Integrand size = 17, antiderivative size = 188 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {\left (c d^2+a e^2\right )^3}{8 e^7 (d+e x)^8}+\frac {6 c d \left (c d^2+a e^2\right )^2}{7 e^7 (d+e x)^7}-\frac {c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^6}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{5 e^7 (d+e x)^5}-\frac {3 c^2 \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac {2 c^3 d}{e^7 (d+e x)^3}-\frac {c^3}{2 e^7 (d+e x)^2} \]
-1/8*(a*e^2+c*d^2)^3/e^7/(e*x+d)^8+6/7*c*d*(a*e^2+c*d^2)^2/e^7/(e*x+d)^7-1 /2*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)/e^7/(e*x+d)^6+4/5*c^2*d*(3*a*e^2+5*c*d^ 2)/e^7/(e*x+d)^5-3/4*c^2*(a*e^2+5*c*d^2)/e^7/(e*x+d)^4+2*c^3*d/e^7/(e*x+d) ^3-1/2*c^3/e^7/(e*x+d)^2
Time = 0.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {35 a^3 e^6+5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+3 a c^2 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{280 e^7 (d+e x)^8} \]
-1/280*(35*a^3*e^6 + 5*a^2*c*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*a*c^2*e^ 2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*c^3*( d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d* e^5*x^5 + 28*e^6*x^6))/(e^7*(d + e*x)^8)
Time = 0.34 (sec) , antiderivative size = 188, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {3 c^2 \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^5}-\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^6 (d+e x)^6}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^7}-\frac {6 c d \left (a e^2+c d^2\right )^2}{e^6 (d+e x)^8}+\frac {\left (a e^2+c d^2\right )^3}{e^6 (d+e x)^9}+\frac {c^3}{e^6 (d+e x)^3}-\frac {6 c^3 d}{e^6 (d+e x)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 c^2 \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}-\frac {c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^6}+\frac {6 c d \left (a e^2+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac {\left (a e^2+c d^2\right )^3}{8 e^7 (d+e x)^8}-\frac {c^3}{2 e^7 (d+e x)^2}+\frac {2 c^3 d}{e^7 (d+e x)^3}\) |
-1/8*(c*d^2 + a*e^2)^3/(e^7*(d + e*x)^8) + (6*c*d*(c*d^2 + a*e^2)^2)/(7*e^ 7*(d + e*x)^7) - (c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(2*e^7*(d + e*x)^6) + (4*c^2*d*(5*c*d^2 + 3*a*e^2))/(5*e^7*(d + e*x)^5) - (3*c^2*(5*c*d^2 + a *e^2))/(4*e^7*(d + e*x)^4) + (2*c^3*d)/(e^7*(d + e*x)^3) - c^3/(2*e^7*(d + e*x)^2)
3.5.86.3.1 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 = 2.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {c^{3} d \,x^{5}}{e^{2}}-\frac {c^{2} \left (3 e^{2} a +5 c \,d^{2}\right ) x^{4}}{4 e^{3}}-\frac {d \,c^{2} \left (3 e^{2} a +5 c \,d^{2}\right ) x^{3}}{5 e^{4}}-\frac {c \left (5 a^{2} e^{4}+3 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x^{2}}{10 e^{5}}-\frac {d c \left (5 a^{2} e^{4}+3 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x}{35 e^{6}}-\frac {35 e^{6} a^{3}+5 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}}{280 e^{7}}}{\left (e x +d \right )^{8}}\) | \(199\) |
| gosper | \(-\frac {140 x^{6} c^{3} e^{6}+280 x^{5} c^{3} d \,e^{5}+210 x^{4} a \,c^{2} e^{6}+350 x^{4} c^{3} d^{2} e^{4}+168 x^{3} a \,c^{2} d \,e^{5}+280 x^{3} c^{3} d^{3} e^{3}+140 x^{2} a^{2} c \,e^{6}+84 x^{2} a \,c^{2} d^{2} e^{4}+140 x^{2} c^{3} d^{4} e^{2}+40 x \,a^{2} c d \,e^{5}+24 x a \,c^{2} d^{3} e^{3}+40 x \,c^{3} d^{5} e +35 e^{6} a^{3}+5 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}}{280 e^{7} \left (e x +d \right )^{8}}\) | \(205\) |
| parallelrisch | \(\frac {-140 c^{3} x^{6} e^{7}-280 c^{3} d \,x^{5} e^{6}-210 a \,c^{2} e^{7} x^{4}-350 c^{3} d^{2} e^{5} x^{4}-168 a \,c^{2} d \,e^{6} x^{3}-280 c^{3} d^{3} e^{4} x^{3}-140 a^{2} c \,e^{7} x^{2}-84 a \,c^{2} d^{2} e^{5} x^{2}-140 c^{3} d^{4} e^{3} x^{2}-40 a^{2} c d \,e^{6} x -24 a \,c^{2} d^{3} e^{4} x -40 c^{3} d^{5} e^{2} x -35 a^{3} e^{7}-5 a^{2} c \,d^{2} e^{5}-3 a \,c^{2} d^{4} e^{3}-5 c^{3} d^{6} e}{280 e^{8} \left (e x +d \right )^{8}}\) | \(208\) |
| norman | \(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {c^{3} d \,x^{5}}{e^{2}}-\frac {\left (3 e^{3} c^{2} a +5 d^{2} e \,c^{3}\right ) x^{4}}{4 e^{4}}-\frac {d \left (3 e^{3} c^{2} a +5 d^{2} e \,c^{3}\right ) x^{3}}{5 e^{5}}-\frac {\left (5 e^{5} a^{2} c +3 d^{2} e^{3} c^{2} a +5 d^{4} e \,c^{3}\right ) x^{2}}{10 e^{6}}-\frac {d \left (5 e^{5} a^{2} c +3 d^{2} e^{3} c^{2} a +5 d^{4} e \,c^{3}\right ) x}{35 e^{7}}-\frac {35 a^{3} e^{7}+5 a^{2} c \,d^{2} e^{5}+3 a \,c^{2} d^{4} e^{3}+5 c^{3} d^{6} e}{280 e^{8}}}{\left (e x +d \right )^{8}}\) | \(212\) |
| default | \(\frac {6 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{6}}+\frac {2 c^{3} d}{e^{7} \left (e x +d \right )^{3}}-\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {c^{3}}{2 e^{7} \left (e x +d \right )^{2}}+\frac {4 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}\) | \(218\) |
(-1/2*c^3*x^6/e-c^3*d*x^5/e^2-1/4*c^2*(3*a*e^2+5*c*d^2)/e^3*x^4-1/5*d*c^2/ e^4*(3*a*e^2+5*c*d^2)*x^3-1/10*c/e^5*(5*a^2*e^4+3*a*c*d^2*e^2+5*c^2*d^4)*x ^2-1/35*d*c/e^6*(5*a^2*e^4+3*a*c*d^2*e^2+5*c^2*d^4)*x-1/280/e^7*(35*a^3*e^ 6+5*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2+5*c^3*d^6))/(e*x+d)^8
Time = 0.43 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 5 \, a^{2} c e^{6}\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{280 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4 + 35*a^3*e^6 + 70*(5*c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 + 56 *(5*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 28*(5*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4 + 5*a^2*c*e^6)*x^2 + 8*(5*c^3*d^5*e + 3*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*x) /(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^1 1*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)
Timed out. \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4} + 5 \, a^{2} c e^{6}\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{280 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4 + 35*a^3*e^6 + 70*(5*c^3*d^2*e^4 + 3*a*c^2*e^6)*x^4 + 56 *(5*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 28*(5*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4 + 5*a^2*c*e^6)*x^2 + 8*(5*c^3*d^5*e + 3*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*x) /(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^1 1*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)
Time = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 350 \, c^{3} d^{2} e^{4} x^{4} + 210 \, a c^{2} e^{6} x^{4} + 280 \, c^{3} d^{3} e^{3} x^{3} + 168 \, a c^{2} d e^{5} x^{3} + 140 \, c^{3} d^{4} e^{2} x^{2} + 84 \, a c^{2} d^{2} e^{4} x^{2} + 140 \, a^{2} c e^{6} x^{2} + 40 \, c^{3} d^{5} e x + 24 \, a c^{2} d^{3} e^{3} x + 40 \, a^{2} c d e^{5} x + 5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 5 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6}}{280 \, {\left (e x + d\right )}^{8} e^{7}} \]
-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 350*c^3*d^2*e^4*x^4 + 210*a* c^2*e^6*x^4 + 280*c^3*d^3*e^3*x^3 + 168*a*c^2*d*e^5*x^3 + 140*c^3*d^4*e^2* x^2 + 84*a*c^2*d^2*e^4*x^2 + 140*a^2*c*e^6*x^2 + 40*c^3*d^5*e*x + 24*a*c^2 *d^3*e^3*x + 40*a^2*c*d*e^5*x + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2* e^4 + 35*a^3*e^6)/((e*x + d)^8*e^7)
Time = 9.40 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {\frac {35\,a^3\,e^6+5\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+5\,c^3\,d^6}{280\,e^7}+\frac {c^3\,x^6}{2\,e}+\frac {c^3\,d\,x^5}{e^2}+\frac {c^2\,x^4\,\left (5\,c\,d^2+3\,a\,e^2\right )}{4\,e^3}+\frac {c\,x^2\,\left (5\,a^2\,e^4+3\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{10\,e^5}+\frac {c\,d\,x\,\left (5\,a^2\,e^4+3\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{35\,e^6}+\frac {c^2\,d\,x^3\,\left (5\,c\,d^2+3\,a\,e^2\right )}{5\,e^4}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
-((35*a^3*e^6 + 5*c^3*d^6 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4)/(280*e^7) + (c^3*x^6)/(2*e) + (c^3*d*x^5)/e^2 + (c^2*x^4*(3*a*e^2 + 5*c*d^2))/(4*e^3) + (c*x^2*(5*a^2*e^4 + 5*c^2*d^4 + 3*a*c*d^2*e^2))/(10*e^5) + (c*d*x*(5*a^ 2*e^4 + 5*c^2*d^4 + 3*a*c*d^2*e^2))/(35*e^6) + (c^2*d*x^3*(3*a*e^2 + 5*c*d ^2))/(5*e^4))/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5*e^3*x ^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)