Integrand size = 26, antiderivative size = 117 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {(b d-a e)^4}{8 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac {b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac {4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac {b^4}{4 e^5 (d+e x)^4} \]
-1/8*(-a*e+b*d)^4/e^5/(e*x+d)^8+4/7*b*(-a*e+b*d)^3/e^5/(e*x+d)^7-b^2*(-a*e +b*d)^2/e^5/(e*x+d)^6+4/5*b^3*(-a*e+b*d)/e^5/(e*x+d)^5-1/4*b^4/e^5/(e*x+d) ^4
Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a b^3 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )}{280 e^5 (d+e x)^8} \]
-1/280*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 + 8*d* e*x + 28*e^2*x^2) + 4*a*b^3*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3 ) + b^4*(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))/(e ^5*(d + e*x)^8)
Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 27, 53, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx\) |
\(\Big \downarrow \) 1098 |
\(\displaystyle \frac {\int \frac {b^4 (a+b x)^4}{(d+e x)^9}dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^4}{(d+e x)^9}dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^6}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^7}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^8}+\frac {(a e-b d)^4}{e^4 (d+e x)^9}+\frac {b^4}{e^4 (d+e x)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac {b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac {4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac {(b d-a e)^4}{8 e^5 (d+e x)^8}-\frac {b^4}{4 e^5 (d+e x)^4}\) |
-1/8*(b*d - a*e)^4/(e^5*(d + e*x)^8) + (4*b*(b*d - a*e)^3)/(7*e^5*(d + e*x )^7) - (b^2*(b*d - a*e)^2)/(e^5*(d + e*x)^6) + (4*b^3*(b*d - a*e))/(5*e^5* (d + e*x)^5) - b^4/(4*e^5*(d + e*x)^4)
3.15.78.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 2.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.46
method | result | size |
risch | \(\frac {-\frac {b^{4} x^{4}}{4 e}-\frac {b^{3} \left (4 a e +b d \right ) x^{3}}{5 e^{2}}-\frac {b^{2} \left (10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) x^{2}}{10 e^{3}}-\frac {b \left (20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{35 e^{4}}-\frac {35 e^{4} a^{4}+20 b \,e^{3} d \,a^{3}+10 b^{2} e^{2} d^{2} a^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}}{280 e^{5}}}{\left (e x +d \right )^{8}}\) | \(171\) |
gosper | \(-\frac {70 b^{4} x^{4} e^{4}+224 x^{3} a \,b^{3} e^{4}+56 x^{3} b^{4} d \,e^{3}+280 x^{2} a^{2} b^{2} e^{4}+112 x^{2} a \,b^{3} d \,e^{3}+28 x^{2} b^{4} d^{2} e^{2}+160 x \,a^{3} b \,e^{4}+80 x \,a^{2} b^{2} d \,e^{3}+32 x a \,b^{3} d^{2} e^{2}+8 x \,b^{4} d^{3} e +35 e^{4} a^{4}+20 b \,e^{3} d \,a^{3}+10 b^{2} e^{2} d^{2} a^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}}{280 e^{5} \left (e x +d \right )^{8}}\) | \(185\) |
default | \(-\frac {4 b^{3} \left (a e -b d \right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{8 e^{5} \left (e x +d \right )^{8}}-\frac {4 b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{4}}{4 e^{5} \left (e x +d \right )^{4}}-\frac {b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{5} \left (e x +d \right )^{6}}\) | \(186\) |
parallelrisch | \(\frac {-70 b^{4} x^{4} e^{7}-224 a \,b^{3} e^{7} x^{3}-56 b^{4} d \,e^{6} x^{3}-280 a^{2} b^{2} e^{7} x^{2}-112 a \,b^{3} d \,e^{6} x^{2}-28 b^{4} d^{2} e^{5} x^{2}-160 a^{3} b \,e^{7} x -80 a^{2} b^{2} d \,e^{6} x -32 a \,b^{3} d^{2} e^{5} x -8 b^{4} d^{3} e^{4} x -35 e^{7} a^{4}-20 a^{3} b d \,e^{6}-10 a^{2} b^{2} d^{2} e^{5}-4 a \,b^{3} d^{3} e^{4}-b^{4} d^{4} e^{3}}{280 e^{8} \left (e x +d \right )^{8}}\) | \(193\) |
norman | \(\frac {-\frac {b^{4} x^{4}}{4 e}-\frac {\left (4 a \,b^{3} e^{4}+b^{4} d \,e^{3}\right ) x^{3}}{5 e^{5}}-\frac {\left (10 a^{2} b^{2} e^{5}+4 a \,b^{3} d \,e^{4}+b^{4} d^{2} e^{3}\right ) x^{2}}{10 e^{6}}-\frac {\left (20 a^{3} b \,e^{6}+10 a^{2} b^{2} d \,e^{5}+4 a \,b^{3} d^{2} e^{4}+b^{4} d^{3} e^{3}\right ) x}{35 e^{7}}-\frac {35 e^{7} a^{4}+20 a^{3} b d \,e^{6}+10 a^{2} b^{2} d^{2} e^{5}+4 a \,b^{3} d^{3} e^{4}+b^{4} d^{4} e^{3}}{280 e^{8}}}{\left (e x +d \right )^{8}}\) | \(197\) |
(-1/4/e*b^4*x^4-1/5*b^3/e^2*(4*a*e+b*d)*x^3-1/10*b^2/e^3*(10*a^2*e^2+4*a*b *d*e+b^2*d^2)*x^2-1/35*b/e^4*(20*a^3*e^3+10*a^2*b*d*e^2+4*a*b^2*d^2*e+b^3* d^3)*x-1/280/e^5*(35*a^4*e^4+20*a^3*b*d*e^3+10*a^2*b^2*d^2*e^2+4*a*b^3*d^3 *e+b^4*d^4))/(e*x+d)^8
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (109) = 218\).
Time = 0.31 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \, {\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \, {\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \]
-1/280*(70*b^4*e^4*x^4 + b^4*d^4 + 4*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 + 20 *a^3*b*d*e^3 + 35*a^4*e^4 + 56*(b^4*d*e^3 + 4*a*b^3*e^4)*x^3 + 28*(b^4*d^2 *e^2 + 4*a*b^3*d*e^3 + 10*a^2*b^2*e^4)*x^2 + 8*(b^4*d^3*e + 4*a*b^3*d^2*e^ 2 + 10*a^2*b^2*d*e^3 + 20*a^3*b*e^4)*x)/(e^13*x^8 + 8*d*e^12*x^7 + 28*d^2* e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^4*e^9*x^4 + 56*d^5*e^8*x^3 + 28*d^6*e^7* x^2 + 8*d^7*e^6*x + d^8*e^5)
Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (109) = 218\).
Time = 0.20 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {70 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \, {\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \, {\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \]
-1/280*(70*b^4*e^4*x^4 + b^4*d^4 + 4*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 + 20 *a^3*b*d*e^3 + 35*a^4*e^4 + 56*(b^4*d*e^3 + 4*a*b^3*e^4)*x^3 + 28*(b^4*d^2 *e^2 + 4*a*b^3*d*e^3 + 10*a^2*b^2*e^4)*x^2 + 8*(b^4*d^3*e + 4*a*b^3*d^2*e^ 2 + 10*a^2*b^2*d*e^3 + 20*a^3*b*e^4)*x)/(e^13*x^8 + 8*d*e^12*x^7 + 28*d^2* e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^4*e^9*x^4 + 56*d^5*e^8*x^3 + 28*d^6*e^7* x^2 + 8*d^7*e^6*x + d^8*e^5)
Time = 0.25 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {70 \, b^{4} e^{4} x^{4} + 56 \, b^{4} d e^{3} x^{3} + 224 \, a b^{3} e^{4} x^{3} + 28 \, b^{4} d^{2} e^{2} x^{2} + 112 \, a b^{3} d e^{3} x^{2} + 280 \, a^{2} b^{2} e^{4} x^{2} + 8 \, b^{4} d^{3} e x + 32 \, a b^{3} d^{2} e^{2} x + 80 \, a^{2} b^{2} d e^{3} x + 160 \, a^{3} b e^{4} x + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}}{280 \, {\left (e x + d\right )}^{8} e^{5}} \]
-1/280*(70*b^4*e^4*x^4 + 56*b^4*d*e^3*x^3 + 224*a*b^3*e^4*x^3 + 28*b^4*d^2 *e^2*x^2 + 112*a*b^3*d*e^3*x^2 + 280*a^2*b^2*e^4*x^2 + 8*b^4*d^3*e*x + 32* a*b^3*d^2*e^2*x + 80*a^2*b^2*d*e^3*x + 160*a^3*b*e^4*x + b^4*d^4 + 4*a*b^3 *d^3*e + 10*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 35*a^4*e^4)/((e*x + d)^8*e^ 5)
Time = 9.68 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {\frac {35\,a^4\,e^4+20\,a^3\,b\,d\,e^3+10\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+b^4\,d^4}{280\,e^5}+\frac {b^4\,x^4}{4\,e}+\frac {b^3\,x^3\,\left (4\,a\,e+b\,d\right )}{5\,e^2}+\frac {b\,x\,\left (20\,a^3\,e^3+10\,a^2\,b\,d\,e^2+4\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{35\,e^4}+\frac {b^2\,x^2\,\left (10\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )}{10\,e^3}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
-((35*a^4*e^4 + b^4*d^4 + 10*a^2*b^2*d^2*e^2 + 4*a*b^3*d^3*e + 20*a^3*b*d* e^3)/(280*e^5) + (b^4*x^4)/(4*e) + (b^3*x^3*(4*a*e + b*d))/(5*e^2) + (b*x* (20*a^3*e^3 + b^3*d^3 + 4*a*b^2*d^2*e + 10*a^2*b*d*e^2))/(35*e^4) + (b^2*x ^2*(10*a^2*e^2 + b^2*d^2 + 4*a*b*d*e))/(10*e^3))/(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)