Integrand size = 22, antiderivative size = 290 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx=\frac {\left (B \left (c d^2+a e^2\right )^3-A c d e \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right )\right ) x}{e^7}-\frac {c (B d-A e) \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right ) x^2}{2 e^6}-\frac {c \left (A c d e \left (c d^2+3 a e^2\right )-B \left (c^2 d^4+3 a c d^2 e^2+3 a^2 e^4\right )\right ) x^3}{3 e^5}-\frac {c^2 (B d-A e) \left (c d^2+3 a e^2\right ) x^4}{4 e^4}+\frac {c^2 \left (B c d^2-A c d e+3 a B e^2\right ) x^5}{5 e^3}-\frac {c^3 (B d-A e) x^6}{6 e^2}+\frac {B c^3 x^7}{7 e}-\frac {(B d-A e) \left (c d^2+a e^2\right )^3 \log (d+e x)}{e^8} \] Output:
(B*(a*e^2+c*d^2)^3-A*c*d*e*(3*a^2*e^4+3*a*c*d^2*e^2+c^2*d^4))*x/e^7-1/2*c* (-A*e+B*d)*(3*a^2*e^4+3*a*c*d^2*e^2+c^2*d^4)*x^2/e^6-1/3*c*(A*c*d*e*(3*a*e ^2+c*d^2)-B*(3*a^2*e^4+3*a*c*d^2*e^2+c^2*d^4))*x^3/e^5-1/4*c^2*(-A*e+B*d)* (3*a*e^2+c*d^2)*x^4/e^4+1/5*c^2*(-A*c*d*e+3*B*a*e^2+B*c*d^2)*x^5/e^3-1/6*c ^3*(-A*e+B*d)*x^6/e^2+1/7*B*c^3*x^7/e-(-A*e+B*d)*(a*e^2+c*d^2)^3*ln(e*x+d) /e^8
Time = 0.17 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx=\frac {e x \left (7 A c e \left (90 a^2 e^4 (-2 d+e x)+15 a c e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+c^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+B \left (420 a^3 e^6+210 a^2 c e^4 \left (6 d^2-3 d e x+2 e^2 x^2\right )+21 a c^2 e^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+c^3 \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 (B d-A e) \left (c d^2+a e^2\right )^3 \log (d+e x)}{420 e^8} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x),x]
Output:
(e*x*(7*A*c*e*(90*a^2*e^4*(-2*d + e*x) + 15*a*c*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + c^2*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 1 5*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)) + B*(420*a^3*e^6 + 210*a^2*c*e ^4*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 21*a*c^2*e^2*(60*d^4 - 30*d^3*e*x + 20* d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + c^3*(420*d^6 - 210*d^5*e*x + 14 0*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x ^6))) - 420*(B*d - A*e)*(c*d^2 + a*e^2)^3*Log[d + e*x])/(420*e^8)
Time = 0.63 (sec) , antiderivative size = 290, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{d+e x} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (\frac {B \left (a e^2+c d^2\right )^3-A c d e \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )}{e^7}+\frac {c x \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right ) (A e-B d)}{e^6}+\frac {c x^2 \left (B \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )-A c d e \left (3 a e^2+c d^2\right )\right )}{e^5}+\frac {c^2 x^3 \left (3 a e^2+c d^2\right ) (A e-B d)}{e^4}-\frac {c^2 x^4 \left (-3 a B e^2+A c d e-B c d^2\right )}{e^3}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)}+\frac {c^3 x^5 (A e-B d)}{e^2}+\frac {B c^3 x^6}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (B \left (a e^2+c d^2\right )^3-A c d e \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{e^7}-\frac {c x^2 \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right ) (B d-A e)}{2 e^6}-\frac {c x^3 \left (A c d e \left (3 a e^2+c d^2\right )-B \left (3 a^2 e^4+3 a c d^2 e^2+c^2 d^4\right )\right )}{3 e^5}-\frac {c^2 x^4 \left (3 a e^2+c d^2\right ) (B d-A e)}{4 e^4}+\frac {c^2 x^5 \left (3 a B e^2-A c d e+B c d^2\right )}{5 e^3}-\frac {\left (a e^2+c d^2\right )^3 (B d-A e) \log (d+e x)}{e^8}-\frac {c^3 x^6 (B d-A e)}{6 e^2}+\frac {B c^3 x^7}{7 e}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x),x]
Output:
((B*(c*d^2 + a*e^2)^3 - A*c*d*e*(c^2*d^4 + 3*a*c*d^2*e^2 + 3*a^2*e^4))*x)/ e^7 - (c*(B*d - A*e)*(c^2*d^4 + 3*a*c*d^2*e^2 + 3*a^2*e^4)*x^2)/(2*e^6) - (c*(A*c*d*e*(c*d^2 + 3*a*e^2) - B*(c^2*d^4 + 3*a*c*d^2*e^2 + 3*a^2*e^4))*x ^3)/(3*e^5) - (c^2*(B*d - A*e)*(c*d^2 + 3*a*e^2)*x^4)/(4*e^4) + (c^2*(B*c* d^2 - A*c*d*e + 3*a*B*e^2)*x^5)/(5*e^3) - (c^3*(B*d - A*e)*x^6)/(6*e^2) + (B*c^3*x^7)/(7*e) - ((B*d - A*e)*(c*d^2 + a*e^2)^3*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 = 0.74 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.43
| method | result | size |
| norman | \(-\frac {\left (3 A \,a^{2} c d \,e^{5}+3 A a \,c^{2} d^{3} e^{3}+A \,c^{3} d^{5} e -B \,a^{3} e^{6}-3 B \,a^{2} c \,d^{2} e^{4}-3 B a \,c^{2} d^{4} e^{2}-B \,c^{3} d^{6}\right ) x}{e^{7}}+\frac {B \,c^{3} x^{7}}{7 e}+\frac {c \left (3 A \,a^{2} e^{5}+3 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-3 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right ) x^{2}}{2 e^{6}}-\frac {c \left (3 A a c d \,e^{3}+A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-3 B a c \,d^{2} e^{2}-B \,c^{2} d^{4}\right ) x^{3}}{3 e^{5}}+\frac {c^{2} \left (3 A a \,e^{3}+A c \,d^{2} e -3 B a d \,e^{2}-B c \,d^{3}\right ) x^{4}}{4 e^{4}}-\frac {c^{2} \left (A c d e -3 B a \,e^{2}-B c \,d^{2}\right ) x^{5}}{5 e^{3}}+\frac {c^{3} \left (A e -B d \right ) x^{6}}{6 e^{2}}+\frac {\left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(414\) |
| default | \(-\frac {3 A \,a^{2} c d \,e^{5} x -3 B a \,c^{2} d^{4} e^{2} x +\frac {3}{2} B a \,c^{2} d^{3} e^{3} x^{2}+\frac {3}{2} B \,a^{2} c d \,e^{5} x^{2}-B \,a^{2} c \,e^{6} x^{3}-\frac {1}{3} B \,c^{3} d^{4} e^{2} x^{3}-\frac {3}{2} A \,a^{2} c \,e^{6} x^{2}+\frac {1}{6} B \,c^{3} d \,e^{5} x^{6}+\frac {1}{5} A \,c^{3} d \,e^{5} x^{5}+3 A a \,c^{2} d^{3} e^{3} x +\frac {1}{4} B \,c^{3} d^{3} e^{3} x^{4}+\frac {1}{3} A \,c^{3} d^{3} e^{3} x^{3}-3 B \,a^{2} c \,d^{2} e^{4} x +A \,c^{3} d^{5} e x -\frac {3}{5} B a \,c^{2} e^{6} x^{5}-\frac {1}{5} B \,c^{3} d^{2} e^{4} x^{5}-\frac {3}{4} A a \,c^{2} e^{6} x^{4}-\frac {1}{4} A \,c^{3} d^{2} e^{4} x^{4}-\frac {1}{2} A \,c^{3} d^{4} e^{2} x^{2}+\frac {1}{2} B \,c^{3} d^{5} e \,x^{2}+A a \,c^{2} d \,e^{5} x^{3}-\frac {3}{2} A a \,c^{2} d^{2} e^{4} x^{2}+\frac {3}{4} B a \,c^{2} d \,e^{5} x^{4}-B a \,c^{2} d^{2} e^{4} x^{3}-B \,a^{3} e^{6} x -B \,c^{3} d^{6} x -\frac {1}{7} B \,c^{3} x^{7} e^{6}-\frac {1}{6} A \,c^{3} e^{6} x^{6}}{e^{7}}+\frac {\left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(485\) |
| risch | \(\frac {3 A \,a^{2} c \,x^{2}}{2 e}-\frac {B \,c^{3} d \,x^{6}}{6 e^{2}}-\frac {A \,c^{3} d \,x^{5}}{5 e^{2}}-\frac {B \,c^{3} d^{3} x^{4}}{4 e^{4}}-\frac {A a \,c^{2} d \,x^{3}}{e^{2}}+\frac {3 A a \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {3 B a \,c^{2} d \,x^{4}}{4 e^{2}}+\frac {B a \,c^{2} d^{2} x^{3}}{e^{3}}+\frac {3 \ln \left (e x +d \right ) A \,a^{2} c \,d^{2}}{e^{3}}+\frac {3 \ln \left (e x +d \right ) A a \,c^{2} d^{4}}{e^{5}}-\frac {3 \ln \left (e x +d \right ) B \,a^{2} c \,d^{3}}{e^{4}}-\frac {3 \ln \left (e x +d \right ) B a \,c^{2} d^{5}}{e^{6}}+\frac {\ln \left (e x +d \right ) A \,c^{3} d^{6}}{e^{7}}-\frac {\ln \left (e x +d \right ) B \,a^{3} d}{e^{2}}-\frac {\ln \left (e x +d \right ) B \,c^{3} d^{7}}{e^{8}}-\frac {3 A \,a^{2} c d x}{e^{2}}+\frac {3 B a \,c^{2} d^{4} x}{e^{5}}-\frac {3 B a \,c^{2} d^{3} x^{2}}{2 e^{4}}-\frac {3 B \,a^{2} c d \,x^{2}}{2 e^{2}}-\frac {3 A a \,c^{2} d^{3} x}{e^{4}}+\frac {3 B \,a^{2} c \,d^{2} x}{e^{3}}-\frac {A \,c^{3} d^{3} x^{3}}{3 e^{4}}-\frac {A \,c^{3} d^{5} x}{e^{6}}+\frac {3 B a \,c^{2} x^{5}}{5 e}+\frac {B \,c^{3} d^{2} x^{5}}{5 e^{3}}+\frac {3 A a \,c^{2} x^{4}}{4 e}+\frac {A \,c^{3} d^{2} x^{4}}{4 e^{3}}+\frac {A \,c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {B \,c^{3} d^{5} x^{2}}{2 e^{6}}+\frac {B \,c^{3} d^{6} x}{e^{7}}+\frac {B \,a^{2} c \,x^{3}}{e}+\frac {B \,c^{3} d^{4} x^{3}}{3 e^{5}}+\frac {\ln \left (e x +d \right ) A \,a^{3}}{e}+\frac {B \,a^{3} x}{e}+\frac {A \,c^{3} x^{6}}{6 e}+\frac {B \,c^{3} x^{7}}{7 e}\) | \(526\) |
| parallelrisch | \(\frac {70 A \,x^{6} c^{3} e^{7}+60 B \,x^{7} c^{3} e^{7}+420 A \ln \left (e x +d \right ) a^{3} e^{7}-420 B \ln \left (e x +d \right ) c^{3} d^{7}+420 B x \,a^{3} e^{7}+1260 B x a \,c^{2} d^{4} e^{3}+105 A \,x^{4} c^{3} d^{2} e^{5}-105 B \,x^{4} c^{3} d^{3} e^{4}-140 A \,x^{3} c^{3} d^{3} e^{4}-420 B \ln \left (e x +d \right ) a^{3} d \,e^{6}-84 A \,x^{5} c^{3} d \,e^{6}-70 B \,x^{6} c^{3} d \,e^{6}+420 A \ln \left (e x +d \right ) c^{3} d^{6} e +420 B \,x^{3} a^{2} c \,e^{7}+140 B \,x^{3} c^{3} d^{4} e^{3}+630 A \,x^{2} a^{2} c \,e^{7}+210 A \,x^{2} c^{3} d^{4} e^{3}-210 B \,x^{2} c^{3} d^{5} e^{2}-420 A x \,c^{3} d^{5} e^{2}+420 B x \,c^{3} d^{6} e +252 B \,x^{5} a \,c^{2} e^{7}+84 B \,x^{5} c^{3} d^{2} e^{5}+315 A \,x^{4} a \,c^{2} e^{7}+1260 A \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{5}+1260 A \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{3}-1260 B \ln \left (e x +d \right ) a^{2} c \,d^{3} e^{4}-1260 B \ln \left (e x +d \right ) a \,c^{2} d^{5} e^{2}+420 B \,x^{3} a \,c^{2} d^{2} e^{5}+630 A \,x^{2} a \,c^{2} d^{2} e^{5}-630 B \,x^{2} a^{2} c d \,e^{6}-630 B \,x^{2} a \,c^{2} d^{3} e^{4}-1260 A x \,a^{2} c d \,e^{6}-1260 A x a \,c^{2} d^{3} e^{4}+1260 B x \,a^{2} c \,d^{2} e^{5}-315 B \,x^{4} a \,c^{2} d \,e^{6}-420 A \,x^{3} a \,c^{2} d \,e^{6}}{420 e^{8}}\) | \(530\) |
Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d),x,method=_RETURNVERBOSE)
Output:
-(3*A*a^2*c*d*e^5+3*A*a*c^2*d^3*e^3+A*c^3*d^5*e-B*a^3*e^6-3*B*a^2*c*d^2*e^ 4-3*B*a*c^2*d^4*e^2-B*c^3*d^6)/e^7*x+1/7*B*c^3*x^7/e+1/2*c/e^6*(3*A*a^2*e^ 5+3*A*a*c*d^2*e^3+A*c^2*d^4*e-3*B*a^2*d*e^4-3*B*a*c*d^3*e^2-B*c^2*d^5)*x^2 -1/3*c/e^5*(3*A*a*c*d*e^3+A*c^2*d^3*e-3*B*a^2*e^4-3*B*a*c*d^2*e^2-B*c^2*d^ 4)*x^3+1/4*c^2/e^4*(3*A*a*e^3+A*c*d^2*e-3*B*a*d*e^2-B*c*d^3)*x^4-1/5*c^2/e ^3*(A*c*d*e-3*B*a*e^2-B*c*d^2)*x^5+1/6*c^3/e^2*(A*e-B*d)*x^6+(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)/e^8*ln(e*x+d)
Time = 0.08 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{7} x^{7} - 70 \, {\left (B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{5} - A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{4} - A c^{3} d^{2} e^{5} + 3 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{3} - A c^{3} d^{3} e^{4} + 3 \, B a c^{2} d^{2} e^{5} - 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e^{2} - A c^{3} d^{4} e^{3} + 3 \, B a c^{2} d^{3} e^{4} - 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} - 3 \, A a^{2} c e^{7}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} e - A c^{3} d^{5} e^{2} + 3 \, B a c^{2} d^{4} e^{3} - 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} - 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x - 420 \, {\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d),x, algorithm="fricas")
Output:
1/420*(60*B*c^3*e^7*x^7 - 70*(B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 84*(B*c^3*d^2 *e^5 - A*c^3*d*e^6 + 3*B*a*c^2*e^7)*x^5 - 105*(B*c^3*d^3*e^4 - A*c^3*d^2*e ^5 + 3*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 140*(B*c^3*d^4*e^3 - A*c^3*d^3 *e^4 + 3*B*a*c^2*d^2*e^5 - 3*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 - 210*(B*c ^3*d^5*e^2 - A*c^3*d^4*e^3 + 3*B*a*c^2*d^3*e^4 - 3*A*a*c^2*d^2*e^5 + 3*B*a ^2*c*d*e^6 - 3*A*a^2*c*e^7)*x^2 + 420*(B*c^3*d^6*e - A*c^3*d^5*e^2 + 3*B*a *c^2*d^4*e^3 - 3*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5 - 3*A*a^2*c*d*e^6 + B *a^3*e^7)*x - 420*(B*c^3*d^7 - A*c^3*d^6*e + 3*B*a*c^2*d^5*e^2 - 3*A*a*c^2 *d^4*e^3 + 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7 )*log(e*x + d))/e^8
Time = 0.50 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx=\frac {B c^{3} x^{7}}{7 e} + x^{6} \left (\frac {A c^{3}}{6 e} - \frac {B c^{3} d}{6 e^{2}}\right ) + x^{5} \left (- \frac {A c^{3} d}{5 e^{2}} + \frac {3 B a c^{2}}{5 e} + \frac {B c^{3} d^{2}}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {3 A a c^{2}}{4 e} + \frac {A c^{3} d^{2}}{4 e^{3}} - \frac {3 B a c^{2} d}{4 e^{2}} - \frac {B c^{3} d^{3}}{4 e^{4}}\right ) + x^{3} \left (- \frac {A a c^{2} d}{e^{2}} - \frac {A c^{3} d^{3}}{3 e^{4}} + \frac {B a^{2} c}{e} + \frac {B a c^{2} d^{2}}{e^{3}} + \frac {B c^{3} d^{4}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} c}{2 e} + \frac {3 A a c^{2} d^{2}}{2 e^{3}} + \frac {A c^{3} d^{4}}{2 e^{5}} - \frac {3 B a^{2} c d}{2 e^{2}} - \frac {3 B a c^{2} d^{3}}{2 e^{4}} - \frac {B c^{3} d^{5}}{2 e^{6}}\right ) + x \left (- \frac {3 A a^{2} c d}{e^{2}} - \frac {3 A a c^{2} d^{3}}{e^{4}} - \frac {A c^{3} d^{5}}{e^{6}} + \frac {B a^{3}}{e} + \frac {3 B a^{2} c d^{2}}{e^{3}} + \frac {3 B a c^{2} d^{4}}{e^{5}} + \frac {B c^{3} d^{6}}{e^{7}}\right ) - \frac {\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{8}} \] Input:
integrate((B*x+A)*(c*x**2+a)**3/(e*x+d),x)
Output:
B*c**3*x**7/(7*e) + x**6*(A*c**3/(6*e) - B*c**3*d/(6*e**2)) + x**5*(-A*c** 3*d/(5*e**2) + 3*B*a*c**2/(5*e) + B*c**3*d**2/(5*e**3)) + x**4*(3*A*a*c**2 /(4*e) + A*c**3*d**2/(4*e**3) - 3*B*a*c**2*d/(4*e**2) - B*c**3*d**3/(4*e** 4)) + x**3*(-A*a*c**2*d/e**2 - A*c**3*d**3/(3*e**4) + B*a**2*c/e + B*a*c** 2*d**2/e**3 + B*c**3*d**4/(3*e**5)) + x**2*(3*A*a**2*c/(2*e) + 3*A*a*c**2* d**2/(2*e**3) + A*c**3*d**4/(2*e**5) - 3*B*a**2*c*d/(2*e**2) - 3*B*a*c**2* d**3/(2*e**4) - B*c**3*d**5/(2*e**6)) + x*(-3*A*a**2*c*d/e**2 - 3*A*a*c**2 *d**3/e**4 - A*c**3*d**5/e**6 + B*a**3/e + 3*B*a**2*c*d**2/e**3 + 3*B*a*c* *2*d**4/e**5 + B*c**3*d**6/e**7) - (-A*e + B*d)*(a*e**2 + c*d**2)**3*log(d + e*x)/e**8
Time = 0.04 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{6} x^{7} - 70 \, {\left (B c^{3} d e^{5} - A c^{3} e^{6}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{4} - A c^{3} d e^{5} + 3 \, B a c^{2} e^{6}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{3} - A c^{3} d^{2} e^{4} + 3 \, B a c^{2} d e^{5} - 3 \, A a c^{2} e^{6}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{2} - A c^{3} d^{3} e^{3} + 3 \, B a c^{2} d^{2} e^{4} - 3 \, A a c^{2} d e^{5} + 3 \, B a^{2} c e^{6}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e - A c^{3} d^{4} e^{2} + 3 \, B a c^{2} d^{3} e^{3} - 3 \, A a c^{2} d^{2} e^{4} + 3 \, B a^{2} c d e^{5} - 3 \, A a^{2} c e^{6}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} - A c^{3} d^{5} e + 3 \, B a c^{2} d^{4} e^{2} - 3 \, A a c^{2} d^{3} e^{3} + 3 \, B a^{2} c d^{2} e^{4} - 3 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d),x, algorithm="maxima")
Output:
1/420*(60*B*c^3*e^6*x^7 - 70*(B*c^3*d*e^5 - A*c^3*e^6)*x^6 + 84*(B*c^3*d^2 *e^4 - A*c^3*d*e^5 + 3*B*a*c^2*e^6)*x^5 - 105*(B*c^3*d^3*e^3 - A*c^3*d^2*e ^4 + 3*B*a*c^2*d*e^5 - 3*A*a*c^2*e^6)*x^4 + 140*(B*c^3*d^4*e^2 - A*c^3*d^3 *e^3 + 3*B*a*c^2*d^2*e^4 - 3*A*a*c^2*d*e^5 + 3*B*a^2*c*e^6)*x^3 - 210*(B*c ^3*d^5*e - A*c^3*d^4*e^2 + 3*B*a*c^2*d^3*e^3 - 3*A*a*c^2*d^2*e^4 + 3*B*a^2 *c*d*e^5 - 3*A*a^2*c*e^6)*x^2 + 420*(B*c^3*d^6 - A*c^3*d^5*e + 3*B*a*c^2*d ^4*e^2 - 3*A*a*c^2*d^3*e^3 + 3*B*a^2*c*d^2*e^4 - 3*A*a^2*c*d*e^5 + B*a^3*e ^6)*x)/e^7 - (B*c^3*d^7 - A*c^3*d^6*e + 3*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4* e^3 + 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7)*log (e*x + d)/e^8
Time = 0.13 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.68 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{6} x^{7} - 70 \, B c^{3} d e^{5} x^{6} + 70 \, A c^{3} e^{6} x^{6} + 84 \, B c^{3} d^{2} e^{4} x^{5} - 84 \, A c^{3} d e^{5} x^{5} + 252 \, B a c^{2} e^{6} x^{5} - 105 \, B c^{3} d^{3} e^{3} x^{4} + 105 \, A c^{3} d^{2} e^{4} x^{4} - 315 \, B a c^{2} d e^{5} x^{4} + 315 \, A a c^{2} e^{6} x^{4} + 140 \, B c^{3} d^{4} e^{2} x^{3} - 140 \, A c^{3} d^{3} e^{3} x^{3} + 420 \, B a c^{2} d^{2} e^{4} x^{3} - 420 \, A a c^{2} d e^{5} x^{3} + 420 \, B a^{2} c e^{6} x^{3} - 210 \, B c^{3} d^{5} e x^{2} + 210 \, A c^{3} d^{4} e^{2} x^{2} - 630 \, B a c^{2} d^{3} e^{3} x^{2} + 630 \, A a c^{2} d^{2} e^{4} x^{2} - 630 \, B a^{2} c d e^{5} x^{2} + 630 \, A a^{2} c e^{6} x^{2} + 420 \, B c^{3} d^{6} x - 420 \, A c^{3} d^{5} e x + 1260 \, B a c^{2} d^{4} e^{2} x - 1260 \, A a c^{2} d^{3} e^{3} x + 1260 \, B a^{2} c d^{2} e^{4} x - 1260 \, A a^{2} c d e^{5} x + 420 \, B a^{3} e^{6} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d),x, algorithm="giac")
Output:
1/420*(60*B*c^3*e^6*x^7 - 70*B*c^3*d*e^5*x^6 + 70*A*c^3*e^6*x^6 + 84*B*c^3 *d^2*e^4*x^5 - 84*A*c^3*d*e^5*x^5 + 252*B*a*c^2*e^6*x^5 - 105*B*c^3*d^3*e^ 3*x^4 + 105*A*c^3*d^2*e^4*x^4 - 315*B*a*c^2*d*e^5*x^4 + 315*A*a*c^2*e^6*x^ 4 + 140*B*c^3*d^4*e^2*x^3 - 140*A*c^3*d^3*e^3*x^3 + 420*B*a*c^2*d^2*e^4*x^ 3 - 420*A*a*c^2*d*e^5*x^3 + 420*B*a^2*c*e^6*x^3 - 210*B*c^3*d^5*e*x^2 + 21 0*A*c^3*d^4*e^2*x^2 - 630*B*a*c^2*d^3*e^3*x^2 + 630*A*a*c^2*d^2*e^4*x^2 - 630*B*a^2*c*d*e^5*x^2 + 630*A*a^2*c*e^6*x^2 + 420*B*c^3*d^6*x - 420*A*c^3* d^5*e*x + 1260*B*a*c^2*d^4*e^2*x - 1260*A*a*c^2*d^3*e^3*x + 1260*B*a^2*c*d ^2*e^4*x - 1260*A*a^2*c*d*e^5*x + 420*B*a^3*e^6*x)/e^7 - (B*c^3*d^7 - A*c^ 3*d^6*e + 3*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 - 3*A* a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7)*log(abs(e*x + d))/e^8
Time = 6.35 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx=x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{e}-\frac {3\,B\,a^2\,c}{e}\right )}{2\,e}+\frac {3\,A\,a^2\,c}{2\,e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{4\,e}+\frac {3\,A\,a\,c^2}{4\,e}\right )+x^6\,\left (\frac {A\,c^3}{6\,e}-\frac {B\,c^3\,d}{6\,e^2}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{3\,e}-\frac {B\,a^2\,c}{e}\right )+x\,\left (\frac {B\,a^3}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,A\,a\,c^2}{e}\right )}{e}-\frac {3\,B\,a^2\,c}{e}\right )}{e}+\frac {3\,A\,a^2\,c}{e}\right )}{e}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,c^3}{e}-\frac {B\,c^3\,d}{e^2}\right )}{5\,e}-\frac {3\,B\,a\,c^2}{5\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^3\,d\,e^6+A\,a^3\,e^7-3\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5-3\,B\,a\,c^2\,d^5\,e^2+3\,A\,a\,c^2\,d^4\,e^3-B\,c^3\,d^7+A\,c^3\,d^6\,e\right )}{e^8}+\frac {B\,c^3\,x^7}{7\,e} \] Input:
int(((a + c*x^2)^3*(A + B*x))/(d + e*x),x)
Output:
x^2*((d*((d*((d*((d*((A*c^3)/e - (B*c^3*d)/e^2))/e - (3*B*a*c^2)/e))/e + ( 3*A*a*c^2)/e))/e - (3*B*a^2*c)/e))/(2*e) + (3*A*a^2*c)/(2*e)) + x^4*((d*(( d*((A*c^3)/e - (B*c^3*d)/e^2))/e - (3*B*a*c^2)/e))/(4*e) + (3*A*a*c^2)/(4* e)) + x^6*((A*c^3)/(6*e) - (B*c^3*d)/(6*e^2)) - x^3*((d*((d*((d*((A*c^3)/e - (B*c^3*d)/e^2))/e - (3*B*a*c^2)/e))/e + (3*A*a*c^2)/e))/(3*e) - (B*a^2* c)/e) + x*((B*a^3)/e - (d*((d*((d*((d*((d*((A*c^3)/e - (B*c^3*d)/e^2))/e - (3*B*a*c^2)/e))/e + (3*A*a*c^2)/e))/e - (3*B*a^2*c)/e))/e + (3*A*a^2*c)/e ))/e) - x^5*((d*((A*c^3)/e - (B*c^3*d)/e^2))/(5*e) - (3*B*a*c^2)/(5*e)) + (log(d + e*x)*(A*a^3*e^7 - B*c^3*d^7 - B*a^3*d*e^6 + A*c^3*d^6*e + 3*A*a*c ^2*d^4*e^3 + 3*A*a^2*c*d^2*e^5 - 3*B*a*c^2*d^5*e^2 - 3*B*a^2*c*d^3*e^4))/e ^8 + (B*c^3*x^7)/(7*e)
Time = 0.22 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.83 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{d+e x} \, dx=\frac {420 \,\mathrm {log}\left (e x +d \right ) a^{4} e^{7}-420 \,\mathrm {log}\left (e x +d \right ) a^{3} b d \,e^{6}+1260 \,\mathrm {log}\left (e x +d \right ) a^{3} c \,d^{2} e^{5}+1260 \,\mathrm {log}\left (e x +d \right ) a^{2} c^{2} d^{4} e^{3}+420 \,\mathrm {log}\left (e x +d \right ) a \,c^{3} d^{6} e -1260 a^{3} c d \,e^{6} x +420 a^{2} b c \,e^{7} x^{3}-1260 \,\mathrm {log}\left (e x +d \right ) a^{2} b c \,d^{3} e^{4}-1260 \,\mathrm {log}\left (e x +d \right ) a b \,c^{2} d^{5} e^{2}+1260 a^{2} b c \,d^{2} e^{5} x -630 a^{2} b c d \,e^{6} x^{2}+1260 a b \,c^{2} d^{4} e^{3} x -630 a b \,c^{2} d^{3} e^{4} x^{2}+420 a b \,c^{2} d^{2} e^{5} x^{3}-315 a b \,c^{2} d \,e^{6} x^{4}-420 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{7}+420 a^{3} b \,e^{7} x +630 a^{3} c \,e^{7} x^{2}+315 a^{2} c^{2} e^{7} x^{4}+70 a \,c^{3} e^{7} x^{6}+60 b \,c^{3} e^{7} x^{7}-1260 a^{2} c^{2} d^{3} e^{4} x +630 a^{2} c^{2} d^{2} e^{5} x^{2}-420 a^{2} c^{2} d \,e^{6} x^{3}+252 a b \,c^{2} e^{7} x^{5}-420 a \,c^{3} d^{5} e^{2} x +210 a \,c^{3} d^{4} e^{3} x^{2}-140 a \,c^{3} d^{3} e^{4} x^{3}+105 a \,c^{3} d^{2} e^{5} x^{4}-84 a \,c^{3} d \,e^{6} x^{5}+420 b \,c^{3} d^{6} e x -210 b \,c^{3} d^{5} e^{2} x^{2}+140 b \,c^{3} d^{4} e^{3} x^{3}-105 b \,c^{3} d^{3} e^{4} x^{4}+84 b \,c^{3} d^{2} e^{5} x^{5}-70 b \,c^{3} d \,e^{6} x^{6}}{420 e^{8}} \] Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d),x)
Output:
(420*log(d + e*x)*a**4*e**7 - 420*log(d + e*x)*a**3*b*d*e**6 + 1260*log(d + e*x)*a**3*c*d**2*e**5 - 1260*log(d + e*x)*a**2*b*c*d**3*e**4 + 1260*log( d + e*x)*a**2*c**2*d**4*e**3 - 1260*log(d + e*x)*a*b*c**2*d**5*e**2 + 420* log(d + e*x)*a*c**3*d**6*e - 420*log(d + e*x)*b*c**3*d**7 + 420*a**3*b*e** 7*x - 1260*a**3*c*d*e**6*x + 630*a**3*c*e**7*x**2 + 1260*a**2*b*c*d**2*e** 5*x - 630*a**2*b*c*d*e**6*x**2 + 420*a**2*b*c*e**7*x**3 - 1260*a**2*c**2*d **3*e**4*x + 630*a**2*c**2*d**2*e**5*x**2 - 420*a**2*c**2*d*e**6*x**3 + 31 5*a**2*c**2*e**7*x**4 + 1260*a*b*c**2*d**4*e**3*x - 630*a*b*c**2*d**3*e**4 *x**2 + 420*a*b*c**2*d**2*e**5*x**3 - 315*a*b*c**2*d*e**6*x**4 + 252*a*b*c **2*e**7*x**5 - 420*a*c**3*d**5*e**2*x + 210*a*c**3*d**4*e**3*x**2 - 140*a *c**3*d**3*e**4*x**3 + 105*a*c**3*d**2*e**5*x**4 - 84*a*c**3*d*e**6*x**5 + 70*a*c**3*e**7*x**6 + 420*b*c**3*d**6*e*x - 210*b*c**3*d**5*e**2*x**2 + 1 40*b*c**3*d**4*e**3*x**3 - 105*b*c**3*d**3*e**4*x**4 + 84*b*c**3*d**2*e**5 *x**5 - 70*b*c**3*d*e**6*x**6 + 60*b*c**3*e**7*x**7)/(420*e**8)