Integrand size = 22, antiderivative size = 270 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx=-\frac {d \left (c d^2+b e^2\right ) \left (c d^4+b d^2 e^2+2 a e^4\right ) x}{e^8}+\frac {\left (c d^2+b e^2\right ) \left (c d^4+b d^2 e^2+2 a e^4\right ) x^2}{2 e^7}-\frac {d \left (c^2 d^4+b^2 e^4+2 c \left (b d^2 e^2+a e^4\right )\right ) x^3}{3 e^6}+\frac {\left (c^2 d^4+b^2 e^4+2 c \left (b d^2 e^2+a e^4\right )\right ) x^4}{4 e^5}-\frac {c d \left (c d^2+2 b e^2\right ) x^5}{5 e^4}+\frac {c \left (c d^2+2 b e^2\right ) x^6}{6 e^3}-\frac {c^2 d x^7}{7 e^2}+\frac {c^2 x^8}{8 e}+\frac {\left (c d^4+b d^2 e^2+a e^4\right )^2 \log (d+e x)}{e^9} \] Output:
-d*(b*e^2+c*d^2)*(2*a*e^4+b*d^2*e^2+c*d^4)*x/e^8+1/2*(b*e^2+c*d^2)*(2*a*e^ 4+b*d^2*e^2+c*d^4)*x^2/e^7-1/3*d*(c^2*d^4+b^2*e^4+2*c*(a*e^4+b*d^2*e^2))*x ^3/e^6+1/4*(c^2*d^4+b^2*e^4+2*c*(a*e^4+b*d^2*e^2))*x^4/e^5-1/5*c*d*(2*b*e^ 2+c*d^2)*x^5/e^4+1/6*c*(2*b*e^2+c*d^2)*x^6/e^3-1/7*c^2*d*x^7/e^2+1/8*c^2*x ^8/e+(a*e^4+b*d^2*e^2+c*d^4)^2*ln(e*x+d)/e^9
Time = 0.18 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx=\frac {x \left (c^2 \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )+70 b e^4 \left (12 a e^2 (-2 d+e x)+b \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+28 c e^2 \left (5 a e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )\right )}{840 e^8}+\frac {\left (c d^4+b d^2 e^2+a e^4\right )^2 \log (d+e x)}{e^9} \] Input:
Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x),x]
Output:
(x*(c^2*(-840*d^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168* d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7) + 70*b*e^4*(1 2*a*e^2*(-2*d + e*x) + b*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 28*c*e^2*(5*a*e^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b*(- 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))))/(840*e^8) + ((c*d^4 + b*d^2*e^2 + a*e^4)^2*Log[d + e*x])/e^9
Time = 0.90 (sec) , antiderivative size = 270, 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.
| \(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (\frac {d x^2 \left (-2 c \left (a e^4+b d^2 e^2\right )-b^2 e^4-c^2 d^4\right )}{e^6}+\frac {x^3 \left (2 c \left (a e^4+b d^2 e^2\right )+b^2 e^4+c^2 d^4\right )}{e^5}+\frac {\left (a e^4+b d^2 e^2+c d^4\right )^2}{e^8 (d+e x)}-\frac {d \left (b e^2+c d^2\right ) \left (2 a e^4+b d^2 e^2+c d^4\right )}{e^8}+\frac {x \left (b e^2+c d^2\right ) \left (2 a e^4+b d^2 e^2+c d^4\right )}{e^7}-\frac {c d x^4 \left (2 b e^2+c d^2\right )}{e^4}+\frac {c x^5 \left (2 b e^2+c d^2\right )}{e^3}-\frac {c^2 d x^6}{e^2}+\frac {c^2 x^7}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d x^3 \left (2 c \left (a e^4+b d^2 e^2\right )+b^2 e^4+c^2 d^4\right )}{3 e^6}+\frac {x^4 \left (2 c \left (a e^4+b d^2 e^2\right )+b^2 e^4+c^2 d^4\right )}{4 e^5}+\frac {\log (d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^2}{e^9}-\frac {d x \left (b e^2+c d^2\right ) \left (2 a e^4+b d^2 e^2+c d^4\right )}{e^8}+\frac {x^2 \left (b e^2+c d^2\right ) \left (2 a e^4+b d^2 e^2+c d^4\right )}{2 e^7}-\frac {c d x^5 \left (2 b e^2+c d^2\right )}{5 e^4}+\frac {c x^6 \left (2 b e^2+c d^2\right )}{6 e^3}-\frac {c^2 d x^7}{7 e^2}+\frac {c^2 x^8}{8 e}\) |
Input:
Int[(a + b*x^2 + c*x^4)^2/(d + e*x),x]
Output:
-((d*(c*d^2 + b*e^2)*(c*d^4 + b*d^2*e^2 + 2*a*e^4)*x)/e^8) + ((c*d^2 + b*e ^2)*(c*d^4 + b*d^2*e^2 + 2*a*e^4)*x^2)/(2*e^7) - (d*(c^2*d^4 + b^2*e^4 + 2 *c*(b*d^2*e^2 + a*e^4))*x^3)/(3*e^6) + ((c^2*d^4 + b^2*e^4 + 2*c*(b*d^2*e^ 2 + a*e^4))*x^4)/(4*e^5) - (c*d*(c*d^2 + 2*b*e^2)*x^5)/(5*e^4) + (c*(c*d^2 + 2*b*e^2)*x^6)/(6*e^3) - (c^2*d*x^7)/(7*e^2) + (c^2*x^8)/(8*e) + ((c*d^4 + b*d^2*e^2 + a*e^4)^2*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 = 0.26 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.18
| method | result | size |
| norman | \(\frac {c^{2} x^{8}}{8 e}+\frac {\left (2 a b \,e^{6}+2 a c \,d^{2} e^{4}+b^{2} d^{2} e^{4}+2 b c \,d^{4} e^{2}+c^{2} d^{6}\right ) x^{2}}{2 e^{7}}+\frac {\left (2 a c \,e^{4}+b^{2} e^{4}+2 b c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{4}}{4 e^{5}}+\frac {c \left (2 b \,e^{2}+c \,d^{2}\right ) x^{6}}{6 e^{3}}-\frac {c^{2} d \,x^{7}}{7 e^{2}}-\frac {d \left (2 a c \,e^{4}+b^{2} e^{4}+2 b c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{3}}{3 e^{6}}-\frac {d \left (2 a b \,e^{6}+2 a c \,d^{2} e^{4}+b^{2} d^{2} e^{4}+2 b c \,d^{4} e^{2}+c^{2} d^{6}\right ) x}{e^{8}}-\frac {c d \left (2 b \,e^{2}+c \,d^{2}\right ) x^{5}}{5 e^{4}}+\frac {\left (a^{2} e^{8}+2 a b \,d^{2} e^{6}+2 a c \,d^{4} e^{4}+b^{2} d^{4} e^{4}+2 b c \,d^{6} e^{2}+d^{8} c^{2}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(319\) |
| default | \(-\frac {-\frac {c^{2} x^{8} e^{7}}{8}+\frac {c^{2} d \,x^{7} e^{6}}{7}+\frac {\left (-e^{5} \left (b \,e^{2}+c \,d^{2}\right ) c -e^{7} c b \right ) x^{6}}{6}+\frac {\left (d \left (b \,e^{2}+c \,d^{2}\right ) c \,e^{4}+c d \,e^{6} b \right ) x^{5}}{5}+\frac {\left (-e^{5} \left (b \,e^{2}+c \,d^{2}\right ) b -e^{3} c \left (2 e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right )\right ) x^{4}}{4}+\frac {\left (d \left (b \,e^{2}+c \,d^{2}\right ) b \,e^{4}+c d \,e^{2} \left (2 e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right )\right ) x^{3}}{3}-\frac {e \left (b \,e^{2}+c \,d^{2}\right ) \left (2 e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) x^{2}}{2}+x d \left (b \,e^{2}+c \,d^{2}\right ) \left (2 e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right )}{e^{8}}+\frac {\left (a^{2} e^{8}+2 a b \,d^{2} e^{6}+2 a c \,d^{4} e^{4}+b^{2} d^{4} e^{4}+2 b c \,d^{6} e^{2}+d^{8} c^{2}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(325\) |
| risch | \(\frac {x^{4} b^{2}}{4 e}-\frac {b^{2} d^{3} x}{e^{4}}-\frac {c^{2} d^{7} x}{e^{8}}+\frac {\ln \left (e x +d \right ) b^{2} d^{4}}{e^{5}}+\frac {\ln \left (e x +d \right ) d^{8} c^{2}}{e^{9}}-\frac {2 a b d x}{e^{2}}-\frac {2 a c \,d^{3} x}{e^{4}}-\frac {2 b c \,d^{5} x}{e^{6}}+\frac {2 \ln \left (e x +d \right ) a b \,d^{2}}{e^{3}}+\frac {2 \ln \left (e x +d \right ) a c \,d^{4}}{e^{5}}+\frac {2 \ln \left (e x +d \right ) b c \,d^{6}}{e^{7}}-\frac {c^{2} d^{3} x^{5}}{5 e^{4}}+\frac {c^{2} d^{2} x^{6}}{6 e^{3}}-\frac {c^{2} d \,x^{7}}{7 e^{2}}+\frac {x^{6} c b}{3 e}+\frac {x^{4} a c}{2 e}+\frac {x^{4} c^{2} d^{4}}{4 e^{5}}-\frac {x^{3} b^{2} d}{3 e^{2}}-\frac {x^{3} c^{2} d^{5}}{3 e^{6}}+\frac {a b \,x^{2}}{e}+\frac {b^{2} d^{2} x^{2}}{2 e^{3}}+\frac {c^{2} d^{6} x^{2}}{2 e^{7}}+\frac {\ln \left (e x +d \right ) a^{2}}{e}+\frac {c^{2} x^{8}}{8 e}-\frac {2 x^{5} c d b}{5 e^{2}}+\frac {x^{4} b c \,d^{2}}{2 e^{3}}-\frac {2 x^{3} b c \,d^{3}}{3 e^{4}}-\frac {2 x^{3} a c d}{3 e^{2}}+\frac {a c \,d^{2} x^{2}}{e^{3}}+\frac {b c \,d^{4} x^{2}}{e^{5}}\) | \(381\) |
| parallelrisch | \(\frac {280 x^{6} b c \,e^{8}-280 x^{3} b^{2} d \,e^{7}+840 x^{2} a b \,e^{8}+420 x^{2} b^{2} d^{2} e^{6}-840 x \,b^{2} d^{3} e^{5}+840 \ln \left (e x +d \right ) b^{2} d^{4} e^{4}+210 x^{4} b^{2} e^{8}-560 x^{3} a c d \,e^{7}+840 x^{2} a c \,d^{2} e^{6}+1680 \ln \left (e x +d \right ) a c \,d^{4} e^{4}-1680 x a c \,d^{3} e^{5}-120 c^{2} d \,x^{7} e^{7}+140 c^{2} d^{2} x^{6} e^{6}-168 c^{2} d^{3} x^{5} e^{5}+420 x^{4} a c \,e^{8}+210 x^{4} c^{2} d^{4} e^{4}-280 x^{3} c^{2} d^{5} e^{3}+420 x^{2} c^{2} d^{6} e^{2}-840 x \,c^{2} d^{7} e +420 x^{4} b c \,d^{2} e^{6}-560 x^{3} b c \,d^{3} e^{5}+840 x^{2} b c \,d^{4} e^{4}-1680 x a b d \,e^{7}-1680 x b c \,d^{5} e^{3}+1680 \ln \left (e x +d \right ) a b \,d^{2} e^{6}+1680 \ln \left (e x +d \right ) b c \,d^{6} e^{2}+840 \ln \left (e x +d \right ) a^{2} e^{8}+840 \ln \left (e x +d \right ) c^{2} d^{8}-336 x^{5} b c d \,e^{7}+105 x^{8} c^{2} e^{8}}{840 e^{9}}\) | \(387\) |
Input:
int((c*x^4+b*x^2+a)^2/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/8*c^2*x^8/e+1/2/e^7*(2*a*b*e^6+2*a*c*d^2*e^4+b^2*d^2*e^4+2*b*c*d^4*e^2+c ^2*d^6)*x^2+1/4/e^5*(2*a*c*e^4+b^2*e^4+2*b*c*d^2*e^2+c^2*d^4)*x^4+1/6*c*(2 *b*e^2+c*d^2)*x^6/e^3-1/7*c^2*d*x^7/e^2-1/3*d/e^6*(2*a*c*e^4+b^2*e^4+2*b*c *d^2*e^2+c^2*d^4)*x^3-d*(2*a*b*e^6+2*a*c*d^2*e^4+b^2*d^2*e^4+2*b*c*d^4*e^2 +c^2*d^6)/e^8*x-1/5*c*d*(2*b*e^2+c*d^2)*x^5/e^4+(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^9*ln(e*x+d)
Time = 0.14 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx=\frac {105 \, c^{2} e^{8} x^{8} - 120 \, c^{2} d e^{7} x^{7} + 140 \, {\left (c^{2} d^{2} e^{6} + 2 \, b c e^{8}\right )} x^{6} - 168 \, {\left (c^{2} d^{3} e^{5} + 2 \, b c d e^{7}\right )} x^{5} + 210 \, {\left (c^{2} d^{4} e^{4} + 2 \, b c d^{2} e^{6} + {\left (b^{2} + 2 \, a c\right )} e^{8}\right )} x^{4} - 280 \, {\left (c^{2} d^{5} e^{3} + 2 \, b c d^{3} e^{5} + {\left (b^{2} + 2 \, a c\right )} d e^{7}\right )} x^{3} + 420 \, {\left (c^{2} d^{6} e^{2} + 2 \, b c d^{4} e^{4} + 2 \, a b e^{8} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{6}\right )} x^{2} - 840 \, {\left (c^{2} d^{7} e + 2 \, b c d^{5} e^{3} + 2 \, a b d e^{7} + {\left (b^{2} + 2 \, a c\right )} d^{3} e^{5}\right )} x + 840 \, {\left (c^{2} d^{8} + 2 \, b c d^{6} e^{2} + 2 \, a b d^{2} e^{6} + a^{2} e^{8} + {\left (b^{2} + 2 \, a c\right )} d^{4} e^{4}\right )} \log \left (e x + d\right )}{840 \, e^{9}} \] Input:
integrate((c*x^4+b*x^2+a)^2/(e*x+d),x, algorithm="fricas")
Output:
1/840*(105*c^2*e^8*x^8 - 120*c^2*d*e^7*x^7 + 140*(c^2*d^2*e^6 + 2*b*c*e^8) *x^6 - 168*(c^2*d^3*e^5 + 2*b*c*d*e^7)*x^5 + 210*(c^2*d^4*e^4 + 2*b*c*d^2* e^6 + (b^2 + 2*a*c)*e^8)*x^4 - 280*(c^2*d^5*e^3 + 2*b*c*d^3*e^5 + (b^2 + 2 *a*c)*d*e^7)*x^3 + 420*(c^2*d^6*e^2 + 2*b*c*d^4*e^4 + 2*a*b*e^8 + (b^2 + 2 *a*c)*d^2*e^6)*x^2 - 840*(c^2*d^7*e + 2*b*c*d^5*e^3 + 2*a*b*d*e^7 + (b^2 + 2*a*c)*d^3*e^5)*x + 840*(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)*log(e*x + d))/e^9
Time = 0.46 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx=- \frac {c^{2} d x^{7}}{7 e^{2}} + \frac {c^{2} x^{8}}{8 e} + x^{6} \left (\frac {b c}{3 e} + \frac {c^{2} d^{2}}{6 e^{3}}\right ) + x^{5} \left (- \frac {2 b c d}{5 e^{2}} - \frac {c^{2} d^{3}}{5 e^{4}}\right ) + x^{4} \left (\frac {a c}{2 e} + \frac {b^{2}}{4 e} + \frac {b c d^{2}}{2 e^{3}} + \frac {c^{2} d^{4}}{4 e^{5}}\right ) + x^{3} \left (- \frac {2 a c d}{3 e^{2}} - \frac {b^{2} d}{3 e^{2}} - \frac {2 b c d^{3}}{3 e^{4}} - \frac {c^{2} d^{5}}{3 e^{6}}\right ) + x^{2} \left (\frac {a b}{e} + \frac {a c d^{2}}{e^{3}} + \frac {b^{2} d^{2}}{2 e^{3}} + \frac {b c d^{4}}{e^{5}} + \frac {c^{2} d^{6}}{2 e^{7}}\right ) + x \left (- \frac {2 a b d}{e^{2}} - \frac {2 a c d^{3}}{e^{4}} - \frac {b^{2} d^{3}}{e^{4}} - \frac {2 b c d^{5}}{e^{6}} - \frac {c^{2} d^{7}}{e^{8}}\right ) + \frac {\left (a e^{4} + b d^{2} e^{2} + c d^{4}\right )^{2} \log {\left (d + e x \right )}}{e^{9}} \] Input:
integrate((c*x**4+b*x**2+a)**2/(e*x+d),x)
Output:
-c**2*d*x**7/(7*e**2) + c**2*x**8/(8*e) + x**6*(b*c/(3*e) + c**2*d**2/(6*e **3)) + x**5*(-2*b*c*d/(5*e**2) - c**2*d**3/(5*e**4)) + x**4*(a*c/(2*e) + b**2/(4*e) + b*c*d**2/(2*e**3) + c**2*d**4/(4*e**5)) + x**3*(-2*a*c*d/(3*e **2) - b**2*d/(3*e**2) - 2*b*c*d**3/(3*e**4) - c**2*d**5/(3*e**6)) + x**2* (a*b/e + a*c*d**2/e**3 + b**2*d**2/(2*e**3) + b*c*d**4/e**5 + c**2*d**6/(2 *e**7)) + x*(-2*a*b*d/e**2 - 2*a*c*d**3/e**4 - b**2*d**3/e**4 - 2*b*c*d**5 /e**6 - c**2*d**7/e**8) + (a*e**4 + b*d**2*e**2 + c*d**4)**2*log(d + e*x)/ e**9
Time = 0.03 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx=\frac {105 \, c^{2} e^{7} x^{8} - 120 \, c^{2} d e^{6} x^{7} + 140 \, {\left (c^{2} d^{2} e^{5} + 2 \, b c e^{7}\right )} x^{6} - 168 \, {\left (c^{2} d^{3} e^{4} + 2 \, b c d e^{6}\right )} x^{5} + 210 \, {\left (c^{2} d^{4} e^{3} + 2 \, b c d^{2} e^{5} + {\left (b^{2} + 2 \, a c\right )} e^{7}\right )} x^{4} - 280 \, {\left (c^{2} d^{5} e^{2} + 2 \, b c d^{3} e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{6}\right )} x^{3} + 420 \, {\left (c^{2} d^{6} e + 2 \, b c d^{4} e^{3} + 2 \, a b e^{7} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{5}\right )} x^{2} - 840 \, {\left (c^{2} d^{7} + 2 \, b c d^{5} e^{2} + 2 \, a b d e^{6} + {\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x}{840 \, e^{8}} + \frac {{\left (c^{2} d^{8} + 2 \, b c d^{6} e^{2} + 2 \, a b d^{2} e^{6} + a^{2} e^{8} + {\left (b^{2} + 2 \, a c\right )} d^{4} e^{4}\right )} \log \left (e x + d\right )}{e^{9}} \] Input:
integrate((c*x^4+b*x^2+a)^2/(e*x+d),x, algorithm="maxima")
Output:
1/840*(105*c^2*e^7*x^8 - 120*c^2*d*e^6*x^7 + 140*(c^2*d^2*e^5 + 2*b*c*e^7) *x^6 - 168*(c^2*d^3*e^4 + 2*b*c*d*e^6)*x^5 + 210*(c^2*d^4*e^3 + 2*b*c*d^2* e^5 + (b^2 + 2*a*c)*e^7)*x^4 - 280*(c^2*d^5*e^2 + 2*b*c*d^3*e^4 + (b^2 + 2 *a*c)*d*e^6)*x^3 + 420*(c^2*d^6*e + 2*b*c*d^4*e^3 + 2*a*b*e^7 + (b^2 + 2*a *c)*d^2*e^5)*x^2 - 840*(c^2*d^7 + 2*b*c*d^5*e^2 + 2*a*b*d*e^6 + (b^2 + 2*a *c)*d^3*e^4)*x)/e^8 + (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)*log(e*x + d)/e^9
Time = 0.13 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx=\frac {105 \, c^{2} e^{7} x^{8} - 120 \, c^{2} d e^{6} x^{7} + 140 \, c^{2} d^{2} e^{5} x^{6} + 280 \, b c e^{7} x^{6} - 168 \, c^{2} d^{3} e^{4} x^{5} - 336 \, b c d e^{6} x^{5} + 210 \, c^{2} d^{4} e^{3} x^{4} + 420 \, b c d^{2} e^{5} x^{4} + 210 \, b^{2} e^{7} x^{4} + 420 \, a c e^{7} x^{4} - 280 \, c^{2} d^{5} e^{2} x^{3} - 560 \, b c d^{3} e^{4} x^{3} - 280 \, b^{2} d e^{6} x^{3} - 560 \, a c d e^{6} x^{3} + 420 \, c^{2} d^{6} e x^{2} + 840 \, b c d^{4} e^{3} x^{2} + 420 \, b^{2} d^{2} e^{5} x^{2} + 840 \, a c d^{2} e^{5} x^{2} + 840 \, a b e^{7} x^{2} - 840 \, c^{2} d^{7} x - 1680 \, b c d^{5} e^{2} x - 840 \, b^{2} d^{3} e^{4} x - 1680 \, a c d^{3} e^{4} x - 1680 \, a b d e^{6} x}{840 \, e^{8}} + \frac {{\left (c^{2} d^{8} + 2 \, b c d^{6} e^{2} + b^{2} d^{4} e^{4} + 2 \, a c d^{4} e^{4} + 2 \, a b d^{2} e^{6} + a^{2} e^{8}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} \] Input:
integrate((c*x^4+b*x^2+a)^2/(e*x+d),x, algorithm="giac")
Output:
1/840*(105*c^2*e^7*x^8 - 120*c^2*d*e^6*x^7 + 140*c^2*d^2*e^5*x^6 + 280*b*c *e^7*x^6 - 168*c^2*d^3*e^4*x^5 - 336*b*c*d*e^6*x^5 + 210*c^2*d^4*e^3*x^4 + 420*b*c*d^2*e^5*x^4 + 210*b^2*e^7*x^4 + 420*a*c*e^7*x^4 - 280*c^2*d^5*e^2 *x^3 - 560*b*c*d^3*e^4*x^3 - 280*b^2*d*e^6*x^3 - 560*a*c*d*e^6*x^3 + 420*c ^2*d^6*e*x^2 + 840*b*c*d^4*e^3*x^2 + 420*b^2*d^2*e^5*x^2 + 840*a*c*d^2*e^5 *x^2 + 840*a*b*e^7*x^2 - 840*c^2*d^7*x - 1680*b*c*d^5*e^2*x - 840*b^2*d^3* e^4*x - 1680*a*c*d^3*e^4*x - 1680*a*b*d*e^6*x)/e^8 + (c^2*d^8 + 2*b*c*d^6* e^2 + b^2*d^4*e^4 + 2*a*c*d^4*e^4 + 2*a*b*d^2*e^6 + a^2*e^8)*log(abs(e*x + d))/e^9
Time = 21.51 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx=x^2\,\left (\frac {d^2\,\left (\frac {b^2+2\,a\,c}{e}+\frac {d^2\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,b\,c}{e}\right )}{e^2}\right )}{2\,e^2}+\frac {a\,b}{e}\right )+x^6\,\left (\frac {c^2\,d^2}{6\,e^3}+\frac {b\,c}{3\,e}\right )+x^4\,\left (\frac {b^2+2\,a\,c}{4\,e}+\frac {d^2\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,b\,c}{e}\right )}{4\,e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^2\,e^8+2\,a\,b\,d^2\,e^6+2\,a\,c\,d^4\,e^4+b^2\,d^4\,e^4+2\,b\,c\,d^6\,e^2+c^2\,d^8\right )}{e^9}+\frac {c^2\,x^8}{8\,e}-\frac {d\,x^3\,\left (\frac {b^2+2\,a\,c}{e}+\frac {d^2\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,b\,c}{e}\right )}{e^2}\right )}{3\,e}-\frac {c^2\,d\,x^7}{7\,e^2}-\frac {d\,x^5\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,b\,c}{e}\right )}{5\,e}-\frac {d\,x\,\left (\frac {d^2\,\left (\frac {b^2+2\,a\,c}{e}+\frac {d^2\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,b\,c}{e}\right )}{e^2}\right )}{e^2}+\frac {2\,a\,b}{e}\right )}{e} \] Input:
int((a + b*x^2 + c*x^4)^2/(d + e*x),x)
Output:
x^2*((d^2*((2*a*c + b^2)/e + (d^2*((c^2*d^2)/e^3 + (2*b*c)/e))/e^2))/(2*e^ 2) + (a*b)/e) + x^6*((c^2*d^2)/(6*e^3) + (b*c)/(3*e)) + x^4*((2*a*c + b^2) /(4*e) + (d^2*((c^2*d^2)/e^3 + (2*b*c)/e))/(4*e^2)) + (log(d + e*x)*(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^9 + (c^2*x^8)/(8*e) - (d*x^3*((2*a*c + b^2)/e + (d^2*((c^2*d^2)/e^3 + (2*b*c)/e))/e^2))/(3*e) - (c^2*d*x^7)/(7*e^2) - (d*x^5*((c^2*d^2)/e^3 + ( 2*b*c)/e))/(5*e) - (d*x*((d^2*((2*a*c + b^2)/e + (d^2*((c^2*d^2)/e^3 + (2* b*c)/e))/e^2))/e^2 + (2*a*b)/e))/e
Time = 0.25 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x} \, dx=\frac {420 a c \,e^{8} x^{4}-840 c^{2} d^{7} e x +420 c^{2} d^{6} e^{2} x^{2}-280 c^{2} d^{5} e^{3} x^{3}+210 c^{2} d^{4} e^{4} x^{4}-168 c^{2} d^{3} e^{5} x^{5}+140 c^{2} d^{2} e^{6} x^{6}-120 c^{2} d \,e^{7} x^{7}+840 \,\mathrm {log}\left (e x +d \right ) a^{2} e^{8}+840 \,\mathrm {log}\left (e x +d \right ) c^{2} d^{8}+105 c^{2} e^{8} x^{8}+1680 \,\mathrm {log}\left (e x +d \right ) a c \,d^{4} e^{4}-1680 a c \,d^{3} e^{5} x +840 a c \,d^{2} e^{6} x^{2}-560 a c d \,e^{7} x^{3}+210 b^{2} e^{8} x^{4}+840 \,\mathrm {log}\left (e x +d \right ) b^{2} d^{4} e^{4}+840 a b \,e^{8} x^{2}-840 b^{2} d^{3} e^{5} x +420 b^{2} d^{2} e^{6} x^{2}-280 b^{2} d \,e^{7} x^{3}+280 b c \,e^{8} x^{6}+1680 \,\mathrm {log}\left (e x +d \right ) a b \,d^{2} e^{6}+1680 \,\mathrm {log}\left (e x +d \right ) b c \,d^{6} e^{2}-1680 a b d \,e^{7} x -1680 b c \,d^{5} e^{3} x +840 b c \,d^{4} e^{4} x^{2}-560 b c \,d^{3} e^{5} x^{3}+420 b c \,d^{2} e^{6} x^{4}-336 b c d \,e^{7} x^{5}}{840 e^{9}} \] Input:
int((c*x^4+b*x^2+a)^2/(e*x+d),x)
Output:
(840*log(d + e*x)*a**2*e**8 + 1680*log(d + e*x)*a*b*d**2*e**6 + 1680*log(d + e*x)*a*c*d**4*e**4 + 840*log(d + e*x)*b**2*d**4*e**4 + 1680*log(d + e*x )*b*c*d**6*e**2 + 840*log(d + e*x)*c**2*d**8 - 1680*a*b*d*e**7*x + 840*a*b *e**8*x**2 - 1680*a*c*d**3*e**5*x + 840*a*c*d**2*e**6*x**2 - 560*a*c*d*e** 7*x**3 + 420*a*c*e**8*x**4 - 840*b**2*d**3*e**5*x + 420*b**2*d**2*e**6*x** 2 - 280*b**2*d*e**7*x**3 + 210*b**2*e**8*x**4 - 1680*b*c*d**5*e**3*x + 840 *b*c*d**4*e**4*x**2 - 560*b*c*d**3*e**5*x**3 + 420*b*c*d**2*e**6*x**4 - 33 6*b*c*d*e**7*x**5 + 280*b*c*e**8*x**6 - 840*c**2*d**7*e*x + 420*c**2*d**6* e**2*x**2 - 280*c**2*d**5*e**3*x**3 + 210*c**2*d**4*e**4*x**4 - 168*c**2*d **3*e**5*x**5 + 140*c**2*d**2*e**6*x**6 - 120*c**2*d*e**7*x**7 + 105*c**2* e**8*x**8)/(840*e**9)