Integrand size = 17, antiderivative size = 178 \[ \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {c d^2 \left (7 c d^4+6 a e^4\right ) x}{e^8}-\frac {c d \left (3 c d^4+2 a e^4\right ) x^2}{e^7}+\frac {c \left (5 c d^4+2 a e^4\right ) x^3}{3 e^6}-\frac {c^2 d^3 x^4}{e^5}+\frac {3 c^2 d^2 x^5}{5 e^4}-\frac {c^2 d x^6}{3 e^3}+\frac {c^2 x^7}{7 e^2}-\frac {\left (c d^4+a e^4\right )^2}{e^9 (d+e x)}-\frac {8 c d^3 \left (c d^4+a e^4\right ) \log (d+e x)}{e^9} \] Output:
c*d^2*(6*a*e^4+7*c*d^4)*x/e^8-c*d*(2*a*e^4+3*c*d^4)*x^2/e^7+1/3*c*(2*a*e^4 +5*c*d^4)*x^3/e^6-c^2*d^3*x^4/e^5+3/5*c^2*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+c*d^4)^2/e^9/(e*x+d)-8*c*d^3*(a*e^4+c*d^4)*ln(e*x+d)/ e^9
Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {c d^2 \left (7 c d^4+6 a e^4\right ) x}{e^8}-\frac {c d \left (3 c d^4+2 a e^4\right ) x^2}{e^7}+\frac {c \left (5 c d^4+2 a e^4\right ) x^3}{3 e^6}-\frac {c^2 d^3 x^4}{e^5}+\frac {3 c^2 d^2 x^5}{5 e^4}-\frac {c^2 d x^6}{3 e^3}+\frac {c^2 x^7}{7 e^2}-\frac {\left (c d^4+a e^4\right )^2}{e^9 (d+e x)}-\frac {8 c d^3 \left (c d^4+a e^4\right ) \log (d+e x)}{e^9} \] Input:
Integrate[(a + c*x^4)^2/(d + e*x)^2,x]
Output:
(c*d^2*(7*c*d^4 + 6*a*e^4)*x)/e^8 - (c*d*(3*c*d^4 + 2*a*e^4)*x^2)/e^7 + (c *(5*c*d^4 + 2*a*e^4)*x^3)/(3*e^6) - (c^2*d^3*x^4)/e^5 + (3*c^2*d^2*x^5)/(5 *e^4) - (c^2*d*x^6)/(3*e^3) + (c^2*x^7)/(7*e^2) - (c*d^4 + a*e^4)^2/(e^9*( d + e*x)) - (8*c*d^3*(c*d^4 + a*e^4)*Log[d + e*x])/e^9
Time = 0.66 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
| \(\displaystyle \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (\frac {\left (a e^4+c d^4\right )^2}{e^8 (d+e x)^2}-\frac {2 c d x \left (2 a e^4+3 c d^4\right )}{e^7}+\frac {c x^2 \left (2 a e^4+5 c d^4\right )}{e^6}-\frac {8 c d^3 \left (a e^4+c d^4\right )}{e^8 (d+e x)}+\frac {c d^2 \left (6 a e^4+7 c d^4\right )}{e^8}-\frac {4 c^2 d^3 x^3}{e^5}+\frac {3 c^2 d^2 x^4}{e^4}-\frac {2 c^2 d x^5}{e^3}+\frac {c^2 x^6}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a e^4+c d^4\right )^2}{e^9 (d+e x)}-\frac {c d x^2 \left (2 a e^4+3 c d^4\right )}{e^7}+\frac {c x^3 \left (2 a e^4+5 c d^4\right )}{3 e^6}-\frac {8 c d^3 \left (a e^4+c d^4\right ) \log (d+e x)}{e^9}+\frac {c d^2 x \left (6 a e^4+7 c d^4\right )}{e^8}-\frac {c^2 d^3 x^4}{e^5}+\frac {3 c^2 d^2 x^5}{5 e^4}-\frac {c^2 d x^6}{3 e^3}+\frac {c^2 x^7}{7 e^2}\) |
Input:
Int[(a + c*x^4)^2/(d + e*x)^2,x]
Output:
(c*d^2*(7*c*d^4 + 6*a*e^4)*x)/e^8 - (c*d*(3*c*d^4 + 2*a*e^4)*x^2)/e^7 + (c *(5*c*d^4 + 2*a*e^4)*x^3)/(3*e^6) - (c^2*d^3*x^4)/e^5 + (3*c^2*d^2*x^5)/(5 *e^4) - (c^2*d*x^6)/(3*e^3) + (c^2*x^7)/(7*e^2) - (c*d^4 + a*e^4)^2/(e^9*( d + e*x)) - (8*c*d^3*(c*d^4 + 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 = 0.17 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {c \left (\frac {1}{7} c \,x^{7} e^{6}-\frac {1}{3} c d \,x^{6} e^{5}+\frac {3}{5} c \,d^{2} x^{5} e^{4}-c \,d^{3} x^{4} e^{3}+\frac {2}{3} a \,e^{6} x^{3}+\frac {5}{3} c \,d^{4} e^{2} x^{3}-2 a d \,e^{5} x^{2}-3 c \,d^{5} e \,x^{2}+6 x a \,d^{2} e^{4}+7 x c \,d^{6}\right )}{e^{8}}-\frac {a^{2} e^{8}+2 a c \,d^{4} e^{4}+d^{8} c^{2}}{e^{9} \left (e x +d \right )}-\frac {8 c \,d^{3} \left (e^{4} a +c \,d^{4}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(172\) |
| norman | \(\frac {-\frac {a^{2} e^{8}+8 a c \,d^{4} e^{4}+8 d^{8} c^{2}}{e^{9}}+\frac {c^{2} x^{8}}{7 e}+\frac {2 c \left (e^{4} a +c \,d^{4}\right ) x^{4}}{3 e^{5}}-\frac {4 c^{2} d \,x^{7}}{21 e^{2}}+\frac {4 c^{2} d^{2} x^{6}}{15 e^{3}}-\frac {2 c^{2} d^{3} x^{5}}{5 e^{4}}-\frac {4 d c \left (e^{4} a +c \,d^{4}\right ) x^{3}}{3 e^{6}}+\frac {4 d^{2} c \left (e^{4} a +c \,d^{4}\right ) x^{2}}{e^{7}}}{e x +d}-\frac {8 c \,d^{3} \left (e^{4} a +c \,d^{4}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(183\) |
| risch | \(\frac {c^{2} x^{7}}{7 e^{2}}-\frac {c^{2} d \,x^{6}}{3 e^{3}}+\frac {3 c^{2} d^{2} x^{5}}{5 e^{4}}-\frac {c^{2} d^{3} x^{4}}{e^{5}}+\frac {2 c a \,x^{3}}{3 e^{2}}+\frac {5 c^{2} d^{4} x^{3}}{3 e^{6}}-\frac {2 c a d \,x^{2}}{e^{3}}-\frac {3 c^{2} d^{5} x^{2}}{e^{7}}+\frac {6 c x a \,d^{2}}{e^{4}}+\frac {7 c^{2} x \,d^{6}}{e^{8}}-\frac {a^{2}}{e \left (e x +d \right )}-\frac {2 a c \,d^{4}}{e^{5} \left (e x +d \right )}-\frac {d^{8} c^{2}}{e^{9} \left (e x +d \right )}-\frac {8 c \,d^{3} \ln \left (e x +d \right ) a}{e^{5}}-\frac {8 c^{2} d^{7} \ln \left (e x +d \right )}{e^{9}}\) | \(208\) |
| parallelrisch | \(-\frac {-15 x^{8} c^{2} e^{8}+20 c^{2} d \,x^{7} e^{7}-28 c^{2} d^{2} x^{6} e^{6}+42 c^{2} d^{3} x^{5} e^{5}-70 x^{4} a c \,e^{8}-70 x^{4} c^{2} d^{4} e^{4}+140 x^{3} a c d \,e^{7}+140 x^{3} c^{2} d^{5} e^{3}+840 \ln \left (e x +d \right ) x a c \,d^{3} e^{5}+840 \ln \left (e x +d \right ) x \,c^{2} d^{7} e -420 x^{2} a c \,d^{2} e^{6}-420 x^{2} c^{2} d^{6} e^{2}+840 \ln \left (e x +d \right ) a c \,d^{4} e^{4}+840 \ln \left (e x +d \right ) c^{2} d^{8}+105 a^{2} e^{8}+840 a c \,d^{4} e^{4}+840 d^{8} c^{2}}{105 e^{9} \left (e x +d \right )}\) | \(230\) |
Input:
int((c*x^4+a)^2/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
c/e^8*(1/7*c*x^7*e^6-1/3*c*d*x^6*e^5+3/5*c*d^2*x^5*e^4-c*d^3*x^4*e^3+2/3*a *e^6*x^3+5/3*c*d^4*e^2*x^3-2*a*d*e^5*x^2-3*c*d^5*e*x^2+6*x*a*d^2*e^4+7*x*c *d^6)-(a^2*e^8+2*a*c*d^4*e^4+c^2*d^8)/e^9/(e*x+d)-8*c*d^3*(a*e^4+c*d^4)*ln (e*x+d)/e^9
Time = 0.08 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {15 \, c^{2} e^{8} x^{8} - 20 \, c^{2} d e^{7} x^{7} + 28 \, c^{2} d^{2} e^{6} x^{6} - 42 \, c^{2} d^{3} e^{5} x^{5} - 105 \, c^{2} d^{8} - 210 \, a c d^{4} e^{4} - 105 \, a^{2} e^{8} + 70 \, {\left (c^{2} d^{4} e^{4} + a c e^{8}\right )} x^{4} - 140 \, {\left (c^{2} d^{5} e^{3} + a c d e^{7}\right )} x^{3} + 420 \, {\left (c^{2} d^{6} e^{2} + a c d^{2} e^{6}\right )} x^{2} + 105 \, {\left (7 \, c^{2} d^{7} e + 6 \, a c d^{3} e^{5}\right )} x - 840 \, {\left (c^{2} d^{8} + a c d^{4} e^{4} + {\left (c^{2} d^{7} e + a c d^{3} e^{5}\right )} x\right )} \log \left (e x + d\right )}{105 \, {\left (e^{10} x + d e^{9}\right )}} \] Input:
integrate((c*x^4+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 + 28*c^2*d^2*e^6*x^6 - 42*c^2*d^3 *e^5*x^5 - 105*c^2*d^8 - 210*a*c*d^4*e^4 - 105*a^2*e^8 + 70*(c^2*d^4*e^4 + a*c*e^8)*x^4 - 140*(c^2*d^5*e^3 + a*c*d*e^7)*x^3 + 420*(c^2*d^6*e^2 + a*c *d^2*e^6)*x^2 + 105*(7*c^2*d^7*e + 6*a*c*d^3*e^5)*x - 840*(c^2*d^8 + a*c*d ^4*e^4 + (c^2*d^7*e + a*c*d^3*e^5)*x)*log(e*x + d))/(e^10*x + d*e^9)
Time = 0.36 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx=- \frac {c^{2} d^{3} x^{4}}{e^{5}} + \frac {3 c^{2} d^{2} x^{5}}{5 e^{4}} - \frac {c^{2} d x^{6}}{3 e^{3}} + \frac {c^{2} x^{7}}{7 e^{2}} - \frac {8 c d^{3} \left (a e^{4} + c d^{4}\right ) \log {\left (d + e x \right )}}{e^{9}} + x^{3} \cdot \left (\frac {2 a c}{3 e^{2}} + \frac {5 c^{2} d^{4}}{3 e^{6}}\right ) + x^{2} \left (- \frac {2 a c d}{e^{3}} - \frac {3 c^{2} d^{5}}{e^{7}}\right ) + x \left (\frac {6 a c d^{2}}{e^{4}} + \frac {7 c^{2} d^{6}}{e^{8}}\right ) + \frac {- a^{2} e^{8} - 2 a c d^{4} e^{4} - c^{2} d^{8}}{d e^{9} + e^{10} x} \] Input:
integrate((c*x**4+a)**2/(e*x+d)**2,x)
Output:
-c**2*d**3*x**4/e**5 + 3*c**2*d**2*x**5/(5*e**4) - c**2*d*x**6/(3*e**3) + c**2*x**7/(7*e**2) - 8*c*d**3*(a*e**4 + c*d**4)*log(d + e*x)/e**9 + x**3*( 2*a*c/(3*e**2) + 5*c**2*d**4/(3*e**6)) + x**2*(-2*a*c*d/e**3 - 3*c**2*d**5 /e**7) + x*(6*a*c*d**2/e**4 + 7*c**2*d**6/e**8) + (-a**2*e**8 - 2*a*c*d**4 *e**4 - c**2*d**8)/(d*e**9 + e**10*x)
Time = 0.03 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx=-\frac {c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}}{e^{10} x + d e^{9}} + \frac {15 \, c^{2} e^{6} x^{7} - 35 \, c^{2} d e^{5} x^{6} + 63 \, c^{2} d^{2} e^{4} x^{5} - 105 \, c^{2} d^{3} e^{3} x^{4} + 35 \, {\left (5 \, c^{2} d^{4} e^{2} + 2 \, a c e^{6}\right )} x^{3} - 105 \, {\left (3 \, c^{2} d^{5} e + 2 \, a c d e^{5}\right )} x^{2} + 105 \, {\left (7 \, c^{2} d^{6} + 6 \, a c d^{2} e^{4}\right )} x}{105 \, e^{8}} - \frac {8 \, {\left (c^{2} d^{7} + a c d^{3} e^{4}\right )} \log \left (e x + d\right )}{e^{9}} \] Input:
integrate((c*x^4+a)^2/(e*x+d)^2,x, algorithm="maxima")
Output:
-(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8)/(e^10*x + d*e^9) + 1/105*(15*c^2*e^6* x^7 - 35*c^2*d*e^5*x^6 + 63*c^2*d^2*e^4*x^5 - 105*c^2*d^3*e^3*x^4 + 35*(5* c^2*d^4*e^2 + 2*a*c*e^6)*x^3 - 105*(3*c^2*d^5*e + 2*a*c*d*e^5)*x^2 + 105*( 7*c^2*d^6 + 6*a*c*d^2*e^4)*x)/e^8 - 8*(c^2*d^7 + a*c*d^3*e^4)*log(e*x + d) /e^9
Time = 0.13 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {{\left (15 \, c^{2} - \frac {140 \, c^{2} d}{e x + d} + \frac {588 \, c^{2} d^{2}}{{\left (e x + d\right )}^{2}} - \frac {1470 \, c^{2} d^{3}}{{\left (e x + d\right )}^{3}} + \frac {70 \, {\left (35 \, c^{2} d^{4} e^{4} + a c e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}} - \frac {420 \, {\left (7 \, c^{2} d^{5} e^{5} + a c d e^{9}\right )}}{{\left (e x + d\right )}^{5} e^{5}} + \frac {420 \, {\left (7 \, c^{2} d^{6} e^{6} + 3 \, a c d^{2} e^{10}\right )}}{{\left (e x + d\right )}^{6} e^{6}}\right )} {\left (e x + d\right )}^{7}}{105 \, e^{9}} + \frac {8 \, {\left (c^{2} d^{7} + a c d^{3} e^{4}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{9}} - \frac {\frac {c^{2} d^{8} e^{7}}{e x + d} + \frac {2 \, a c d^{4} e^{11}}{e x + d} + \frac {a^{2} e^{15}}{e x + d}}{e^{16}} \] Input:
integrate((c*x^4+a)^2/(e*x+d)^2,x, algorithm="giac")
Output:
1/105*(15*c^2 - 140*c^2*d/(e*x + d) + 588*c^2*d^2/(e*x + d)^2 - 1470*c^2*d ^3/(e*x + d)^3 + 70*(35*c^2*d^4*e^4 + a*c*e^8)/((e*x + d)^4*e^4) - 420*(7* c^2*d^5*e^5 + a*c*d*e^9)/((e*x + d)^5*e^5) + 420*(7*c^2*d^6*e^6 + 3*a*c*d^ 2*e^10)/((e*x + d)^6*e^6))*(e*x + d)^7/e^9 + 8*(c^2*d^7 + a*c*d^3*e^4)*log (abs(e*x + d)/((e*x + d)^2*abs(e)))/e^9 - (c^2*d^8*e^7/(e*x + d) + 2*a*c*d ^4*e^11/(e*x + d) + a^2*e^15/(e*x + d))/e^16
Time = 21.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx=x\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {5\,c^2\,d^4}{e^6}+\frac {2\,a\,c}{e^2}\right )}{e}-\frac {4\,c^2\,d^5}{e^7}\right )}{e}-\frac {d^2\,\left (\frac {5\,c^2\,d^4}{e^6}+\frac {2\,a\,c}{e^2}\right )}{e^2}\right )+x^3\,\left (\frac {5\,c^2\,d^4}{3\,e^6}+\frac {2\,a\,c}{3\,e^2}\right )-x^2\,\left (\frac {d\,\left (\frac {5\,c^2\,d^4}{e^6}+\frac {2\,a\,c}{e^2}\right )}{e}-\frac {2\,c^2\,d^5}{e^7}\right )-\frac {\ln \left (d+e\,x\right )\,\left (8\,c^2\,d^7+8\,a\,c\,d^3\,e^4\right )}{e^9}+\frac {c^2\,x^7}{7\,e^2}-\frac {a^2\,e^8+2\,a\,c\,d^4\,e^4+c^2\,d^8}{e\,\left (x\,e^9+d\,e^8\right )}-\frac {c^2\,d\,x^6}{3\,e^3}+\frac {3\,c^2\,d^2\,x^5}{5\,e^4}-\frac {c^2\,d^3\,x^4}{e^5} \] Input:
int((a + c*x^4)^2/(d + e*x)^2,x)
Output:
x*((2*d*((2*d*((5*c^2*d^4)/e^6 + (2*a*c)/e^2))/e - (4*c^2*d^5)/e^7))/e - ( d^2*((5*c^2*d^4)/e^6 + (2*a*c)/e^2))/e^2) + x^3*((5*c^2*d^4)/(3*e^6) + (2* a*c)/(3*e^2)) - x^2*((d*((5*c^2*d^4)/e^6 + (2*a*c)/e^2))/e - (2*c^2*d^5)/e ^7) - (log(d + e*x)*(8*c^2*d^7 + 8*a*c*d^3*e^4))/e^9 + (c^2*x^7)/(7*e^2) - (a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)/(e*(d*e^8 + e^9*x)) - (c^2*d*x^6)/(3* e^3) + (3*c^2*d^2*x^5)/(5*e^4) - (c^2*d^3*x^4)/e^5
Time = 0.14 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+c x^4\right )^2}{(d+e x)^2} \, dx=\frac {-840 \,\mathrm {log}\left (e x +d \right ) a c \,d^{5} e^{4}-840 \,\mathrm {log}\left (e x +d \right ) a c \,d^{4} e^{5} x -840 \,\mathrm {log}\left (e x +d \right ) c^{2} d^{9}-840 \,\mathrm {log}\left (e x +d \right ) c^{2} d^{8} e x +105 a^{2} e^{9} x +840 a c \,d^{4} e^{5} x +420 a c \,d^{3} e^{6} x^{2}-140 a c \,d^{2} e^{7} x^{3}+70 a c d \,e^{8} x^{4}+840 c^{2} d^{8} e x +420 c^{2} d^{7} e^{2} x^{2}-140 c^{2} d^{6} e^{3} x^{3}+70 c^{2} d^{5} e^{4} x^{4}-42 c^{2} d^{4} e^{5} x^{5}+28 c^{2} d^{3} e^{6} x^{6}-20 c^{2} d^{2} e^{7} x^{7}+15 c^{2} d \,e^{8} x^{8}}{105 d \,e^{9} \left (e x +d \right )} \] Input:
int((c*x^4+a)^2/(e*x+d)^2,x)
Output:
( - 840*log(d + e*x)*a*c*d**5*e**4 - 840*log(d + e*x)*a*c*d**4*e**5*x - 84 0*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 + 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 + 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))