Integrand size = 22, antiderivative size = 286 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {\left (7 c^2 d^6+10 b c d^4 e^2+3 b^2 d^2 e^4+6 a c d^2 e^4+2 a b e^6\right ) x}{e^8}-\frac {d \left (3 c^2 d^4+4 b c d^2 e^2+b^2 e^4+2 a c e^4\right ) x^2}{e^7}+\frac {\left (5 c^2 d^4+6 b c d^2 e^2+b^2 e^4+2 a c e^4\right ) x^3}{3 e^6}-\frac {c d \left (c d^2+b e^2\right ) x^4}{e^5}+\frac {c \left (3 c d^2+2 b e^2\right ) x^5}{5 e^4}-\frac {c^2 d x^6}{3 e^3}+\frac {c^2 x^7}{7 e^2}-\frac {\left (c d^4+b d^2 e^2+a e^4\right )^2}{e^9 (d+e x)}-\frac {4 d \left (2 c d^2+b e^2\right ) \left (c d^4+b d^2 e^2+a e^4\right ) \log (d+e x)}{e^9} \] Output:
(2*a*b*e^6+6*a*c*d^2*e^4+3*b^2*d^2*e^4+10*b*c*d^4*e^2+7*c^2*d^6)*x/e^8-d*( 2*a*c*e^4+b^2*e^4+4*b*c*d^2*e^2+3*c^2*d^4)*x^2/e^7+1/3*(2*a*c*e^4+b^2*e^4+ 6*b*c*d^2*e^2+5*c^2*d^4)*x^3/e^6-c*d*(b*e^2+c*d^2)*x^4/e^5+1/5*c*(2*b*e^2+ 3*c*d^2)*x^5/e^4-1/3*c^2*d*x^6/e^3+1/7*c^2*x^7/e^2-(a*e^4+b*d^2*e^2+c*d^4) ^2/e^9/(e*x+d)-4*d*(b*e^2+2*c*d^2)*(a*e^4+b*d^2*e^2+c*d^4)*ln(e*x+d)/e^9
Time = 0.19 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {\left (7 c^2 d^6+10 b c d^4 e^2+3 b^2 d^2 e^4+6 a c d^2 e^4+2 a b e^6\right ) x}{e^8}-\frac {d \left (3 c^2 d^4+4 b c d^2 e^2+b^2 e^4+2 a c e^4\right ) x^2}{e^7}+\frac {\left (5 c^2 d^4+6 b c d^2 e^2+b^2 e^4+2 a c e^4\right ) x^3}{3 e^6}-\frac {c d \left (c d^2+b e^2\right ) x^4}{e^5}+\frac {c \left (3 c d^2+2 b e^2\right ) x^5}{5 e^4}-\frac {c^2 d x^6}{3 e^3}+\frac {c^2 x^7}{7 e^2}-\frac {\left (c d^4+b d^2 e^2+a e^4\right )^2}{e^9 (d+e x)}-\frac {4 d \left (2 c d^2+b e^2\right ) \left (c d^4+b d^2 e^2+a e^4\right ) \log (d+e x)}{e^9} \] Input:
Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x)^2,x]
Output:
((7*c^2*d^6 + 10*b*c*d^4*e^2 + 3*b^2*d^2*e^4 + 6*a*c*d^2*e^4 + 2*a*b*e^6)* x)/e^8 - (d*(3*c^2*d^4 + 4*b*c*d^2*e^2 + b^2*e^4 + 2*a*c*e^4)*x^2)/e^7 + ( (5*c^2*d^4 + 6*b*c*d^2*e^2 + b^2*e^4 + 2*a*c*e^4)*x^3)/(3*e^6) - (c*d*(c*d ^2 + b*e^2)*x^4)/e^5 + (c*(3*c*d^2 + 2*b*e^2)*x^5)/(5*e^4) - (c^2*d*x^6)/( 3*e^3) + (c^2*x^7)/(7*e^2) - (c*d^4 + b*d^2*e^2 + a*e^4)^2/(e^9*(d + e*x)) - (4*d*(2*c*d^2 + b*e^2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Log[d + e*x])/e^9
Time = 0.89 (sec) , antiderivative size = 286, 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 = {2389, 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.
| \(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (-\frac {2 d x \left (2 a c e^4+b^2 e^4+4 b c d^2 e^2+3 c^2 d^4\right )}{e^7}+\frac {x^2 \left (2 a c e^4+b^2 e^4+6 b c d^2 e^2+5 c^2 d^4\right )}{e^6}+\frac {2 a b e^6+6 a c d^2 e^4+3 b^2 d^2 e^4+10 b c d^4 e^2+7 c^2 d^6}{e^8}-\frac {4 d \left (b e^2+2 c d^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^8 (d+e x)}+\frac {\left (a e^4+b d^2 e^2+c d^4\right )^2}{e^8 (d+e x)^2}-\frac {4 c d x^3 \left (b e^2+c d^2\right )}{e^5}+\frac {c x^4 \left (2 b e^2+3 c d^2\right )}{e^4}-\frac {2 c^2 d x^5}{e^3}+\frac {c^2 x^6}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d x^2 \left (2 a c e^4+b^2 e^4+4 b c d^2 e^2+3 c^2 d^4\right )}{e^7}+\frac {x^3 \left (2 a c e^4+b^2 e^4+6 b c d^2 e^2+5 c^2 d^4\right )}{3 e^6}+\frac {x \left (2 a b e^6+6 a c d^2 e^4+3 b^2 d^2 e^4+10 b c d^4 e^2+7 c^2 d^6\right )}{e^8}-\frac {\left (a e^4+b d^2 e^2+c d^4\right )^2}{e^9 (d+e x)}-\frac {4 d \left (b e^2+2 c d^2\right ) \log (d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}{e^9}-\frac {c d x^4 \left (b e^2+c d^2\right )}{e^5}+\frac {c x^5 \left (2 b e^2+3 c d^2\right )}{5 e^4}-\frac {c^2 d x^6}{3 e^3}+\frac {c^2 x^7}{7 e^2}\) |
Input:
Int[(a + b*x^2 + c*x^4)^2/(d + e*x)^2,x]
Output:
((7*c^2*d^6 + 10*b*c*d^4*e^2 + 3*b^2*d^2*e^4 + 6*a*c*d^2*e^4 + 2*a*b*e^6)* x)/e^8 - (d*(3*c^2*d^4 + 4*b*c*d^2*e^2 + b^2*e^4 + 2*a*c*e^4)*x^2)/e^7 + ( (5*c^2*d^4 + 6*b*c*d^2*e^2 + b^2*e^4 + 2*a*c*e^4)*x^3)/(3*e^6) - (c*d*(c*d ^2 + b*e^2)*x^4)/e^5 + (c*(3*c*d^2 + 2*b*e^2)*x^5)/(5*e^4) - (c^2*d*x^6)/( 3*e^3) + (c^2*x^7)/(7*e^2) - (c*d^4 + b*d^2*e^2 + a*e^4)^2/(e^9*(d + e*x)) - (4*d*(2*c*d^2 + b*e^2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Log[d + e*x])/e^9
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.16 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.17
| method | result | size |
| norman | \(\frac {-\frac {a^{2} e^{8}+4 a b \,d^{2} e^{6}+8 a c \,d^{4} e^{4}+4 b^{2} d^{4} e^{4}+12 b c \,d^{6} e^{2}+8 d^{8} c^{2}}{e^{9}}+\frac {c^{2} x^{8}}{7 e}+\frac {2 \left (a b \,e^{6}+2 a c \,d^{2} e^{4}+b^{2} d^{2} e^{4}+3 b c \,d^{4} e^{2}+2 c^{2} d^{6}\right ) x^{2}}{e^{7}}+\frac {\left (2 a c \,e^{4}+b^{2} e^{4}+3 b c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) x^{4}}{3 e^{5}}+\frac {2 c \left (3 b \,e^{2}+2 c \,d^{2}\right ) x^{6}}{15 e^{3}}-\frac {4 c^{2} d \,x^{7}}{21 e^{2}}-\frac {2 d \left (2 a c \,e^{4}+b^{2} e^{4}+3 b c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) x^{3}}{3 e^{6}}-\frac {d c \left (3 b \,e^{2}+2 c \,d^{2}\right ) x^{5}}{5 e^{4}}}{e x +d}-\frac {4 d \left (a b \,e^{6}+2 a c \,d^{2} e^{4}+b^{2} d^{2} e^{4}+3 b c \,d^{4} e^{2}+2 c^{2} d^{6}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(334\) |
| default | \(\frac {\frac {1}{7} c^{2} x^{7} e^{6}-\frac {1}{3} c^{2} d \,x^{6} e^{5}+\frac {2}{5} b c \,e^{6} x^{5}+\frac {3}{5} c^{2} d^{2} e^{4} x^{5}-b c d \,e^{5} x^{4}-c^{2} d^{3} e^{3} x^{4}+\frac {2}{3} a c \,e^{6} x^{3}+\frac {1}{3} b^{2} e^{6} x^{3}+2 b c \,d^{2} e^{4} x^{3}+\frac {5}{3} c^{2} d^{4} e^{2} x^{3}-2 a c d \,e^{5} x^{2}-b^{2} d \,e^{5} x^{2}-4 b c \,d^{3} e^{3} x^{2}-3 c^{2} d^{5} e \,x^{2}+2 x a b \,e^{6}+6 x a c \,d^{2} e^{4}+3 x \,b^{2} d^{2} e^{4}+10 x b c \,d^{4} e^{2}+7 x \,c^{2} d^{6}}{e^{8}}-\frac {a^{2} e^{8}+2 a b \,d^{2} e^{6}+2 a c \,d^{4} e^{4}+b^{2} d^{4} e^{4}+2 b c \,d^{6} e^{2}+d^{8} c^{2}}{e^{9} \left (e x +d \right )}-\frac {4 d \left (a b \,e^{6}+2 a c \,d^{2} e^{4}+b^{2} d^{2} e^{4}+3 b c \,d^{4} e^{2}+2 c^{2} d^{6}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(350\) |
| risch | \(-\frac {4 d \ln \left (e x +d \right ) a b}{e^{3}}-\frac {d^{8} c^{2}}{e^{9} \left (e x +d \right )}-\frac {4 d^{3} \ln \left (e x +d \right ) b^{2}}{e^{5}}-\frac {8 d^{7} \ln \left (e x +d \right ) c^{2}}{e^{9}}+\frac {2 b c \,x^{5}}{5 e^{2}}+\frac {2 a c \,x^{3}}{3 e^{2}}+\frac {5 c^{2} d^{4} x^{3}}{3 e^{6}}-\frac {b^{2} d \,x^{2}}{e^{3}}-\frac {3 c^{2} d^{5} x^{2}}{e^{7}}+\frac {2 x a b}{e^{2}}+\frac {3 x \,b^{2} d^{2}}{e^{4}}+\frac {7 x \,c^{2} d^{6}}{e^{8}}-\frac {b^{2} d^{4}}{e^{5} \left (e x +d \right )}+\frac {3 c^{2} d^{2} x^{5}}{5 e^{4}}-\frac {c^{2} d \,x^{6}}{3 e^{3}}-\frac {c^{2} d^{3} x^{4}}{e^{5}}-\frac {8 d^{3} \ln \left (e x +d \right ) a c}{e^{5}}-\frac {12 d^{5} \ln \left (e x +d \right ) b c}{e^{7}}-\frac {2 a c d \,x^{2}}{e^{3}}-\frac {b c d \,x^{4}}{e^{3}}+\frac {2 b c \,d^{2} x^{3}}{e^{4}}-\frac {4 b c \,d^{3} x^{2}}{e^{5}}+\frac {6 x a c \,d^{2}}{e^{4}}+\frac {10 x b c \,d^{4}}{e^{6}}-\frac {2 a b \,d^{2}}{e^{3} \left (e x +d \right )}-\frac {2 a c \,d^{4}}{e^{5} \left (e x +d \right )}-\frac {2 b c \,d^{6}}{e^{7} \left (e x +d \right )}+\frac {b^{2} x^{3}}{3 e^{2}}+\frac {c^{2} x^{7}}{7 e^{2}}-\frac {a^{2}}{e \left (e x +d \right )}\) | \(408\) |
| parallelrisch | \(-\frac {-42 x^{6} b c \,e^{8}+70 x^{3} b^{2} d \,e^{7}-210 x^{2} a b \,e^{8}-210 x^{2} b^{2} d^{2} e^{6}+420 \ln \left (e x +d \right ) b^{2} d^{4} e^{4}+420 \ln \left (e x +d \right ) x \,b^{2} d^{3} e^{5}+420 a b \,d^{2} e^{6}+1260 b c \,d^{6} e^{2}+840 a c \,d^{4} e^{4}-35 x^{4} b^{2} e^{8}+840 \ln \left (e x +d \right ) x a c \,d^{3} e^{5}+420 \ln \left (e x +d \right ) x a b d \,e^{7}+840 \ln \left (e x +d \right ) x \,c^{2} d^{7} e +1260 \ln \left (e x +d \right ) x b c \,d^{5} e^{3}+140 x^{3} a c d \,e^{7}-420 x^{2} a c \,d^{2} e^{6}+840 \ln \left (e x +d \right ) a c \,d^{4} e^{4}+20 c^{2} d \,x^{7} e^{7}-28 c^{2} d^{2} x^{6} e^{6}+42 c^{2} d^{3} x^{5} e^{5}-70 x^{4} a c \,e^{8}-70 x^{4} c^{2} d^{4} e^{4}+140 x^{3} c^{2} d^{5} e^{3}-420 x^{2} c^{2} d^{6} e^{2}-105 x^{4} b c \,d^{2} e^{6}+210 x^{3} b c \,d^{3} e^{5}-630 x^{2} b c \,d^{4} e^{4}+420 \ln \left (e x +d \right ) a b \,d^{2} e^{6}+1260 \ln \left (e x +d \right ) b c \,d^{6} e^{2}+420 b^{2} d^{4} e^{4}+840 d^{8} c^{2}+840 \ln \left (e x +d \right ) c^{2} d^{8}+105 a^{2} e^{8}+63 x^{5} b c d \,e^{7}-15 x^{8} c^{2} e^{8}}{105 e^{9} \left (e x +d \right )}\) | \(467\) |
Input:
int((c*x^4+b*x^2+a)^2/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
(-(a^2*e^8+4*a*b*d^2*e^6+8*a*c*d^4*e^4+4*b^2*d^4*e^4+12*b*c*d^6*e^2+8*c^2* d^8)/e^9+1/7*c^2*x^8/e+2/e^7*(a*b*e^6+2*a*c*d^2*e^4+b^2*d^2*e^4+3*b*c*d^4* e^2+2*c^2*d^6)*x^2+1/3*(2*a*c*e^4+b^2*e^4+3*b*c*d^2*e^2+2*c^2*d^4)/e^5*x^4 +2/15*c*(3*b*e^2+2*c*d^2)/e^3*x^6-4/21*c^2*d*x^7/e^2-2/3*d*(2*a*c*e^4+b^2* e^4+3*b*c*d^2*e^2+2*c^2*d^4)/e^6*x^3-1/5*d*c*(3*b*e^2+2*c*d^2)/e^4*x^5)/(e *x+d)-4*d/e^9*(a*b*e^6+2*a*c*d^2*e^4+b^2*d^2*e^4+3*b*c*d^4*e^2+2*c^2*d^6)* ln(e*x+d)
Time = 0.07 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {15 \, c^{2} e^{8} x^{8} - 20 \, c^{2} d e^{7} x^{7} - 105 \, c^{2} d^{8} - 210 \, b c d^{6} e^{2} - 210 \, a b d^{2} e^{6} - 105 \, a^{2} e^{8} - 105 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{4} + 14 \, {\left (2 \, c^{2} d^{2} e^{6} + 3 \, b c e^{8}\right )} x^{6} - 21 \, {\left (2 \, c^{2} d^{3} e^{5} + 3 \, b c d e^{7}\right )} x^{5} + 35 \, {\left (2 \, c^{2} d^{4} e^{4} + 3 \, b c d^{2} e^{6} + {\left (b^{2} + 2 \, a c\right )} e^{8}\right )} x^{4} - 70 \, {\left (2 \, c^{2} d^{5} e^{3} + 3 \, b c d^{3} e^{5} + {\left (b^{2} + 2 \, a c\right )} d e^{7}\right )} x^{3} + 210 \, {\left (2 \, c^{2} d^{6} e^{2} + 3 \, b c d^{4} e^{4} + a b e^{8} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{6}\right )} x^{2} + 105 \, {\left (7 \, c^{2} d^{7} e + 10 \, b c d^{5} e^{3} + 2 \, a b d e^{7} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{5}\right )} x - 420 \, {\left (2 \, c^{2} d^{8} + 3 \, b c d^{6} e^{2} + a b d^{2} e^{6} + {\left (b^{2} + 2 \, a c\right )} d^{4} e^{4} + {\left (2 \, c^{2} d^{7} e + 3 \, b c d^{5} e^{3} + a b d e^{7} + {\left (b^{2} + 2 \, a c\right )} d^{3} e^{5}\right )} x\right )} \log \left (e x + d\right )}{105 \, {\left (e^{10} x + d e^{9}\right )}} \] Input:
integrate((c*x^4+b*x^2+a)^2/(e*x+d)^2,x, algorithm="fricas")
Output:
1/105*(15*c^2*e^8*x^8 - 20*c^2*d*e^7*x^7 - 105*c^2*d^8 - 210*b*c*d^6*e^2 - 210*a*b*d^2*e^6 - 105*a^2*e^8 - 105*(b^2 + 2*a*c)*d^4*e^4 + 14*(2*c^2*d^2 *e^6 + 3*b*c*e^8)*x^6 - 21*(2*c^2*d^3*e^5 + 3*b*c*d*e^7)*x^5 + 35*(2*c^2*d ^4*e^4 + 3*b*c*d^2*e^6 + (b^2 + 2*a*c)*e^8)*x^4 - 70*(2*c^2*d^5*e^3 + 3*b* c*d^3*e^5 + (b^2 + 2*a*c)*d*e^7)*x^3 + 210*(2*c^2*d^6*e^2 + 3*b*c*d^4*e^4 + a*b*e^8 + (b^2 + 2*a*c)*d^2*e^6)*x^2 + 105*(7*c^2*d^7*e + 10*b*c*d^5*e^3 + 2*a*b*d*e^7 + 3*(b^2 + 2*a*c)*d^3*e^5)*x - 420*(2*c^2*d^8 + 3*b*c*d^6*e ^2 + a*b*d^2*e^6 + (b^2 + 2*a*c)*d^4*e^4 + (2*c^2*d^7*e + 3*b*c*d^5*e^3 + a*b*d*e^7 + (b^2 + 2*a*c)*d^3*e^5)*x)*log(e*x + d))/(e^10*x + d*e^9)
Time = 0.75 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx=- \frac {c^{2} d x^{6}}{3 e^{3}} + \frac {c^{2} x^{7}}{7 e^{2}} - \frac {4 d \left (b e^{2} + 2 c d^{2}\right ) \left (a e^{4} + b d^{2} e^{2} + c d^{4}\right ) \log {\left (d + e x \right )}}{e^{9}} + x^{5} \cdot \left (\frac {2 b c}{5 e^{2}} + \frac {3 c^{2} d^{2}}{5 e^{4}}\right ) + x^{4} \left (- \frac {b c d}{e^{3}} - \frac {c^{2} d^{3}}{e^{5}}\right ) + x^{3} \cdot \left (\frac {2 a c}{3 e^{2}} + \frac {b^{2}}{3 e^{2}} + \frac {2 b c d^{2}}{e^{4}} + \frac {5 c^{2} d^{4}}{3 e^{6}}\right ) + x^{2} \left (- \frac {2 a c d}{e^{3}} - \frac {b^{2} d}{e^{3}} - \frac {4 b c d^{3}}{e^{5}} - \frac {3 c^{2} d^{5}}{e^{7}}\right ) + x \left (\frac {2 a b}{e^{2}} + \frac {6 a c d^{2}}{e^{4}} + \frac {3 b^{2} d^{2}}{e^{4}} + \frac {10 b c d^{4}}{e^{6}} + \frac {7 c^{2} d^{6}}{e^{8}}\right ) + \frac {- a^{2} e^{8} - 2 a b d^{2} e^{6} - 2 a c d^{4} e^{4} - b^{2} d^{4} e^{4} - 2 b c d^{6} e^{2} - c^{2} d^{8}}{d e^{9} + e^{10} x} \] Input:
integrate((c*x**4+b*x**2+a)**2/(e*x+d)**2,x)
Output:
-c**2*d*x**6/(3*e**3) + c**2*x**7/(7*e**2) - 4*d*(b*e**2 + 2*c*d**2)*(a*e* *4 + b*d**2*e**2 + c*d**4)*log(d + e*x)/e**9 + x**5*(2*b*c/(5*e**2) + 3*c* *2*d**2/(5*e**4)) + x**4*(-b*c*d/e**3 - c**2*d**3/e**5) + x**3*(2*a*c/(3*e **2) + b**2/(3*e**2) + 2*b*c*d**2/e**4 + 5*c**2*d**4/(3*e**6)) + x**2*(-2* a*c*d/e**3 - b**2*d/e**3 - 4*b*c*d**3/e**5 - 3*c**2*d**5/e**7) + x*(2*a*b/ e**2 + 6*a*c*d**2/e**4 + 3*b**2*d**2/e**4 + 10*b*c*d**4/e**6 + 7*c**2*d**6 /e**8) + (-a**2*e**8 - 2*a*b*d**2*e**6 - 2*a*c*d**4*e**4 - b**2*d**4*e**4 - 2*b*c*d**6*e**2 - c**2*d**8)/(d*e**9 + e**10*x)
Time = 0.03 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx=-\frac {c^{2} d^{8} + 2 \, b c d^{6} e^{2} + 2 \, a b d^{2} e^{6} + a^{2} e^{8} + {\left (b^{2} + 2 \, a c\right )} d^{4} e^{4}}{e^{10} x + d e^{9}} + \frac {15 \, c^{2} e^{6} x^{7} - 35 \, c^{2} d e^{5} x^{6} + 21 \, {\left (3 \, c^{2} d^{2} e^{4} + 2 \, b c e^{6}\right )} x^{5} - 105 \, {\left (c^{2} d^{3} e^{3} + b c d e^{5}\right )} x^{4} + 35 \, {\left (5 \, c^{2} d^{4} e^{2} + 6 \, b c d^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} e^{6}\right )} x^{3} - 105 \, {\left (3 \, c^{2} d^{5} e + 4 \, b c d^{3} e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{5}\right )} x^{2} + 105 \, {\left (7 \, c^{2} d^{6} + 10 \, b c d^{4} e^{2} + 2 \, a b e^{6} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x}{105 \, e^{8}} - \frac {4 \, {\left (2 \, c^{2} d^{7} + 3 \, b c d^{5} e^{2} + a b d e^{6} + {\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} \log \left (e x + d\right )}{e^{9}} \] Input:
integrate((c*x^4+b*x^2+a)^2/(e*x+d)^2,x, algorithm="maxima")
Output:
-(c^2*d^8 + 2*b*c*d^6*e^2 + 2*a*b*d^2*e^6 + a^2*e^8 + (b^2 + 2*a*c)*d^4*e^ 4)/(e^10*x + d*e^9) + 1/105*(15*c^2*e^6*x^7 - 35*c^2*d*e^5*x^6 + 21*(3*c^2 *d^2*e^4 + 2*b*c*e^6)*x^5 - 105*(c^2*d^3*e^3 + b*c*d*e^5)*x^4 + 35*(5*c^2* d^4*e^2 + 6*b*c*d^2*e^4 + (b^2 + 2*a*c)*e^6)*x^3 - 105*(3*c^2*d^5*e + 4*b* c*d^3*e^3 + (b^2 + 2*a*c)*d*e^5)*x^2 + 105*(7*c^2*d^6 + 10*b*c*d^4*e^2 + 2 *a*b*e^6 + 3*(b^2 + 2*a*c)*d^2*e^4)*x)/e^8 - 4*(2*c^2*d^7 + 3*b*c*d^5*e^2 + a*b*d*e^6 + (b^2 + 2*a*c)*d^3*e^4)*log(e*x + d)/e^9
Time = 0.12 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {{\left (15 \, c^{2} - \frac {140 \, c^{2} d}{e x + d} + \frac {42 \, {\left (14 \, c^{2} d^{2} e^{2} + b c e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {105 \, {\left (14 \, c^{2} d^{3} e^{3} + 3 \, b c d e^{5}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {35 \, {\left (70 \, c^{2} d^{4} e^{4} + 30 \, b c d^{2} e^{6} + b^{2} e^{8} + 2 \, a c e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}} - \frac {210 \, {\left (14 \, c^{2} d^{5} e^{5} + 10 \, b c d^{3} e^{7} + b^{2} d e^{9} + 2 \, a c d e^{9}\right )}}{{\left (e x + d\right )}^{5} e^{5}} + \frac {210 \, {\left (14 \, c^{2} d^{6} e^{6} + 15 \, b c d^{4} e^{8} + 3 \, b^{2} d^{2} e^{10} + 6 \, a c d^{2} e^{10} + a b e^{12}\right )}}{{\left (e x + d\right )}^{6} e^{6}}\right )} {\left (e x + d\right )}^{7}}{105 \, e^{9}} + \frac {4 \, {\left (2 \, c^{2} d^{7} + 3 \, b c d^{5} e^{2} + b^{2} d^{3} e^{4} + 2 \, a c d^{3} e^{4} + a b d e^{6}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{9}} - \frac {\frac {c^{2} d^{8} e^{7}}{e x + d} + \frac {2 \, b c d^{6} e^{9}}{e x + d} + \frac {b^{2} d^{4} e^{11}}{e x + d} + \frac {2 \, a c d^{4} e^{11}}{e x + d} + \frac {2 \, a b d^{2} e^{13}}{e x + d} + \frac {a^{2} e^{15}}{e x + d}}{e^{16}} \] Input:
integrate((c*x^4+b*x^2+a)^2/(e*x+d)^2,x, algorithm="giac")
Output:
1/105*(15*c^2 - 140*c^2*d/(e*x + d) + 42*(14*c^2*d^2*e^2 + b*c*e^4)/((e*x + d)^2*e^2) - 105*(14*c^2*d^3*e^3 + 3*b*c*d*e^5)/((e*x + d)^3*e^3) + 35*(7 0*c^2*d^4*e^4 + 30*b*c*d^2*e^6 + b^2*e^8 + 2*a*c*e^8)/((e*x + d)^4*e^4) - 210*(14*c^2*d^5*e^5 + 10*b*c*d^3*e^7 + b^2*d*e^9 + 2*a*c*d*e^9)/((e*x + d) ^5*e^5) + 210*(14*c^2*d^6*e^6 + 15*b*c*d^4*e^8 + 3*b^2*d^2*e^10 + 6*a*c*d^ 2*e^10 + a*b*e^12)/((e*x + d)^6*e^6))*(e*x + d)^7/e^9 + 4*(2*c^2*d^7 + 3*b *c*d^5*e^2 + b^2*d^3*e^4 + 2*a*c*d^3*e^4 + a*b*d*e^6)*log(abs(e*x + d)/((e *x + d)^2*abs(e)))/e^9 - (c^2*d^8*e^7/(e*x + d) + 2*b*c*d^6*e^9/(e*x + d) + b^2*d^4*e^11/(e*x + d) + 2*a*c*d^4*e^11/(e*x + d) + 2*a*b*d^2*e^13/(e*x + d) + a^2*e^15/(e*x + d))/e^16
Time = 0.11 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.41 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx=x^5\,\left (\frac {3\,c^2\,d^2}{5\,e^4}+\frac {2\,b\,c}{5\,e^2}\right )-x^2\,\left (\frac {d\,\left (\frac {b^2+2\,a\,c}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e}-\frac {2\,c^2\,d^3}{e^5}\right )}{e}-\frac {d^2\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {2\,d\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e}-\frac {2\,c^2\,d^3}{e^5}\right )}{2\,e^2}\right )+x\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {b^2+2\,a\,c}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e}-\frac {2\,c^2\,d^3}{e^5}\right )}{e}-\frac {d^2\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {2\,d\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e}-\frac {2\,c^2\,d^3}{e^5}\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {b^2+2\,a\,c}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e}-\frac {2\,c^2\,d^3}{e^5}\right )}{e}-\frac {d^2\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e^2}\right )}{e^2}+\frac {2\,a\,b}{e^2}\right )-x^4\,\left (\frac {d\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{2\,e}-\frac {c^2\,d^3}{2\,e^5}\right )+x^3\,\left (\frac {b^2+2\,a\,c}{3\,e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{e}-\frac {2\,c^2\,d^3}{e^5}\right )}{3\,e}-\frac {d^2\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,b\,c}{e^2}\right )}{3\,e^2}\right )-\frac {\ln \left (d+e\,x\right )\,\left (4\,b^2\,d^3\,e^4+12\,b\,c\,d^5\,e^2+4\,a\,b\,d\,e^6+8\,c^2\,d^7+8\,a\,c\,d^3\,e^4\right )}{e^9}+\frac {c^2\,x^7}{7\,e^2}-\frac {a^2\,e^8+2\,a\,b\,d^2\,e^6+2\,a\,c\,d^4\,e^4+b^2\,d^4\,e^4+2\,b\,c\,d^6\,e^2+c^2\,d^8}{e\,\left (x\,e^9+d\,e^8\right )}-\frac {c^2\,d\,x^6}{3\,e^3} \] Input:
int((a + b*x^2 + c*x^4)^2/(d + e*x)^2,x)
Output:
x^5*((3*c^2*d^2)/(5*e^4) + (2*b*c)/(5*e^2)) - x^2*((d*((2*a*c + b^2)/e^2 + (2*d*((2*d*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/e - (2*c^2*d^3)/e^5))/e - (d^ 2*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/e^2))/e - (d^2*((2*d*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/e - (2*c^2*d^3)/e^5))/(2*e^2)) + x*((2*d*((2*d*((2*a*c + b^ 2)/e^2 + (2*d*((2*d*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/e - (2*c^2*d^3)/e^5)) /e - (d^2*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/e^2))/e - (d^2*((2*d*((3*c^2*d^ 2)/e^4 + (2*b*c)/e^2))/e - (2*c^2*d^3)/e^5))/e^2))/e - (d^2*((2*a*c + b^2) /e^2 + (2*d*((2*d*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/e - (2*c^2*d^3)/e^5))/e - (d^2*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/e^2))/e^2 + (2*a*b)/e^2) - x^4*(( d*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/(2*e) - (c^2*d^3)/(2*e^5)) + x^3*((2*a* c + b^2)/(3*e^2) + (2*d*((2*d*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/e - (2*c^2* d^3)/e^5))/(3*e) - (d^2*((3*c^2*d^2)/e^4 + (2*b*c)/e^2))/(3*e^2)) - (log(d + e*x)*(8*c^2*d^7 + 4*b^2*d^3*e^4 + 4*a*b*d*e^6 + 8*a*c*d^3*e^4 + 12*b*c* d^5*e^2))/e^9 + (c^2*x^7)/(7*e^2) - (a^2*e^8 + c^2*d^8 + b^2*d^4*e^4 + 2*a *b*d^2*e^6 + 2*a*c*d^4*e^4 + 2*b*c*d^6*e^2)/(e*(d*e^8 + e^9*x)) - (c^2*d*x ^6)/(3*e^3)
Time = 0.24 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {-420 \,\mathrm {log}\left (e x +d \right ) b^{2} d^{5} e^{4}+420 b^{2} d^{4} e^{5} x +210 b^{2} d^{3} e^{6} x^{2}-70 b^{2} d^{2} e^{7} x^{3}+35 b^{2} d \,e^{8} x^{4}+105 b c \,d^{3} e^{6} x^{4}-63 b c \,d^{2} e^{7} x^{5}+42 b c d \,e^{8} x^{6}-420 \,\mathrm {log}\left (e x +d \right ) a b \,d^{2} e^{7} x -1260 \,\mathrm {log}\left (e x +d \right ) b c \,d^{6} e^{3} x -840 \,\mathrm {log}\left (e x +d \right ) a c \,d^{4} e^{5} x -840 \,\mathrm {log}\left (e x +d \right ) c^{2} d^{9}+105 a^{2} e^{9} x -420 \,\mathrm {log}\left (e x +d \right ) a b \,d^{3} e^{6}-420 \,\mathrm {log}\left (e x +d \right ) b^{2} d^{4} e^{5} x -1260 \,\mathrm {log}\left (e x +d \right ) b c \,d^{7} e^{2}+420 a b \,d^{2} e^{7} x +210 a b d \,e^{8} x^{2}+1260 b c \,d^{6} e^{3} x +630 b c \,d^{5} e^{4} x^{2}-210 b c \,d^{4} e^{5} x^{3}+840 c^{2} d^{8} e x +420 c^{2} d^{7} e^{2} x^{2}-140 c^{2} d^{6} e^{3} x^{3}+70 c^{2} d^{5} e^{4} x^{4}-42 c^{2} d^{4} e^{5} x^{5}+28 c^{2} d^{3} e^{6} x^{6}-20 c^{2} d^{2} e^{7} x^{7}+15 c^{2} d \,e^{8} x^{8}-840 \,\mathrm {log}\left (e x +d \right ) a c \,d^{5} e^{4}-840 \,\mathrm {log}\left (e x +d \right ) c^{2} d^{8} e x +840 a c \,d^{4} e^{5} x +420 a c \,d^{3} e^{6} x^{2}-140 a c \,d^{2} e^{7} x^{3}+70 a c d \,e^{8} x^{4}}{105 d \,e^{9} \left (e x +d \right )} \] Input:
int((c*x^4+b*x^2+a)^2/(e*x+d)^2,x)
Output:
( - 420*log(d + e*x)*a*b*d**3*e**6 - 420*log(d + e*x)*a*b*d**2*e**7*x - 84 0*log(d + e*x)*a*c*d**5*e**4 - 840*log(d + e*x)*a*c*d**4*e**5*x - 420*log( d + e*x)*b**2*d**5*e**4 - 420*log(d + e*x)*b**2*d**4*e**5*x - 1260*log(d + e*x)*b*c*d**7*e**2 - 1260*log(d + e*x)*b*c*d**6*e**3*x - 840*log(d + e*x) *c**2*d**9 - 840*log(d + e*x)*c**2*d**8*e*x + 105*a**2*e**9*x + 420*a*b*d* *2*e**7*x + 210*a*b*d*e**8*x**2 + 840*a*c*d**4*e**5*x + 420*a*c*d**3*e**6* x**2 - 140*a*c*d**2*e**7*x**3 + 70*a*c*d*e**8*x**4 + 420*b**2*d**4*e**5*x + 210*b**2*d**3*e**6*x**2 - 70*b**2*d**2*e**7*x**3 + 35*b**2*d*e**8*x**4 + 1260*b*c*d**6*e**3*x + 630*b*c*d**5*e**4*x**2 - 210*b*c*d**4*e**5*x**3 + 105*b*c*d**3*e**6*x**4 - 63*b*c*d**2*e**7*x**5 + 42*b*c*d*e**8*x**6 + 840* c**2*d**8*e*x + 420*c**2*d**7*e**2*x**2 - 140*c**2*d**6*e**3*x**3 + 70*c** 2*d**5*e**4*x**4 - 42*c**2*d**4*e**5*x**5 + 28*c**2*d**3*e**6*x**6 - 20*c* *2*d**2*e**7*x**7 + 15*c**2*d*e**8*x**8)/(105*d*e**9*(d + e*x))