Integrand size = 17, antiderivative size = 178 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {\left (c d^2+a e^2\right )^3}{7 e^7 (d+e x)^7}+\frac {c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^6}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{5 e^7 (d+e x)^5}+\frac {c^2 d \left (5 c d^2+3 a e^2\right )}{e^7 (d+e x)^4}-\frac {c^2 \left (5 c d^2+a e^2\right )}{e^7 (d+e x)^3}+\frac {3 c^3 d}{e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)} \]
-1/7*(a*e^2+c*d^2)^3/e^7/(e*x+d)^7+c*d*(a*e^2+c*d^2)^2/e^7/(e*x+d)^6-3/5*c *(a*e^2+c*d^2)*(a*e^2+5*c*d^2)/e^7/(e*x+d)^5+c^2*d*(3*a*e^2+5*c*d^2)/e^7/( e*x+d)^4-c^2*(a*e^2+5*c*d^2)/e^7/(e*x+d)^3+3*c^3*d/e^7/(e*x+d)^2-c^3/e^7/( e*x+d)
Time = 0.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {5 a^3 e^6+a^2 c e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a c^2 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )}{35 e^7 (d+e x)^7} \]
-1/35*(5*a^3*e^6 + a^2*c*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a*c^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^ 5 + 7*e^6*x^6))/(e^7*(d + e*x)^7)
Time = 0.35 (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 = {476, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {3 c^2 \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^4}-\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^6 (d+e x)^5}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^6}-\frac {6 c d \left (a e^2+c d^2\right )^2}{e^6 (d+e x)^7}+\frac {\left (a e^2+c d^2\right )^3}{e^6 (d+e x)^8}+\frac {c^3}{e^6 (d+e x)^2}-\frac {6 c^3 d}{e^6 (d+e x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^2 \left (a e^2+5 c d^2\right )}{e^7 (d+e x)^3}+\frac {c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)^4}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac {c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^6}-\frac {\left (a e^2+c d^2\right )^3}{7 e^7 (d+e x)^7}-\frac {c^3}{e^7 (d+e x)}+\frac {3 c^3 d}{e^7 (d+e x)^2}\) |
-1/7*(c*d^2 + a*e^2)^3/(e^7*(d + e*x)^7) + (c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)^6) - (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(5*e^7*(d + e*x)^5) + (c^2*d*(5*c*d^2 + 3*a*e^2))/(e^7*(d + e*x)^4) - (c^2*(5*c*d^2 + a*e^2))/( e^7*(d + e*x)^3) + (3*c^3*d)/(e^7*(d + e*x)^2) - c^3/(e^7*(d + e*x))
3.5.85.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 2.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 c^{3} d \,x^{5}}{e^{2}}-\frac {c^{2} \left (e^{2} a +5 c \,d^{2}\right ) x^{4}}{e^{3}}-\frac {d \,c^{2} \left (e^{2} a +5 c \,d^{2}\right ) x^{3}}{e^{4}}-\frac {3 c \left (a^{2} e^{4}+a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x^{2}}{5 e^{5}}-\frac {d c \left (a^{2} e^{4}+a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x}{5 e^{6}}-\frac {5 e^{6} a^{3}+d^{2} e^{4} a^{2} c +d^{4} e^{2} c^{2} a +5 c^{3} d^{6}}{35 e^{7}}}{\left (e x +d \right )^{7}}\) | \(191\) |
| norman | \(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 c^{3} d \,x^{5}}{e^{2}}-\frac {\left (e^{2} c^{2} a +5 c^{3} d^{2}\right ) x^{4}}{e^{3}}-\frac {d \left (e^{2} c^{2} a +5 c^{3} d^{2}\right ) x^{3}}{e^{4}}-\frac {3 \left (e^{4} a^{2} c +d^{2} e^{2} c^{2} a +5 d^{4} c^{3}\right ) x^{2}}{5 e^{5}}-\frac {d \left (e^{4} a^{2} c +d^{2} e^{2} c^{2} a +5 d^{4} c^{3}\right ) x}{5 e^{6}}-\frac {5 e^{6} a^{3}+d^{2} e^{4} a^{2} c +d^{4} e^{2} c^{2} a +5 c^{3} d^{6}}{35 e^{7}}}{\left (e x +d \right )^{7}}\) | \(199\) |
| gosper | \(-\frac {35 x^{6} c^{3} e^{6}+105 x^{5} c^{3} d \,e^{5}+35 x^{4} a \,c^{2} e^{6}+175 x^{4} c^{3} d^{2} e^{4}+35 x^{3} a \,c^{2} d \,e^{5}+175 x^{3} c^{3} d^{3} e^{3}+21 x^{2} a^{2} c \,e^{6}+21 x^{2} a \,c^{2} d^{2} e^{4}+105 x^{2} c^{3} d^{4} e^{2}+7 x \,a^{2} c d \,e^{5}+7 x a \,c^{2} d^{3} e^{3}+35 x \,c^{3} d^{5} e +5 e^{6} a^{3}+d^{2} e^{4} a^{2} c +d^{4} e^{2} c^{2} a +5 c^{3} d^{6}}{35 \left (e x +d \right )^{7} e^{7}}\) | \(203\) |
| parallelrisch | \(\frac {-35 x^{6} c^{3} e^{6}-105 x^{5} c^{3} d \,e^{5}-35 x^{4} a \,c^{2} e^{6}-175 x^{4} c^{3} d^{2} e^{4}-35 x^{3} a \,c^{2} d \,e^{5}-175 x^{3} c^{3} d^{3} e^{3}-21 x^{2} a^{2} c \,e^{6}-21 x^{2} a \,c^{2} d^{2} e^{4}-105 x^{2} c^{3} d^{4} e^{2}-7 x \,a^{2} c d \,e^{5}-7 x a \,c^{2} d^{3} e^{3}-35 x \,c^{3} d^{5} e -5 e^{6} a^{3}-d^{2} e^{4} a^{2} c -d^{4} e^{2} c^{2} a -5 c^{3} d^{6}}{35 e^{7} \left (e x +d \right )^{7}}\) | \(205\) |
| default | \(-\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{7 e^{7} \left (e x +d \right )^{7}}+\frac {c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{7} \left (e x +d \right )^{6}}-\frac {c^{3}}{e^{7} \left (e x +d \right )}-\frac {c^{2} \left (e^{2} a +5 c \,d^{2}\right )}{e^{7} \left (e x +d \right )^{3}}+\frac {c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{e^{7} \left (e x +d \right )^{4}}+\frac {3 c^{3} d}{e^{7} \left (e x +d \right )^{2}}-\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{5 e^{7} \left (e x +d \right )^{5}}\) | \(216\) |
(-c^3*x^6/e-3*c^3*d*x^5/e^2-c^2*(a*e^2+5*c*d^2)/e^3*x^4-d*c^2*(a*e^2+5*c*d ^2)/e^4*x^3-3/5*c*(a^2*e^4+a*c*d^2*e^2+5*c^2*d^4)/e^5*x^2-1/5*d*c*(a^2*e^4 +a*c*d^2*e^2+5*c^2*d^4)/e^6*x-1/35*(5*a^3*e^6+a^2*c*d^2*e^4+a*c^2*d^4*e^2+ 5*c^3*d^6)/e^7)/(e*x+d)^7
Time = 0.27 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \, {\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \, {\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \, {\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \, {\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]
-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^ 2*c*d^2*e^4 + 5*a^3*e^6 + 35*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^4 + 35*(5*c^3*d ^3*e^3 + a*c^2*d*e^5)*x^3 + 21*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4 + a^2*c*e^6) *x^2 + 7*(5*c^3*d^5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^14*x^7 + 7*d*e^ 13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9* x^2 + 7*d^6*e^8*x + d^7*e^7)
Time = 125.02 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {- 5 a^{3} e^{6} - a^{2} c d^{2} e^{4} - a c^{2} d^{4} e^{2} - 5 c^{3} d^{6} - 105 c^{3} d e^{5} x^{5} - 35 c^{3} e^{6} x^{6} + x^{4} \left (- 35 a c^{2} e^{6} - 175 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 35 a c^{2} d e^{5} - 175 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 21 a^{2} c e^{6} - 21 a c^{2} d^{2} e^{4} - 105 c^{3} d^{4} e^{2}\right ) + x \left (- 7 a^{2} c d e^{5} - 7 a c^{2} d^{3} e^{3} - 35 c^{3} d^{5} e\right )}{35 d^{7} e^{7} + 245 d^{6} e^{8} x + 735 d^{5} e^{9} x^{2} + 1225 d^{4} e^{10} x^{3} + 1225 d^{3} e^{11} x^{4} + 735 d^{2} e^{12} x^{5} + 245 d e^{13} x^{6} + 35 e^{14} x^{7}} \]
(-5*a**3*e**6 - a**2*c*d**2*e**4 - a*c**2*d**4*e**2 - 5*c**3*d**6 - 105*c* *3*d*e**5*x**5 - 35*c**3*e**6*x**6 + x**4*(-35*a*c**2*e**6 - 175*c**3*d**2 *e**4) + x**3*(-35*a*c**2*d*e**5 - 175*c**3*d**3*e**3) + x**2*(-21*a**2*c* e**6 - 21*a*c**2*d**2*e**4 - 105*c**3*d**4*e**2) + x*(-7*a**2*c*d*e**5 - 7 *a*c**2*d**3*e**3 - 35*c**3*d**5*e))/(35*d**7*e**7 + 245*d**6*e**8*x + 735 *d**5*e**9*x**2 + 1225*d**4*e**10*x**3 + 1225*d**3*e**11*x**4 + 735*d**2*e **12*x**5 + 245*d*e**13*x**6 + 35*e**14*x**7)
Time = 0.21 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 35 \, {\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{4} + 35 \, {\left (5 \, c^{3} d^{3} e^{3} + a c^{2} d e^{5}\right )} x^{3} + 21 \, {\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 7 \, {\left (5 \, c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{35 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]
-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^ 2*c*d^2*e^4 + 5*a^3*e^6 + 35*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^4 + 35*(5*c^3*d ^3*e^3 + a*c^2*d*e^5)*x^3 + 21*(5*c^3*d^4*e^2 + a*c^2*d^2*e^4 + a^2*c*e^6) *x^2 + 7*(5*c^3*d^5*e + a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/(e^14*x^7 + 7*d*e^ 13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9* x^2 + 7*d^6*e^8*x + d^7*e^7)
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {35 \, c^{3} e^{6} x^{6} + 105 \, c^{3} d e^{5} x^{5} + 175 \, c^{3} d^{2} e^{4} x^{4} + 35 \, a c^{2} e^{6} x^{4} + 175 \, c^{3} d^{3} e^{3} x^{3} + 35 \, a c^{2} d e^{5} x^{3} + 105 \, c^{3} d^{4} e^{2} x^{2} + 21 \, a c^{2} d^{2} e^{4} x^{2} + 21 \, a^{2} c e^{6} x^{2} + 35 \, c^{3} d^{5} e x + 7 \, a c^{2} d^{3} e^{3} x + 7 \, a^{2} c d e^{5} x + 5 \, c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6}}{35 \, {\left (e x + d\right )}^{7} e^{7}} \]
-1/35*(35*c^3*e^6*x^6 + 105*c^3*d*e^5*x^5 + 175*c^3*d^2*e^4*x^4 + 35*a*c^2 *e^6*x^4 + 175*c^3*d^3*e^3*x^3 + 35*a*c^2*d*e^5*x^3 + 105*c^3*d^4*e^2*x^2 + 21*a*c^2*d^2*e^4*x^2 + 21*a^2*c*e^6*x^2 + 35*c^3*d^5*e*x + 7*a*c^2*d^3*e ^3*x + 7*a^2*c*d*e^5*x + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + 5*a^3 *e^6)/((e*x + d)^7*e^7)
Time = 0.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {\frac {5\,a^3\,e^6+a^2\,c\,d^2\,e^4+a\,c^2\,d^4\,e^2+5\,c^3\,d^6}{35\,e^7}+\frac {c^3\,x^6}{e}+\frac {3\,c^3\,d\,x^5}{e^2}+\frac {c^2\,x^4\,\left (5\,c\,d^2+a\,e^2\right )}{e^3}+\frac {3\,c\,x^2\,\left (a^2\,e^4+a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{5\,e^5}+\frac {c\,d\,x\,\left (a^2\,e^4+a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{5\,e^6}+\frac {c^2\,d\,x^3\,\left (5\,c\,d^2+a\,e^2\right )}{e^4}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]
-((5*a^3*e^6 + 5*c^3*d^6 + a*c^2*d^4*e^2 + a^2*c*d^2*e^4)/(35*e^7) + (c^3* x^6)/e + (3*c^3*d*x^5)/e^2 + (c^2*x^4*(a*e^2 + 5*c*d^2))/e^3 + (3*c*x^2*(a ^2*e^4 + 5*c^2*d^4 + a*c*d^2*e^2))/(5*e^5) + (c*d*x*(a^2*e^4 + 5*c^2*d^4 + a*c*d^2*e^2))/(5*e^6) + (c^2*d*x^3*(a*e^2 + 5*c*d^2))/e^4)/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2 *e^5*x^5 + 7*d^6*e*x)