Integrand size = 17, antiderivative size = 178 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx=-\frac {c d^3 \left (c d^4+2 a e^4\right ) x}{e^8}+\frac {c d^2 \left (c d^4+2 a e^4\right ) x^2}{2 e^7}-\frac {c d \left (c d^4+2 a e^4\right ) x^3}{3 e^6}+\frac {c \left (c d^4+2 a e^4\right ) x^4}{4 e^5}-\frac {c^2 d^3 x^5}{5 e^4}+\frac {c^2 d^2 x^6}{6 e^3}-\frac {c^2 d x^7}{7 e^2}+\frac {c^2 x^8}{8 e}+\frac {\left (c d^4+a e^4\right )^2 \log (d+e x)}{e^9} \] Output:
-c*d^3*(2*a*e^4+c*d^4)*x/e^8+1/2*c*d^2*(2*a*e^4+c*d^4)*x^2/e^7-1/3*c*d*(2* a*e^4+c*d^4)*x^3/e^6+1/4*c*(2*a*e^4+c*d^4)*x^4/e^5-1/5*c^2*d^3*x^5/e^4+1/6 *c^2*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+c*d^4)^2*ln(e*x+d) /e^9
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx=\frac {c x \left (140 a e^4 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+c \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )\right )}{840 e^8}+\frac {\left (c d^4+a e^4\right )^2 \log (d+e x)}{e^9} \] Input:
Integrate[(a + c*x^4)^2/(d + e*x),x]
Output:
(c*x*(140*a*e^4*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + c*(-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)))/(840*e^8) + ((c*d^4 + a*e ^4)^2*Log[d + e*x])/e^9
Time = 0.61 (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.
| \(\displaystyle \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (\frac {\left (a e^4+c d^4\right )^2}{e^8 (d+e x)}-\frac {c d x^2 \left (2 a e^4+c d^4\right )}{e^6}+\frac {c x^3 \left (2 a e^4+c d^4\right )}{e^5}-\frac {c d^3 \left (2 a e^4+c d^4\right )}{e^8}+\frac {c d^2 x \left (2 a e^4+c d^4\right )}{e^7}-\frac {c^2 d^3 x^4}{e^4}+\frac {c^2 d^2 x^5}{e^3}-\frac {c^2 d x^6}{e^2}+\frac {c^2 x^7}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a e^4+c d^4\right )^2 \log (d+e x)}{e^9}-\frac {c d x^3 \left (2 a e^4+c d^4\right )}{3 e^6}+\frac {c x^4 \left (2 a e^4+c d^4\right )}{4 e^5}-\frac {c d^3 x \left (2 a e^4+c d^4\right )}{e^8}+\frac {c d^2 x^2 \left (2 a e^4+c d^4\right )}{2 e^7}-\frac {c^2 d^3 x^5}{5 e^4}+\frac {c^2 d^2 x^6}{6 e^3}-\frac {c^2 d x^7}{7 e^2}+\frac {c^2 x^8}{8 e}\) |
Input:
Int[(a + c*x^4)^2/(d + e*x),x]
Output:
-((c*d^3*(c*d^4 + 2*a*e^4)*x)/e^8) + (c*d^2*(c*d^4 + 2*a*e^4)*x^2)/(2*e^7) - (c*d*(c*d^4 + 2*a*e^4)*x^3)/(3*e^6) + (c*(c*d^4 + 2*a*e^4)*x^4)/(4*e^5) - (c^2*d^3*x^5)/(5*e^4) + (c^2*d^2*x^6)/(6*e^3) - (c^2*d*x^7)/(7*e^2) + ( c^2*x^8)/(8*e) + ((c*d^4 + 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 = 0.23 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(-\frac {c \left (-\frac {c \,e^{7} x^{8}}{8}+\frac {c d \,e^{6} x^{7}}{7}-\frac {c \,d^{2} x^{6} e^{5}}{6}+\frac {c \,d^{3} x^{5} e^{4}}{5}-\frac {\left (2 e^{4} a +c \,d^{4}\right ) x^{4} e^{3}}{4}+\frac {d \left (2 e^{4} a +c \,d^{4}\right ) x^{3} e^{2}}{3}-\frac {d^{2} \left (2 e^{4} a +c \,d^{4}\right ) x^{2} e}{2}+x \,d^{3} \left (2 e^{4} a +c \,d^{4}\right )\right )}{e^{8}}+\frac {\left (a^{2} e^{8}+2 a c \,d^{4} e^{4}+d^{8} c^{2}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(166\) |
| norman | \(\frac {c^{2} x^{8}}{8 e}+\frac {c \left (2 e^{4} a +c \,d^{4}\right ) x^{4}}{4 e^{5}}-\frac {c^{2} d \,x^{7}}{7 e^{2}}+\frac {c^{2} d^{2} x^{6}}{6 e^{3}}-\frac {c^{2} d^{3} x^{5}}{5 e^{4}}-\frac {c d \left (2 e^{4} a +c \,d^{4}\right ) x^{3}}{3 e^{6}}+\frac {c \,d^{2} \left (2 e^{4} a +c \,d^{4}\right ) x^{2}}{2 e^{7}}-\frac {c \,d^{3} \left (2 e^{4} a +c \,d^{4}\right ) x}{e^{8}}+\frac {\left (a^{2} e^{8}+2 a c \,d^{4} e^{4}+d^{8} c^{2}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(177\) |
| risch | \(\frac {c^{2} x^{8}}{8 e}-\frac {c^{2} d \,x^{7}}{7 e^{2}}+\frac {c^{2} d^{2} x^{6}}{6 e^{3}}-\frac {c^{2} d^{3} x^{5}}{5 e^{4}}+\frac {c a \,x^{4}}{2 e}+\frac {c^{2} d^{4} x^{4}}{4 e^{5}}-\frac {2 c a d \,x^{3}}{3 e^{2}}-\frac {c^{2} d^{5} x^{3}}{3 e^{6}}+\frac {c a \,d^{2} x^{2}}{e^{3}}+\frac {c^{2} d^{6} x^{2}}{2 e^{7}}-\frac {2 c a \,d^{3} x}{e^{4}}-\frac {c^{2} d^{7} x}{e^{8}}+\frac {\ln \left (e x +d \right ) a^{2}}{e}+\frac {2 \ln \left (e x +d \right ) a c \,d^{4}}{e^{5}}+\frac {\ln \left (e x +d \right ) d^{8} c^{2}}{e^{9}}\) | \(196\) |
| parallelrisch | \(\frac {105 x^{8} c^{2} e^{8}-120 c^{2} d \,x^{7} e^{7}+140 c^{2} d^{2} x^{6} e^{6}-168 c^{2} d^{3} x^{5} e^{5}+420 x^{4} a c \,e^{8}+210 x^{4} c^{2} d^{4} e^{4}-560 x^{3} a c d \,e^{7}-280 x^{3} c^{2} d^{5} e^{3}+840 x^{2} a c \,d^{2} e^{6}+420 x^{2} c^{2} d^{6} e^{2}+840 \ln \left (e x +d \right ) a^{2} e^{8}+1680 \ln \left (e x +d \right ) a c \,d^{4} e^{4}+840 \ln \left (e x +d \right ) c^{2} d^{8}-1680 x a c \,d^{3} e^{5}-840 x \,c^{2} d^{7} e}{840 e^{9}}\) | \(199\) |
Input:
int((c*x^4+a)^2/(e*x+d),x,method=_RETURNVERBOSE)
Output:
-c/e^8*(-1/8*c*e^7*x^8+1/7*c*d*e^6*x^7-1/6*c*d^2*x^6*e^5+1/5*c*d^3*x^5*e^4 -1/4*(2*a*e^4+c*d^4)*x^4*e^3+1/3*d*(2*a*e^4+c*d^4)*x^3*e^2-1/2*d^2*(2*a*e^ 4+c*d^4)*x^2*e+x*d^3*(2*a*e^4+c*d^4))+(a^2*e^8+2*a*c*d^4*e^4+c^2*d^8)/e^9* ln(e*x+d)
Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx=\frac {105 \, c^{2} e^{8} x^{8} - 120 \, c^{2} d e^{7} x^{7} + 140 \, c^{2} d^{2} e^{6} x^{6} - 168 \, c^{2} d^{3} e^{5} x^{5} + 210 \, {\left (c^{2} d^{4} e^{4} + 2 \, a c e^{8}\right )} x^{4} - 280 \, {\left (c^{2} d^{5} e^{3} + 2 \, a c d e^{7}\right )} x^{3} + 420 \, {\left (c^{2} d^{6} e^{2} + 2 \, a c d^{2} e^{6}\right )} x^{2} - 840 \, {\left (c^{2} d^{7} e + 2 \, a c d^{3} e^{5}\right )} x + 840 \, {\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )} \log \left (e x + d\right )}{840 \, e^{9}} \] Input:
integrate((c*x^4+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*x^6 - 168*c^2 *d^3*e^5*x^5 + 210*(c^2*d^4*e^4 + 2*a*c*e^8)*x^4 - 280*(c^2*d^5*e^3 + 2*a* c*d*e^7)*x^3 + 420*(c^2*d^6*e^2 + 2*a*c*d^2*e^6)*x^2 - 840*(c^2*d^7*e + 2* a*c*d^3*e^5)*x + 840*(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8)*log(e*x + d))/e^9
Time = 0.22 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx=- \frac {c^{2} d^{3} x^{5}}{5 e^{4}} + \frac {c^{2} d^{2} x^{6}}{6 e^{3}} - \frac {c^{2} d x^{7}}{7 e^{2}} + \frac {c^{2} x^{8}}{8 e} + x^{4} \left (\frac {a c}{2 e} + \frac {c^{2} d^{4}}{4 e^{5}}\right ) + x^{3} \left (- \frac {2 a c d}{3 e^{2}} - \frac {c^{2} d^{5}}{3 e^{6}}\right ) + x^{2} \left (\frac {a c d^{2}}{e^{3}} + \frac {c^{2} d^{6}}{2 e^{7}}\right ) + x \left (- \frac {2 a c d^{3}}{e^{4}} - \frac {c^{2} d^{7}}{e^{8}}\right ) + \frac {\left (a e^{4} + c d^{4}\right )^{2} \log {\left (d + e x \right )}}{e^{9}} \] Input:
integrate((c*x**4+a)**2/(e*x+d),x)
Output:
-c**2*d**3*x**5/(5*e**4) + c**2*d**2*x**6/(6*e**3) - c**2*d*x**7/(7*e**2) + c**2*x**8/(8*e) + x**4*(a*c/(2*e) + c**2*d**4/(4*e**5)) + x**3*(-2*a*c*d /(3*e**2) - c**2*d**5/(3*e**6)) + x**2*(a*c*d**2/e**3 + c**2*d**6/(2*e**7) ) + x*(-2*a*c*d**3/e**4 - c**2*d**7/e**8) + (a*e**4 + c*d**4)**2*log(d + e *x)/e**9
Time = 0.03 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx=\frac {105 \, c^{2} e^{7} x^{8} - 120 \, c^{2} d e^{6} x^{7} + 140 \, c^{2} d^{2} e^{5} x^{6} - 168 \, c^{2} d^{3} e^{4} x^{5} + 210 \, {\left (c^{2} d^{4} e^{3} + 2 \, a c e^{7}\right )} x^{4} - 280 \, {\left (c^{2} d^{5} e^{2} + 2 \, a c d e^{6}\right )} x^{3} + 420 \, {\left (c^{2} d^{6} e + 2 \, a c d^{2} e^{5}\right )} x^{2} - 840 \, {\left (c^{2} d^{7} + 2 \, a c d^{3} e^{4}\right )} x}{840 \, e^{8}} + \frac {{\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )} \log \left (e x + d\right )}{e^{9}} \] Input:
integrate((c*x^4+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*x^6 - 168*c^2 *d^3*e^4*x^5 + 210*(c^2*d^4*e^3 + 2*a*c*e^7)*x^4 - 280*(c^2*d^5*e^2 + 2*a* c*d*e^6)*x^3 + 420*(c^2*d^6*e + 2*a*c*d^2*e^5)*x^2 - 840*(c^2*d^7 + 2*a*c* d^3*e^4)*x)/e^8 + (c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8)*log(e*x + d)/e^9
Time = 0.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx=\frac {105 \, c^{2} e^{7} x^{8} - 120 \, c^{2} d e^{6} x^{7} + 140 \, c^{2} d^{2} e^{5} x^{6} - 168 \, c^{2} d^{3} e^{4} x^{5} + 210 \, c^{2} d^{4} e^{3} x^{4} + 420 \, a c e^{7} x^{4} - 280 \, c^{2} d^{5} e^{2} x^{3} - 560 \, a c d e^{6} x^{3} + 420 \, c^{2} d^{6} e x^{2} + 840 \, a c d^{2} e^{5} x^{2} - 840 \, c^{2} d^{7} x - 1680 \, a c d^{3} e^{4} x}{840 \, e^{8}} + \frac {{\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} \] Input:
integrate((c*x^4+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 - 168*c^2 *d^3*e^4*x^5 + 210*c^2*d^4*e^3*x^4 + 420*a*c*e^7*x^4 - 280*c^2*d^5*e^2*x^3 - 560*a*c*d*e^6*x^3 + 420*c^2*d^6*e*x^2 + 840*a*c*d^2*e^5*x^2 - 840*c^2*d ^7*x - 1680*a*c*d^3*e^4*x)/e^8 + (c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8)*log(a bs(e*x + d))/e^9
Time = 0.06 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx=x^4\,\left (\frac {c^2\,d^4}{4\,e^5}+\frac {a\,c}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^2\,e^8+2\,a\,c\,d^4\,e^4+c^2\,d^8\right )}{e^9}+\frac {c^2\,x^8}{8\,e}-\frac {c^2\,d\,x^7}{7\,e^2}-\frac {d\,x^3\,\left (\frac {c^2\,d^4}{e^5}+\frac {2\,a\,c}{e}\right )}{3\,e}-\frac {d^3\,x\,\left (\frac {c^2\,d^4}{e^5}+\frac {2\,a\,c}{e}\right )}{e^3}+\frac {c^2\,d^2\,x^6}{6\,e^3}-\frac {c^2\,d^3\,x^5}{5\,e^4}+\frac {d^2\,x^2\,\left (\frac {c^2\,d^4}{e^5}+\frac {2\,a\,c}{e}\right )}{2\,e^2} \] Input:
int((a + c*x^4)^2/(d + e*x),x)
Output:
x^4*((c^2*d^4)/(4*e^5) + (a*c)/(2*e)) + (log(d + e*x)*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4))/e^9 + (c^2*x^8)/(8*e) - (c^2*d*x^7)/(7*e^2) - (d*x^3*((c^ 2*d^4)/e^5 + (2*a*c)/e))/(3*e) - (d^3*x*((c^2*d^4)/e^5 + (2*a*c)/e))/e^3 + (c^2*d^2*x^6)/(6*e^3) - (c^2*d^3*x^5)/(5*e^4) + (d^2*x^2*((c^2*d^4)/e^5 + (2*a*c)/e))/(2*e^2)
Time = 0.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+c x^4\right )^2}{d+e x} \, dx=\frac {840 \,\mathrm {log}\left (e x +d \right ) a^{2} e^{8}+1680 \,\mathrm {log}\left (e x +d \right ) a c \,d^{4} e^{4}+840 \,\mathrm {log}\left (e x +d \right ) c^{2} d^{8}-1680 a c \,d^{3} e^{5} x +840 a c \,d^{2} e^{6} x^{2}-560 a c d \,e^{7} x^{3}+420 a c \,e^{8} x^{4}-840 c^{2} d^{7} e x +420 c^{2} d^{6} e^{2} x^{2}-280 c^{2} d^{5} e^{3} x^{3}+210 c^{2} d^{4} e^{4} x^{4}-168 c^{2} d^{3} e^{5} x^{5}+140 c^{2} d^{2} e^{6} x^{6}-120 c^{2} d \,e^{7} x^{7}+105 c^{2} e^{8} x^{8}}{840 e^{9}} \] Input:
int((c*x^4+a)^2/(e*x+d),x)
Output:
(840*log(d + e*x)*a**2*e**8 + 1680*log(d + e*x)*a*c*d**4*e**4 + 840*log(d + e*x)*c**2*d**8 - 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*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)