Integrand size = 26, antiderivative size = 89 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=\frac {(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac {b (a+b x)^5}{21 (b d-a e)^2 (d+e x)^6}+\frac {b^2 (a+b x)^5}{105 (b d-a e)^3 (d+e x)^5} \]
1/7*(b*x+a)^5/(-a*e+b*d)/(e*x+d)^7+1/21*b*(b*x+a)^5/(-a*e+b*d)^2/(e*x+d)^6 +1/105*b^2*(b*x+a)^5/(-a*e+b*d)^3/(e*x+d)^5
Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b^3 e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{105 e^5 (d+e x)^7} \]
-1/105*(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b^2*e^2*(d^2 + 7*d*e *x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(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))/(e^ 5*(d + e*x)^7)
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1098, 27, 55, 55, 48}
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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx\) |
\(\Big \downarrow \) 1098 |
\(\displaystyle \frac {\int \frac {b^4 (a+b x)^4}{(d+e x)^8}dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^4}{(d+e x)^8}dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2 b \int \frac {(a+b x)^4}{(d+e x)^7}dx}{7 (b d-a e)}+\frac {(a+b x)^5}{7 (d+e x)^7 (b d-a e)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2 b \left (\frac {b \int \frac {(a+b x)^4}{(d+e x)^6}dx}{6 (b d-a e)}+\frac {(a+b x)^5}{6 (d+e x)^6 (b d-a e)}\right )}{7 (b d-a e)}+\frac {(a+b x)^5}{7 (d+e x)^7 (b d-a e)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(a+b x)^5}{7 (d+e x)^7 (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac {(a+b x)^5}{6 (d+e x)^6 (b d-a e)}\right )}{7 (b d-a e)}\) |
(a + b*x)^5/(7*(b*d - a*e)*(d + e*x)^7) + (2*b*((a + b*x)^5/(6*(b*d - a*e) *(d + e*x)^6) + (b*(a + b*x)^5)/(30*(b*d - a*e)^2*(d + e*x)^5)))/(7*(b*d - a*e))
3.15.77.3.1 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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(83)=166\).
Time = 2.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.92
method | result | size |
risch | \(\frac {-\frac {b^{4} x^{4}}{3 e}-\frac {b^{3} \left (3 a e +b d \right ) x^{3}}{3 e^{2}}-\frac {b^{2} \left (6 a^{2} e^{2}+3 a b d e +b^{2} d^{2}\right ) x^{2}}{5 e^{3}}-\frac {b \left (10 a^{3} e^{3}+6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{15 e^{4}}-\frac {15 e^{4} a^{4}+10 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}+3 a \,b^{3} d^{3} e +b^{4} d^{4}}{105 e^{5}}}{\left (e x +d \right )^{7}}\) | \(171\) |
gosper | \(-\frac {35 b^{4} x^{4} e^{4}+105 x^{3} a \,b^{3} e^{4}+35 x^{3} b^{4} d \,e^{3}+126 x^{2} a^{2} b^{2} e^{4}+63 x^{2} a \,b^{3} d \,e^{3}+21 x^{2} b^{4} d^{2} e^{2}+70 x \,a^{3} b \,e^{4}+42 x \,a^{2} b^{2} d \,e^{3}+21 x a \,b^{3} d^{2} e^{2}+7 x \,b^{4} d^{3} e +15 e^{4} a^{4}+10 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}+3 a \,b^{3} d^{3} e +b^{4} d^{4}}{105 e^{5} \left (e x +d \right )^{7}}\) | \(185\) |
default | \(-\frac {6 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{4}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{3} \left (a e -b d \right )}{e^{5} \left (e x +d \right )^{4}}-\frac {2 b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 e^{5} \left (e x +d \right )^{6}}\) | \(186\) |
parallelrisch | \(\frac {-35 b^{4} x^{4} e^{6}-105 a \,b^{3} e^{6} x^{3}-35 b^{4} d \,e^{5} x^{3}-126 a^{2} b^{2} e^{6} x^{2}-63 a \,b^{3} d \,e^{5} x^{2}-21 b^{4} d^{2} e^{4} x^{2}-70 a^{3} b \,e^{6} x -42 a^{2} b^{2} d \,e^{5} x -21 a \,b^{3} d^{2} e^{4} x -7 b^{4} d^{3} e^{3} x -15 a^{4} e^{6}-10 a^{3} b d \,e^{5}-6 a^{2} b^{2} d^{2} e^{4}-3 a \,b^{3} d^{3} e^{3}-b^{4} d^{4} e^{2}}{105 e^{7} \left (e x +d \right )^{7}}\) | \(193\) |
norman | \(\frac {-\frac {b^{4} x^{4}}{3 e}-\frac {\left (3 a \,b^{3} e^{3}+b^{4} d \,e^{2}\right ) x^{3}}{3 e^{4}}-\frac {\left (6 a^{2} b^{2} e^{4}+3 a \,b^{3} d \,e^{3}+b^{4} d^{2} e^{2}\right ) x^{2}}{5 e^{5}}-\frac {\left (10 a^{3} b \,e^{5}+6 a^{2} b^{2} d \,e^{4}+3 a \,b^{3} d^{2} e^{3}+b^{4} d^{3} e^{2}\right ) x}{15 e^{6}}-\frac {15 a^{4} e^{6}+10 a^{3} b d \,e^{5}+6 a^{2} b^{2} d^{2} e^{4}+3 a \,b^{3} d^{3} e^{3}+b^{4} d^{4} e^{2}}{105 e^{7}}}{\left (e x +d \right )^{7}}\) | \(197\) |
(-1/3/e*b^4*x^4-1/3/e^2*b^3*(3*a*e+b*d)*x^3-1/5*b^2/e^3*(6*a^2*e^2+3*a*b*d *e+b^2*d^2)*x^2-1/15/e^4*b*(10*a^3*e^3+6*a^2*b*d*e^2+3*a*b^2*d^2*e+b^3*d^3 )*x-1/105/e^5*(15*a^4*e^4+10*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2+3*a*b^3*d^3*e+b ^4*d^4))/(e*x+d)^7
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (83) = 166\).
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.78 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \, {\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \, {\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
-1/105*(35*b^4*e^4*x^4 + b^4*d^4 + 3*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 10* a^3*b*d*e^3 + 15*a^4*e^4 + 35*(b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 21*(b^4*d^2* e^2 + 3*a*b^3*d*e^3 + 6*a^2*b^2*e^4)*x^2 + 7*(b^4*d^3*e + 3*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3 + 10*a^3*b*e^4)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^1 0*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d ^7*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (73) = 146\).
Time = 114.33 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=\frac {- 15 a^{4} e^{4} - 10 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 3 a b^{3} d^{3} e - b^{4} d^{4} - 35 b^{4} e^{4} x^{4} + x^{3} \left (- 105 a b^{3} e^{4} - 35 b^{4} d e^{3}\right ) + x^{2} \left (- 126 a^{2} b^{2} e^{4} - 63 a b^{3} d e^{3} - 21 b^{4} d^{2} e^{2}\right ) + x \left (- 70 a^{3} b e^{4} - 42 a^{2} b^{2} d e^{3} - 21 a b^{3} d^{2} e^{2} - 7 b^{4} d^{3} e\right )}{105 d^{7} e^{5} + 735 d^{6} e^{6} x + 2205 d^{5} e^{7} x^{2} + 3675 d^{4} e^{8} x^{3} + 3675 d^{3} e^{9} x^{4} + 2205 d^{2} e^{10} x^{5} + 735 d e^{11} x^{6} + 105 e^{12} x^{7}} \]
(-15*a**4*e**4 - 10*a**3*b*d*e**3 - 6*a**2*b**2*d**2*e**2 - 3*a*b**3*d**3* e - b**4*d**4 - 35*b**4*e**4*x**4 + x**3*(-105*a*b**3*e**4 - 35*b**4*d*e** 3) + x**2*(-126*a**2*b**2*e**4 - 63*a*b**3*d*e**3 - 21*b**4*d**2*e**2) + x *(-70*a**3*b*e**4 - 42*a**2*b**2*d*e**3 - 21*a*b**3*d**2*e**2 - 7*b**4*d** 3*e))/(105*d**7*e**5 + 735*d**6*e**6*x + 2205*d**5*e**7*x**2 + 3675*d**4*e **8*x**3 + 3675*d**3*e**9*x**4 + 2205*d**2*e**10*x**5 + 735*d*e**11*x**6 + 105*e**12*x**7)
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (83) = 166\).
Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.78 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \, {\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \, {\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
-1/105*(35*b^4*e^4*x^4 + b^4*d^4 + 3*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 10* a^3*b*d*e^3 + 15*a^4*e^4 + 35*(b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 21*(b^4*d^2* e^2 + 3*a*b^3*d*e^3 + 6*a^2*b^2*e^4)*x^2 + 7*(b^4*d^3*e + 3*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3 + 10*a^3*b*e^4)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^1 0*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d ^7*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (83) = 166\).
Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {35 \, b^{4} e^{4} x^{4} + 35 \, b^{4} d e^{3} x^{3} + 105 \, a b^{3} e^{4} x^{3} + 21 \, b^{4} d^{2} e^{2} x^{2} + 63 \, a b^{3} d e^{3} x^{2} + 126 \, a^{2} b^{2} e^{4} x^{2} + 7 \, b^{4} d^{3} e x + 21 \, a b^{3} d^{2} e^{2} x + 42 \, a^{2} b^{2} d e^{3} x + 70 \, a^{3} b e^{4} x + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4}}{105 \, {\left (e x + d\right )}^{7} e^{5}} \]
-1/105*(35*b^4*e^4*x^4 + 35*b^4*d*e^3*x^3 + 105*a*b^3*e^4*x^3 + 21*b^4*d^2 *e^2*x^2 + 63*a*b^3*d*e^3*x^2 + 126*a^2*b^2*e^4*x^2 + 7*b^4*d^3*e*x + 21*a *b^3*d^2*e^2*x + 42*a^2*b^2*d*e^3*x + 70*a^3*b*e^4*x + b^4*d^4 + 3*a*b^3*d ^3*e + 6*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + 15*a^4*e^4)/((e*x + d)^7*e^5)
Time = 9.68 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.66 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\frac {15\,a^4\,e^4+10\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+3\,a\,b^3\,d^3\,e+b^4\,d^4}{105\,e^5}+\frac {b^4\,x^4}{3\,e}+\frac {b^3\,x^3\,\left (3\,a\,e+b\,d\right )}{3\,e^2}+\frac {b\,x\,\left (10\,a^3\,e^3+6\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{15\,e^4}+\frac {b^2\,x^2\,\left (6\,a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^3}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]
-((15*a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 + 3*a*b^3*d^3*e + 10*a^3*b*d*e ^3)/(105*e^5) + (b^4*x^4)/(3*e) + (b^3*x^3*(3*a*e + b*d))/(3*e^2) + (b*x*( 10*a^3*e^3 + b^3*d^3 + 3*a*b^2*d^2*e + 6*a^2*b*d*e^2))/(15*e^4) + (b^2*x^2 *(6*a^2*e^2 + b^2*d^2 + 3*a*b*d*e))/(5*e^3))/(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)