a c^{2} e^{3}\right )} x^{2} - 12 \, {\left (20 \, B c^{3} d^{3} - 10 \, A c^{3} d^{2} e + 12 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} x}{12 \, e^{7}} + \frac {{\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^4,x, algorithm="maxima")
Output:
1/6*(107*B*c^3*d^7 - 74*A*c^3*d^6*e + 141*B*a*c^2*d^5*e^2 - 78*A*a*c^2*d^4 *e^3 + 33*B*a^2*c*d^3*e^4 - 6*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 2*A*a^3*e^7 + 18*(7*B*c^3*d^5*e^2 - 5*A*c^3*d^4*e^3 + 10*B*a*c^2*d^3*e^4 - 6*A*a*c^2*d ^2*e^5 + 3*B*a^2*c*d*e^6 - A*a^2*c*e^7)*x^2 + 3*(77*B*c^3*d^6*e - 54*A*c^3 *d^5*e^2 + 105*B*a*c^2*d^4*e^3 - 60*A*a*c^2*d^3*e^4 + 27*B*a^2*c*d^2*e^5 - 6*A*a^2*c*d*e^6 - B*a^3*e^7)*x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8) + 1/12*(3*B*c^3*e^3*x^4 - 4*(4*B*c^3*d*e^2 - A*c^3*e^3)*x^3 + 6*( 10*B*c^3*d^2*e - 4*A*c^3*d*e^2 + 3*B*a*c^2*e^3)*x^2 - 12*(20*B*c^3*d^3 - 1 0*A*c^3*d^2*e + 12*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3)*x)/e^7 + (35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*l og(e*x + d)/e^8
Time = 0.14 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {{\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {107 \, B c^{3} d^{7} - 74 \, A c^{3} d^{6} e + 141 \, B a c^{2} d^{5} e^{2} - 78 \, A a c^{2} d^{4} e^{3} + 33 \, B a^{2} c d^{3} e^{4} - 6 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 2 \, A a^{3} e^{7} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B a c^{2} d^{3} e^{4} - 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} - A a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e - 54 \, A c^{3} d^{5} e^{2} + 105 \, B a c^{2} d^{4} e^{3} - 60 \, A a c^{2} d^{3} e^{4} + 27 \, B a^{2} c d^{2} e^{5} - 6 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{8}} + \frac {3 \, B c^{3} e^{12} x^{4} - 16 \, B c^{3} d e^{11} x^{3} + 4 \, A c^{3} e^{12} x^{3} + 60 \, B c^{3} d^{2} e^{10} x^{2} - 24 \, A c^{3} d e^{11} x^{2} + 18 \, B a c^{2} e^{12} x^{2} - 240 \, B c^{3} d^{3} e^{9} x + 120 \, A c^{3} d^{2} e^{10} x - 144 \, B a c^{2} d e^{11} x + 36 \, A a c^{2} e^{12} x}{12 \, e^{16}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^4,x, algorithm="giac")
Output:
(35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3 + 3 *B*a^2*c*e^4)*log(abs(e*x + d))/e^8 + 1/6*(107*B*c^3*d^7 - 74*A*c^3*d^6*e + 141*B*a*c^2*d^5*e^2 - 78*A*a*c^2*d^4*e^3 + 33*B*a^2*c*d^3*e^4 - 6*A*a^2* c*d^2*e^5 - B*a^3*d*e^6 - 2*A*a^3*e^7 + 18*(7*B*c^3*d^5*e^2 - 5*A*c^3*d^4* e^3 + 10*B*a*c^2*d^3*e^4 - 6*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 - A*a^2*c*e ^7)*x^2 + 3*(77*B*c^3*d^6*e - 54*A*c^3*d^5*e^2 + 105*B*a*c^2*d^4*e^3 - 60* A*a*c^2*d^3*e^4 + 27*B*a^2*c*d^2*e^5 - 6*A*a^2*c*d*e^6 - B*a^3*e^7)*x)/((e *x + d)^3*e^8) + 1/12*(3*B*c^3*e^12*x^4 - 16*B*c^3*d*e^11*x^3 + 4*A*c^3*e^ 12*x^3 + 60*B*c^3*d^2*e^10*x^2 - 24*A*c^3*d*e^11*x^2 + 18*B*a*c^2*e^12*x^2 - 240*B*c^3*d^3*e^9*x + 120*A*c^3*d^2*e^10*x - 144*B*a*c^2*d*e^11*x + 36* A*a*c^2*e^12*x)/e^16
Time = 6.18 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.77 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^4} \, dx=x^3\,\left (\frac {A\,c^3}{3\,e^4}-\frac {4\,B\,c^3\,d}{3\,e^5}\right )-\frac {\frac {B\,a^3\,d\,e^6+2\,A\,a^3\,e^7-33\,B\,a^2\,c\,d^3\,e^4+6\,A\,a^2\,c\,d^2\,e^5-141\,B\,a\,c^2\,d^5\,e^2+78\,A\,a\,c^2\,d^4\,e^3-107\,B\,c^3\,d^7+74\,A\,c^3\,d^6\,e}{6\,e}+x^2\,\left (-9\,B\,a^2\,c\,d\,e^5+3\,A\,a^2\,c\,e^6-30\,B\,a\,c^2\,d^3\,e^3+18\,A\,a\,c^2\,d^2\,e^4-21\,B\,c^3\,d^5\,e+15\,A\,c^3\,d^4\,e^2\right )+x\,\left (\frac {B\,a^3\,e^6}{2}-\frac {27\,B\,a^2\,c\,d^2\,e^4}{2}+3\,A\,a^2\,c\,d\,e^5-\frac {105\,B\,a\,c^2\,d^4\,e^2}{2}+30\,A\,a\,c^2\,d^3\,e^3-\frac {77\,B\,c^3\,d^6}{2}+27\,A\,c^3\,d^5\,e\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x\,\left (\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,c^3}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,a\,c^2}{e^4}+\frac {6\,B\,c^3\,d^2}{e^6}\right )}{e}-\frac {6\,d^2\,\left (\frac {A\,c^3}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e^2}+\frac {3\,A\,a\,c^2}{e^4}-\frac {4\,B\,c^3\,d^3}{e^7}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,a\,c^2}{2\,e^4}+\frac {3\,B\,c^3\,d^2}{e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,a^2\,c\,e^4+30\,B\,a\,c^2\,d^2\,e^2-12\,A\,a\,c^2\,d\,e^3+35\,B\,c^3\,d^4-20\,A\,c^3\,d^3\,e\right )}{e^8}+\frac {B\,c^3\,x^4}{4\,e^4} \] Input:
int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^4,x)
Output:
x^3*((A*c^3)/(3*e^4) - (4*B*c^3*d)/(3*e^5)) - ((2*A*a^3*e^7 - 107*B*c^3*d^ 7 + B*a^3*d*e^6 + 74*A*c^3*d^6*e + 78*A*a*c^2*d^4*e^3 + 6*A*a^2*c*d^2*e^5 - 141*B*a*c^2*d^5*e^2 - 33*B*a^2*c*d^3*e^4)/(6*e) + x^2*(3*A*a^2*c*e^6 - 2 1*B*c^3*d^5*e + 15*A*c^3*d^4*e^2 + 18*A*a*c^2*d^2*e^4 - 30*B*a*c^2*d^3*e^3 - 9*B*a^2*c*d*e^5) + x*((B*a^3*e^6)/2 - (77*B*c^3*d^6)/2 + 27*A*c^3*d^5*e + 30*A*a*c^2*d^3*e^3 - (105*B*a*c^2*d^4*e^2)/2 - (27*B*a^2*c*d^2*e^4)/2 + 3*A*a^2*c*d*e^5))/(d^3*e^7 + e^10*x^3 + 3*d^2*e^8*x + 3*d*e^9*x^2) + x*(( 4*d*((4*d*((A*c^3)/e^4 - (4*B*c^3*d)/e^5))/e - (3*B*a*c^2)/e^4 + (6*B*c^3* d^2)/e^6))/e - (6*d^2*((A*c^3)/e^4 - (4*B*c^3*d)/e^5))/e^2 + (3*A*a*c^2)/e ^4 - (4*B*c^3*d^3)/e^7) - x^2*((2*d*((A*c^3)/e^4 - (4*B*c^3*d)/e^5))/e - ( 3*B*a*c^2)/(2*e^4) + (3*B*c^3*d^2)/e^6) + (log(d + e*x)*(35*B*c^3*d^4 + 3* B*a^2*c*e^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3))/e^8 + (B*c^3*x^4)/(4*e^4)
Time = 0.25 (sec) , antiderivative size = 833, normalized size of antiderivative = 2.69 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^4,x)
Output:
(36*log(d + e*x)*a**2*b*c*d**4*e**4 + 108*log(d + e*x)*a**2*b*c*d**3*e**5* x + 108*log(d + e*x)*a**2*b*c*d**2*e**6*x**2 + 36*log(d + e*x)*a**2*b*c*d* e**7*x**3 - 144*log(d + e*x)*a**2*c**2*d**5*e**3 - 432*log(d + e*x)*a**2*c **2*d**4*e**4*x - 432*log(d + e*x)*a**2*c**2*d**3*e**5*x**2 - 144*log(d + e*x)*a**2*c**2*d**2*e**6*x**3 + 360*log(d + e*x)*a*b*c**2*d**6*e**2 + 1080 *log(d + e*x)*a*b*c**2*d**5*e**3*x + 1080*log(d + e*x)*a*b*c**2*d**4*e**4* x**2 + 360*log(d + e*x)*a*b*c**2*d**3*e**5*x**3 - 240*log(d + e*x)*a*c**3* d**7*e - 720*log(d + e*x)*a*c**3*d**6*e**2*x - 720*log(d + e*x)*a*c**3*d** 5*e**3*x**2 - 240*log(d + e*x)*a*c**3*d**4*e**4*x**3 + 420*log(d + e*x)*b* c**3*d**8 + 1260*log(d + e*x)*b*c**3*d**7*e*x + 1260*log(d + e*x)*b*c**3*d **6*e**2*x**2 + 420*log(d + e*x)*b*c**3*d**5*e**3*x**3 - 4*a**4*d*e**7 - 2 *a**3*b*d**2*e**6 - 6*a**3*b*d*e**7*x + 12*a**3*c*e**8*x**3 + 30*a**2*b*c* d**4*e**4 + 54*a**2*b*c*d**3*e**5*x - 36*a**2*b*c*d*e**7*x**3 - 120*a**2*c **2*d**5*e**3 - 216*a**2*c**2*d**4*e**4*x + 144*a**2*c**2*d**2*e**6*x**3 + 36*a**2*c**2*d*e**7*x**4 + 300*a*b*c**2*d**6*e**2 + 540*a*b*c**2*d**5*e** 3*x - 360*a*b*c**2*d**3*e**5*x**3 - 90*a*b*c**2*d**2*e**6*x**4 + 18*a*b*c* *2*d*e**7*x**5 - 200*a*c**3*d**7*e - 360*a*c**3*d**6*e**2*x + 240*a*c**3*d **4*e**4*x**3 + 60*a*c**3*d**3*e**5*x**4 - 12*a*c**3*d**2*e**6*x**5 + 4*a* c**3*d*e**7*x**6 + 350*b*c**3*d**8 + 630*b*c**3*d**7*e*x - 420*b*c**3*d**5 *e**3*x**3 - 105*b*c**3*d**4*e**4*x**4 + 21*b*c**3*d**3*e**5*x**5 - 7*b...