Integrand size = 22, antiderivative size = 169 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{d+e x} \, dx=\frac {\left (B \left (c d^2+a e^2\right )^2-A c d e \left (c d^2+2 a e^2\right )\right ) x}{e^5}-\frac {c (B d-A e) \left (c d^2+2 a e^2\right ) x^2}{2 e^4}+\frac {c \left (B c d^2-A c d e+2 a B e^2\right ) x^3}{3 e^3}-\frac {c^2 (B d-A e) x^4}{4 e^2}+\frac {B c^2 x^5}{5 e}-\frac {(B d-A e) \left (c d^2+a e^2\right )^2 \log (d+e x)}{e^6} \]
(B*(a*e^2+c*d^2)^2-A*c*d*e*(2*a*e^2+c*d^2))*x/e^5-1/2*c*(-A*e+B*d)*(2*a*e^ 2+c*d^2)*x^2/e^4+1/3*c*(-A*c*d*e+2*B*a*e^2+B*c*d^2)*x^3/e^3-1/4*c^2*(-A*e+ B*d)*x^4/e^2+1/5*B*c^2*x^5/e-(-A*e+B*d)*(a*e^2+c*d^2)^2*ln(e*x+d)/e^6
Time = 0.06 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{d+e x} \, dx=\frac {e x \left (5 A c e \left (12 a e^2 (-2 d+e x)+c \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+B \left (60 a^2 e^4+20 a c e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+c^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )\right )-60 (B d-A e) \left (c d^2+a e^2\right )^2 \log (d+e x)}{60 e^6} \]
(e*x*(5*A*c*e*(12*a*e^2*(-2*d + e*x) + c*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^ 2 + 3*e^3*x^3)) + B*(60*a^2*e^4 + 20*a*c*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + c^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)) ) - 60*(B*d - A*e)*(c*d^2 + a*e^2)^2*Log[d + e*x])/(60*e^6)
Time = 0.41 (sec) , antiderivative size = 169, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^2 (A+B x)}{d+e x} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (\frac {\left (a e^2+c d^2\right )^2 (A e-B d)}{e^5 (d+e x)}+\frac {B \left (a e^2+c d^2\right )^2-A c d e \left (2 a e^2+c d^2\right )}{e^5}+\frac {c x \left (2 a e^2+c d^2\right ) (A e-B d)}{e^4}-\frac {c x^2 \left (-2 a B e^2+A c d e-B c d^2\right )}{e^3}+\frac {c^2 x^3 (A e-B d)}{e^2}+\frac {B c^2 x^4}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a e^2+c d^2\right )^2 (B d-A e) \log (d+e x)}{e^6}+\frac {x \left (B \left (a e^2+c d^2\right )^2-A c d e \left (2 a e^2+c d^2\right )\right )}{e^5}-\frac {c x^2 \left (2 a e^2+c d^2\right ) (B d-A e)}{2 e^4}+\frac {c x^3 \left (2 a B e^2-A c d e+B c d^2\right )}{3 e^3}-\frac {c^2 x^4 (B d-A e)}{4 e^2}+\frac {B c^2 x^5}{5 e}\) |
((B*(c*d^2 + a*e^2)^2 - A*c*d*e*(c*d^2 + 2*a*e^2))*x)/e^5 - (c*(B*d - A*e) *(c*d^2 + 2*a*e^2)*x^2)/(2*e^4) + (c*(B*c*d^2 - A*c*d*e + 2*a*B*e^2)*x^3)/ (3*e^3) - (c^2*(B*d - A*e)*x^4)/(4*e^2) + (B*c^2*x^5)/(5*e) - ((B*d - A*e) *(c*d^2 + a*e^2)^2*Log[d + e*x])/e^6
3.14.3.3.1 Defintions of rubi rules used
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 = 0.25 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.33
| method | result | size |
| norman | \(-\frac {\left (2 A a c d \,e^{3}+A \,c^{2} d^{3} e -B \,e^{4} a^{2}-2 B a c \,d^{2} e^{2}-B \,c^{2} d^{4}\right ) x}{e^{5}}+\frac {B \,c^{2} x^{5}}{5 e}+\frac {c \left (2 A a \,e^{3}+A c \,d^{2} e -2 B a d \,e^{2}-B c \,d^{3}\right ) x^{2}}{2 e^{4}}-\frac {c \left (A c d e -2 B a \,e^{2}-B c \,d^{2}\right ) x^{3}}{3 e^{3}}+\frac {c^{2} \left (A e -B d \right ) x^{4}}{4 e^{2}}+\frac {\left (A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(224\) |
| default | \(-\frac {-\frac {1}{5} B \,c^{2} x^{5} e^{4}-\frac {1}{4} A \,c^{2} e^{4} x^{4}+\frac {1}{4} B \,c^{2} d \,e^{3} x^{4}+\frac {1}{3} A \,c^{2} d \,e^{3} x^{3}-\frac {2}{3} B a c \,e^{4} x^{3}-\frac {1}{3} B \,c^{2} d^{2} e^{2} x^{3}-A a c \,e^{4} x^{2}-\frac {1}{2} A \,c^{2} d^{2} e^{2} x^{2}+B a c d \,e^{3} x^{2}+\frac {1}{2} B \,c^{2} d^{3} e \,x^{2}+2 A a c d \,e^{3} x +A \,c^{2} d^{3} e x -B \,a^{2} e^{4} x -2 B a c \,d^{2} e^{2} x -B \,c^{2} d^{4} x}{e^{5}}+\frac {\left (A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(255\) |
| risch | \(\frac {B \,c^{2} x^{5}}{5 e}+\frac {A \,c^{2} x^{4}}{4 e}-\frac {B \,c^{2} d \,x^{4}}{4 e^{2}}-\frac {A \,c^{2} d \,x^{3}}{3 e^{2}}+\frac {2 B a c \,x^{3}}{3 e}+\frac {B \,c^{2} d^{2} x^{3}}{3 e^{3}}+\frac {A a c \,x^{2}}{e}+\frac {A \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {B a c d \,x^{2}}{e^{2}}-\frac {B \,c^{2} d^{3} x^{2}}{2 e^{4}}-\frac {2 A a c d x}{e^{2}}-\frac {A \,c^{2} d^{3} x}{e^{4}}+\frac {B \,a^{2} x}{e}+\frac {2 B a c \,d^{2} x}{e^{3}}+\frac {B \,c^{2} d^{4} x}{e^{5}}+\frac {\ln \left (e x +d \right ) A \,a^{2}}{e}+\frac {2 \ln \left (e x +d \right ) A a c \,d^{2}}{e^{3}}+\frac {\ln \left (e x +d \right ) A \,c^{2} d^{4}}{e^{5}}-\frac {\ln \left (e x +d \right ) B \,a^{2} d}{e^{2}}-\frac {2 \ln \left (e x +d \right ) B a c \,d^{3}}{e^{4}}-\frac {\ln \left (e x +d \right ) B \,c^{2} d^{5}}{e^{6}}\) | \(285\) |
| parallelrisch | \(\frac {12 B \,x^{5} c^{2} e^{5}+15 A \,x^{4} c^{2} e^{5}-15 B \,x^{4} c^{2} d \,e^{4}-20 A \,x^{3} c^{2} d \,e^{4}+40 B \,x^{3} a c \,e^{5}+20 B \,x^{3} c^{2} d^{2} e^{3}+60 A \,x^{2} a c \,e^{5}+30 A \,x^{2} c^{2} d^{2} e^{3}-60 B \,x^{2} a c d \,e^{4}-30 B \,x^{2} c^{2} d^{3} e^{2}+60 A \ln \left (e x +d \right ) a^{2} e^{5}+120 A \ln \left (e x +d \right ) a c \,d^{2} e^{3}+60 A \ln \left (e x +d \right ) c^{2} d^{4} e -120 A x a c d \,e^{4}-60 A x \,c^{2} d^{3} e^{2}-60 B \ln \left (e x +d \right ) a^{2} d \,e^{4}-120 B \ln \left (e x +d \right ) a c \,d^{3} e^{2}-60 B \ln \left (e x +d \right ) c^{2} d^{5}+60 B x \,a^{2} e^{5}+120 B x a c \,d^{2} e^{3}+60 B x \,c^{2} d^{4} e}{60 e^{6}}\) | \(288\) |
-(2*A*a*c*d*e^3+A*c^2*d^3*e-B*a^2*e^4-2*B*a*c*d^2*e^2-B*c^2*d^4)/e^5*x+1/5 *B*c^2*x^5/e+1/2*c/e^4*(2*A*a*e^3+A*c*d^2*e-2*B*a*d*e^2-B*c*d^3)*x^2-1/3*c /e^3*(A*c*d*e-2*B*a*e^2-B*c*d^2)*x^3+1/4*c^2/e^2*(A*e-B*d)*x^4+(A*a^2*e^5+ 2*A*a*c*d^2*e^3+A*c^2*d^4*e-B*a^2*d*e^4-2*B*a*c*d^3*e^2-B*c^2*d^5)/e^6*ln( e*x+d)
Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.44 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{d+e x} \, dx=\frac {12 \, B c^{2} e^{5} x^{5} - 15 \, {\left (B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 20 \, {\left (B c^{2} d^{2} e^{3} - A c^{2} d e^{4} + 2 \, B a c e^{5}\right )} x^{3} - 30 \, {\left (B c^{2} d^{3} e^{2} - A c^{2} d^{2} e^{3} + 2 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 60 \, {\left (B c^{2} d^{4} e - A c^{2} d^{3} e^{2} + 2 \, B a c d^{2} e^{3} - 2 \, A a c d e^{4} + B a^{2} e^{5}\right )} x - 60 \, {\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \]
1/60*(12*B*c^2*e^5*x^5 - 15*(B*c^2*d*e^4 - A*c^2*e^5)*x^4 + 20*(B*c^2*d^2* e^3 - A*c^2*d*e^4 + 2*B*a*c*e^5)*x^3 - 30*(B*c^2*d^3*e^2 - A*c^2*d^2*e^3 + 2*B*a*c*d*e^4 - 2*A*a*c*e^5)*x^2 + 60*(B*c^2*d^4*e - A*c^2*d^3*e^2 + 2*B* a*c*d^2*e^3 - 2*A*a*c*d*e^4 + B*a^2*e^5)*x - 60*(B*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)*log(e*x + d) )/e^6
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.22 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{d+e x} \, dx=\frac {B c^{2} x^{5}}{5 e} + x^{4} \left (\frac {A c^{2}}{4 e} - \frac {B c^{2} d}{4 e^{2}}\right ) + x^{3} \left (- \frac {A c^{2} d}{3 e^{2}} + \frac {2 B a c}{3 e} + \frac {B c^{2} d^{2}}{3 e^{3}}\right ) + x^{2} \left (\frac {A a c}{e} + \frac {A c^{2} d^{2}}{2 e^{3}} - \frac {B a c d}{e^{2}} - \frac {B c^{2} d^{3}}{2 e^{4}}\right ) + x \left (- \frac {2 A a c d}{e^{2}} - \frac {A c^{2} d^{3}}{e^{4}} + \frac {B a^{2}}{e} + \frac {2 B a c d^{2}}{e^{3}} + \frac {B c^{2} d^{4}}{e^{5}}\right ) - \frac {\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{6}} \]
B*c**2*x**5/(5*e) + x**4*(A*c**2/(4*e) - B*c**2*d/(4*e**2)) + x**3*(-A*c** 2*d/(3*e**2) + 2*B*a*c/(3*e) + B*c**2*d**2/(3*e**3)) + x**2*(A*a*c/e + A*c **2*d**2/(2*e**3) - B*a*c*d/e**2 - B*c**2*d**3/(2*e**4)) + x*(-2*A*a*c*d/e **2 - A*c**2*d**3/e**4 + B*a**2/e + 2*B*a*c*d**2/e**3 + B*c**2*d**4/e**5) - (-A*e + B*d)*(a*e**2 + c*d**2)**2*log(d + e*x)/e**6
Time = 0.20 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{d+e x} \, dx=\frac {12 \, B c^{2} e^{4} x^{5} - 15 \, {\left (B c^{2} d e^{3} - A c^{2} e^{4}\right )} x^{4} + 20 \, {\left (B c^{2} d^{2} e^{2} - A c^{2} d e^{3} + 2 \, B a c e^{4}\right )} x^{3} - 30 \, {\left (B c^{2} d^{3} e - A c^{2} d^{2} e^{2} + 2 \, B a c d e^{3} - 2 \, A a c e^{4}\right )} x^{2} + 60 \, {\left (B c^{2} d^{4} - A c^{2} d^{3} e + 2 \, B a c d^{2} e^{2} - 2 \, A a c d e^{3} + B a^{2} e^{4}\right )} x}{60 \, e^{5}} - \frac {{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} \log \left (e x + d\right )}{e^{6}} \]
1/60*(12*B*c^2*e^4*x^5 - 15*(B*c^2*d*e^3 - A*c^2*e^4)*x^4 + 20*(B*c^2*d^2* e^2 - A*c^2*d*e^3 + 2*B*a*c*e^4)*x^3 - 30*(B*c^2*d^3*e - A*c^2*d^2*e^2 + 2 *B*a*c*d*e^3 - 2*A*a*c*e^4)*x^2 + 60*(B*c^2*d^4 - A*c^2*d^3*e + 2*B*a*c*d^ 2*e^2 - 2*A*a*c*d*e^3 + B*a^2*e^4)*x)/e^5 - (B*c^2*d^5 - A*c^2*d^4*e + 2*B *a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)*log(e*x + d)/e^6
Time = 0.26 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.53 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{d+e x} \, dx=\frac {12 \, B c^{2} e^{4} x^{5} - 15 \, B c^{2} d e^{3} x^{4} + 15 \, A c^{2} e^{4} x^{4} + 20 \, B c^{2} d^{2} e^{2} x^{3} - 20 \, A c^{2} d e^{3} x^{3} + 40 \, B a c e^{4} x^{3} - 30 \, B c^{2} d^{3} e x^{2} + 30 \, A c^{2} d^{2} e^{2} x^{2} - 60 \, B a c d e^{3} x^{2} + 60 \, A a c e^{4} x^{2} + 60 \, B c^{2} d^{4} x - 60 \, A c^{2} d^{3} e x + 120 \, B a c d^{2} e^{2} x - 120 \, A a c d e^{3} x + 60 \, B a^{2} e^{4} x}{60 \, e^{5}} - \frac {{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} \]
1/60*(12*B*c^2*e^4*x^5 - 15*B*c^2*d*e^3*x^4 + 15*A*c^2*e^4*x^4 + 20*B*c^2* d^2*e^2*x^3 - 20*A*c^2*d*e^3*x^3 + 40*B*a*c*e^4*x^3 - 30*B*c^2*d^3*e*x^2 + 30*A*c^2*d^2*e^2*x^2 - 60*B*a*c*d*e^3*x^2 + 60*A*a*c*e^4*x^2 + 60*B*c^2*d ^4*x - 60*A*c^2*d^3*e*x + 120*B*a*c*d^2*e^2*x - 120*A*a*c*d*e^3*x + 60*B*a ^2*e^4*x)/e^5 - (B*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e ^3 + B*a^2*d*e^4 - A*a^2*e^5)*log(abs(e*x + d))/e^6
Time = 0.07 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{d+e x} \, dx=x\,\left (\frac {B\,a^2}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^2}{e}-\frac {B\,c^2\,d}{e^2}\right )}{e}-\frac {2\,B\,a\,c}{e}\right )}{e}+\frac {2\,A\,a\,c}{e}\right )}{e}\right )+x^4\,\left (\frac {A\,c^2}{4\,e}-\frac {B\,c^2\,d}{4\,e^2}\right )-x^3\,\left (\frac {d\,\left (\frac {A\,c^2}{e}-\frac {B\,c^2\,d}{e^2}\right )}{3\,e}-\frac {2\,B\,a\,c}{3\,e}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^2}{e}-\frac {B\,c^2\,d}{e^2}\right )}{e}-\frac {2\,B\,a\,c}{e}\right )}{2\,e}+\frac {A\,a\,c}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^2\,d\,e^4+A\,a^2\,e^5-2\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3-B\,c^2\,d^5+A\,c^2\,d^4\,e\right )}{e^6}+\frac {B\,c^2\,x^5}{5\,e} \]
x*((B*a^2)/e - (d*((d*((d*((A*c^2)/e - (B*c^2*d)/e^2))/e - (2*B*a*c)/e))/e + (2*A*a*c)/e))/e) + x^4*((A*c^2)/(4*e) - (B*c^2*d)/(4*e^2)) - x^3*((d*(( A*c^2)/e - (B*c^2*d)/e^2))/(3*e) - (2*B*a*c)/(3*e)) + x^2*((d*((d*((A*c^2) /e - (B*c^2*d)/e^2))/e - (2*B*a*c)/e))/(2*e) + (A*a*c)/e) + (log(d + e*x)* (A*a^2*e^5 - B*c^2*d^5 - B*a^2*d*e^4 + A*c^2*d^4*e + 2*A*a*c*d^2*e^3 - 2*B *a*c*d^3*e^2))/e^6 + (B*c^2*x^5)/(5*e)