Integrand size = 33, antiderivative size = 254 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(b d-a e)^4 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {3 b^2 (b d-a e)^2 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {b^4 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 e^5 (a+b x)} \]
1/8*(-a*e+b*d)^4*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-4/9*b*(-a*e+b*d)^ 3*(e*x+d)^9*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+3/5*b^2*(-a*e+b*d)^2*(e*x+d)^10* ((b*x+a)^2)^(1/2)/e^5/(b*x+a)-4/11*b^3*(-a*e+b*d)*(e*x+d)^11*((b*x+a)^2)^( 1/2)/e^5/(b*x+a)+1/12*b^4*(e*x+d)^12*((b*x+a)^2)^(1/2)/e^5/(b*x+a)
Time = 1.09 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.70 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (495 a^4 \left (8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right )+220 a^3 b x \left (36 d^7+168 d^6 e x+378 d^5 e^2 x^2+504 d^4 e^3 x^3+420 d^3 e^4 x^4+216 d^2 e^5 x^5+63 d e^6 x^6+8 e^7 x^7\right )+66 a^2 b^2 x^2 \left (120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right )+12 a b^3 x^3 \left (330 d^7+1848 d^6 e x+4620 d^5 e^2 x^2+6600 d^4 e^3 x^3+5775 d^3 e^4 x^4+3080 d^2 e^5 x^5+924 d e^6 x^6+120 e^7 x^7\right )+b^4 x^4 \left (792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right )\right )}{3960 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(495*a^4*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^ 4*e^3*x^3 + 56*d^3*e^4*x^4 + 28*d^2*e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7) + 220 *a^3*b*x*(36*d^7 + 168*d^6*e*x + 378*d^5*e^2*x^2 + 504*d^4*e^3*x^3 + 420*d ^3*e^4*x^4 + 216*d^2*e^5*x^5 + 63*d*e^6*x^6 + 8*e^7*x^7) + 66*a^2*b^2*x^2* (120*d^7 + 630*d^6*e*x + 1512*d^5*e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^ 4*x^4 + 945*d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7) + 12*a*b^3*x^3*(330* d^7 + 1848*d^6*e*x + 4620*d^5*e^2*x^2 + 6600*d^4*e^3*x^3 + 5775*d^3*e^4*x^ 4 + 3080*d^2*e^5*x^5 + 924*d*e^6*x^6 + 120*e^7*x^7) + b^4*x^4*(792*d^7 + 4 620*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*x^3 + 15400*d^3*e^4*x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7)))/(3960*(a + b*x))
Time = 0.50 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 49, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^7 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^4 (d+e x)^7dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^4 (d+e x)^7dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4 (d+e x)^{11}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{10}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^9}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^8}{e^4}+\frac {(a e-b d)^4 (d+e x)^7}{e^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {4 b^3 (d+e x)^{11} (b d-a e)}{11 e^5}+\frac {3 b^2 (d+e x)^{10} (b d-a e)^2}{5 e^5}-\frac {4 b (d+e x)^9 (b d-a e)^3}{9 e^5}+\frac {(d+e x)^8 (b d-a e)^4}{8 e^5}+\frac {b^4 (d+e x)^{12}}{12 e^5}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^4*(d + e*x)^8)/(8*e^5) - (4*b *(b*d - a*e)^3*(d + e*x)^9)/(9*e^5) + (3*b^2*(b*d - a*e)^2*(d + e*x)^10)/( 5*e^5) - (4*b^3*(b*d - a*e)*(d + e*x)^11)/(11*e^5) + (b^4*(d + e*x)^12)/(1 2*e^5)))/(a + b*x)
3.20.68.3.1 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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(563\) vs. \(2(189)=378\).
Time = 0.98 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.22
| method | result | size |
| gosper | \(\frac {x \left (330 b^{4} e^{7} x^{11}+1440 x^{10} b^{3} a \,e^{7}+2520 x^{10} b^{4} d \,e^{6}+2376 x^{9} b^{2} a^{2} e^{7}+11088 x^{9} b^{3} a d \,e^{6}+8316 x^{9} b^{4} d^{2} e^{5}+1760 x^{8} a^{3} b \,e^{7}+18480 x^{8} a^{2} b^{2} d \,e^{6}+36960 x^{8} a \,b^{3} d^{2} e^{5}+15400 x^{8} b^{4} d^{3} e^{4}+495 x^{7} a^{4} e^{7}+13860 x^{7} b \,a^{3} d \,e^{6}+62370 x^{7} b^{2} a^{2} d^{2} e^{5}+69300 x^{7} b^{3} a \,d^{3} e^{4}+17325 x^{7} b^{4} d^{4} e^{3}+3960 a^{4} d \,e^{6} x^{6}+47520 a^{3} b \,d^{2} e^{5} x^{6}+118800 a^{2} b^{2} d^{3} e^{4} x^{6}+79200 a \,b^{3} d^{4} e^{3} x^{6}+11880 b^{4} d^{5} e^{2} x^{6}+13860 x^{5} a^{4} d^{2} e^{5}+92400 x^{5} b \,a^{3} d^{3} e^{4}+138600 x^{5} b^{2} a^{2} d^{4} e^{3}+55440 x^{5} b^{3} a \,d^{5} e^{2}+4620 x^{5} b^{4} d^{6} e +27720 x^{4} a^{4} d^{3} e^{4}+110880 x^{4} b \,a^{3} d^{4} e^{3}+99792 x^{4} b^{2} a^{2} d^{5} e^{2}+22176 x^{4} b^{3} a \,d^{6} e +792 x^{4} b^{4} d^{7}+34650 x^{3} a^{4} d^{4} e^{3}+83160 x^{3} b \,a^{3} d^{5} e^{2}+41580 x^{3} b^{2} a^{2} d^{6} e +3960 x^{3} b^{3} a \,d^{7}+27720 x^{2} a^{4} d^{5} e^{2}+36960 x^{2} b \,a^{3} d^{6} e +7920 x^{2} b^{2} a^{2} d^{7}+13860 x \,a^{4} d^{6} e +7920 x b \,a^{3} d^{7}+3960 a^{4} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3960 \left (b x +a \right )^{3}}\) | \(564\) |
| default | \(\frac {x \left (330 b^{4} e^{7} x^{11}+1440 x^{10} b^{3} a \,e^{7}+2520 x^{10} b^{4} d \,e^{6}+2376 x^{9} b^{2} a^{2} e^{7}+11088 x^{9} b^{3} a d \,e^{6}+8316 x^{9} b^{4} d^{2} e^{5}+1760 x^{8} a^{3} b \,e^{7}+18480 x^{8} a^{2} b^{2} d \,e^{6}+36960 x^{8} a \,b^{3} d^{2} e^{5}+15400 x^{8} b^{4} d^{3} e^{4}+495 x^{7} a^{4} e^{7}+13860 x^{7} b \,a^{3} d \,e^{6}+62370 x^{7} b^{2} a^{2} d^{2} e^{5}+69300 x^{7} b^{3} a \,d^{3} e^{4}+17325 x^{7} b^{4} d^{4} e^{3}+3960 a^{4} d \,e^{6} x^{6}+47520 a^{3} b \,d^{2} e^{5} x^{6}+118800 a^{2} b^{2} d^{3} e^{4} x^{6}+79200 a \,b^{3} d^{4} e^{3} x^{6}+11880 b^{4} d^{5} e^{2} x^{6}+13860 x^{5} a^{4} d^{2} e^{5}+92400 x^{5} b \,a^{3} d^{3} e^{4}+138600 x^{5} b^{2} a^{2} d^{4} e^{3}+55440 x^{5} b^{3} a \,d^{5} e^{2}+4620 x^{5} b^{4} d^{6} e +27720 x^{4} a^{4} d^{3} e^{4}+110880 x^{4} b \,a^{3} d^{4} e^{3}+99792 x^{4} b^{2} a^{2} d^{5} e^{2}+22176 x^{4} b^{3} a \,d^{6} e +792 x^{4} b^{4} d^{7}+34650 x^{3} a^{4} d^{4} e^{3}+83160 x^{3} b \,a^{3} d^{5} e^{2}+41580 x^{3} b^{2} a^{2} d^{6} e +3960 x^{3} b^{3} a \,d^{7}+27720 x^{2} a^{4} d^{5} e^{2}+36960 x^{2} b \,a^{3} d^{6} e +7920 x^{2} b^{2} a^{2} d^{7}+13860 x \,a^{4} d^{6} e +7920 x b \,a^{3} d^{7}+3960 a^{4} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3960 \left (b x +a \right )^{3}}\) | \(564\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} e^{7} x^{12}}{12 b x +12 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 b^{3} a \,e^{7}+7 b^{4} d \,e^{6}\right ) x^{11}}{11 b x +11 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 b^{2} a^{2} e^{7}+28 b^{3} a d \,e^{6}+21 b^{4} d^{2} e^{5}\right ) x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 a^{3} b \,e^{7}+42 a^{2} b^{2} d \,e^{6}+84 a \,b^{3} d^{2} e^{5}+35 b^{4} d^{3} e^{4}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{4} e^{7}+28 b \,a^{3} d \,e^{6}+126 b^{2} a^{2} d^{2} e^{5}+140 b^{3} a \,d^{3} e^{4}+35 b^{4} d^{4} e^{3}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (7 a^{4} d \,e^{6}+84 b \,a^{3} d^{2} e^{5}+210 b^{2} a^{2} d^{3} e^{4}+140 b^{3} a \,d^{4} e^{3}+21 b^{4} d^{5} e^{2}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (21 a^{4} d^{2} e^{5}+140 b \,a^{3} d^{3} e^{4}+210 b^{2} a^{2} d^{4} e^{3}+84 b^{3} a \,d^{5} e^{2}+7 b^{4} d^{6} e \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (35 a^{4} d^{3} e^{4}+140 b \,a^{3} d^{4} e^{3}+126 b^{2} a^{2} d^{5} e^{2}+28 b^{3} a \,d^{6} e +b^{4} d^{7}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (35 a^{4} d^{4} e^{3}+84 b \,a^{3} d^{5} e^{2}+42 b^{2} a^{2} d^{6} e +4 b^{3} a \,d^{7}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (21 a^{4} d^{5} e^{2}+28 b \,a^{3} d^{6} e +6 b^{2} a^{2} d^{7}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (7 a^{4} d^{6} e +4 b \,a^{3} d^{7}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{4} d^{7} x}{b x +a}\) | \(685\) |
1/3960*x*(330*b^4*e^7*x^11+1440*a*b^3*e^7*x^10+2520*b^4*d*e^6*x^10+2376*a^ 2*b^2*e^7*x^9+11088*a*b^3*d*e^6*x^9+8316*b^4*d^2*e^5*x^9+1760*a^3*b*e^7*x^ 8+18480*a^2*b^2*d*e^6*x^8+36960*a*b^3*d^2*e^5*x^8+15400*b^4*d^3*e^4*x^8+49 5*a^4*e^7*x^7+13860*a^3*b*d*e^6*x^7+62370*a^2*b^2*d^2*e^5*x^7+69300*a*b^3* d^3*e^4*x^7+17325*b^4*d^4*e^3*x^7+3960*a^4*d*e^6*x^6+47520*a^3*b*d^2*e^5*x ^6+118800*a^2*b^2*d^3*e^4*x^6+79200*a*b^3*d^4*e^3*x^6+11880*b^4*d^5*e^2*x^ 6+13860*a^4*d^2*e^5*x^5+92400*a^3*b*d^3*e^4*x^5+138600*a^2*b^2*d^4*e^3*x^5 +55440*a*b^3*d^5*e^2*x^5+4620*b^4*d^6*e*x^5+27720*a^4*d^3*e^4*x^4+110880*a ^3*b*d^4*e^3*x^4+99792*a^2*b^2*d^5*e^2*x^4+22176*a*b^3*d^6*e*x^4+792*b^4*d ^7*x^4+34650*a^4*d^4*e^3*x^3+83160*a^3*b*d^5*e^2*x^3+41580*a^2*b^2*d^6*e*x ^3+3960*a*b^3*d^7*x^3+27720*a^4*d^5*e^2*x^2+36960*a^3*b*d^6*e*x^2+7920*a^2 *b^2*d^7*x^2+13860*a^4*d^6*e*x+7920*a^3*b*d^7*x+3960*a^4*d^7)*((b*x+a)^2)^ (3/2)/(b*x+a)^3
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (189) = 378\).
Time = 0.40 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.93 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{12} \, b^{4} e^{7} x^{12} + a^{4} d^{7} x + \frac {1}{11} \, {\left (7 \, b^{4} d e^{6} + 4 \, a b^{3} e^{7}\right )} x^{11} + \frac {1}{10} \, {\left (21 \, b^{4} d^{2} e^{5} + 28 \, a b^{3} d e^{6} + 6 \, a^{2} b^{2} e^{7}\right )} x^{10} + \frac {1}{9} \, {\left (35 \, b^{4} d^{3} e^{4} + 84 \, a b^{3} d^{2} e^{5} + 42 \, a^{2} b^{2} d e^{6} + 4 \, a^{3} b e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{4} d^{4} e^{3} + 140 \, a b^{3} d^{3} e^{4} + 126 \, a^{2} b^{2} d^{2} e^{5} + 28 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{8} + {\left (3 \, b^{4} d^{5} e^{2} + 20 \, a b^{3} d^{4} e^{3} + 30 \, a^{2} b^{2} d^{3} e^{4} + 12 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{7} + \frac {7}{6} \, {\left (b^{4} d^{6} e + 12 \, a b^{3} d^{5} e^{2} + 30 \, a^{2} b^{2} d^{4} e^{3} + 20 \, a^{3} b d^{3} e^{4} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{7} + 28 \, a b^{3} d^{6} e + 126 \, a^{2} b^{2} d^{5} e^{2} + 140 \, a^{3} b d^{4} e^{3} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{7} + 42 \, a^{2} b^{2} d^{6} e + 84 \, a^{3} b d^{5} e^{2} + 35 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{7} + 28 \, a^{3} b d^{6} e + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{7} + 7 \, a^{4} d^{6} e\right )} x^{2} \]
1/12*b^4*e^7*x^12 + a^4*d^7*x + 1/11*(7*b^4*d*e^6 + 4*a*b^3*e^7)*x^11 + 1/ 10*(21*b^4*d^2*e^5 + 28*a*b^3*d*e^6 + 6*a^2*b^2*e^7)*x^10 + 1/9*(35*b^4*d^ 3*e^4 + 84*a*b^3*d^2*e^5 + 42*a^2*b^2*d*e^6 + 4*a^3*b*e^7)*x^9 + 1/8*(35*b ^4*d^4*e^3 + 140*a*b^3*d^3*e^4 + 126*a^2*b^2*d^2*e^5 + 28*a^3*b*d*e^6 + a^ 4*e^7)*x^8 + (3*b^4*d^5*e^2 + 20*a*b^3*d^4*e^3 + 30*a^2*b^2*d^3*e^4 + 12*a ^3*b*d^2*e^5 + a^4*d*e^6)*x^7 + 7/6*(b^4*d^6*e + 12*a*b^3*d^5*e^2 + 30*a^2 *b^2*d^4*e^3 + 20*a^3*b*d^3*e^4 + 3*a^4*d^2*e^5)*x^6 + 1/5*(b^4*d^7 + 28*a *b^3*d^6*e + 126*a^2*b^2*d^5*e^2 + 140*a^3*b*d^4*e^3 + 35*a^4*d^3*e^4)*x^5 + 1/4*(4*a*b^3*d^7 + 42*a^2*b^2*d^6*e + 84*a^3*b*d^5*e^2 + 35*a^4*d^4*e^3 )*x^4 + 1/3*(6*a^2*b^2*d^7 + 28*a^3*b*d^6*e + 21*a^4*d^5*e^2)*x^3 + 1/2*(4 *a^3*b*d^7 + 7*a^4*d^6*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 48164 vs. \(2 (184) = 368\).
Time = 1.88 (sec) , antiderivative size = 48164, normalized size of antiderivative = 189.62 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**3*e**7*x**11/12 + x**10*(3 7*a*b**4*e**7/12 + 7*b**5*d*e**6)/(11*b**2) + x**9*(109*a**2*b**3*e**7/12 + 35*a*b**4*d*e**6 - 21*a*(37*a*b**4*e**7/12 + 7*b**5*d*e**6)/(11*b) + 21* b**5*d**2*e**5)/(10*b**2) + x**8*(10*a**3*b**2*e**7 + 70*a**2*b**3*d*e**6 - 10*a**2*(37*a*b**4*e**7/12 + 7*b**5*d*e**6)/(11*b**2) + 105*a*b**4*d**2* e**5 - 19*a*(109*a**2*b**3*e**7/12 + 35*a*b**4*d*e**6 - 21*a*(37*a*b**4*e* *7/12 + 7*b**5*d*e**6)/(11*b) + 21*b**5*d**2*e**5)/(10*b) + 35*b**5*d**3*e **4)/(9*b**2) + x**7*(5*a**4*b*e**7 + 70*a**3*b**2*d*e**6 + 210*a**2*b**3* d**2*e**5 - 9*a**2*(109*a**2*b**3*e**7/12 + 35*a*b**4*d*e**6 - 21*a*(37*a* b**4*e**7/12 + 7*b**5*d*e**6)/(11*b) + 21*b**5*d**2*e**5)/(10*b**2) + 175* a*b**4*d**3*e**4 - 17*a*(10*a**3*b**2*e**7 + 70*a**2*b**3*d*e**6 - 10*a**2 *(37*a*b**4*e**7/12 + 7*b**5*d*e**6)/(11*b**2) + 105*a*b**4*d**2*e**5 - 19 *a*(109*a**2*b**3*e**7/12 + 35*a*b**4*d*e**6 - 21*a*(37*a*b**4*e**7/12 + 7 *b**5*d*e**6)/(11*b) + 21*b**5*d**2*e**5)/(10*b) + 35*b**5*d**3*e**4)/(9*b ) + 35*b**5*d**4*e**3)/(8*b**2) + x**6*(a**5*e**7 + 35*a**4*b*d*e**6 + 210 *a**3*b**2*d**2*e**5 + 350*a**2*b**3*d**3*e**4 - 8*a**2*(10*a**3*b**2*e**7 + 70*a**2*b**3*d*e**6 - 10*a**2*(37*a*b**4*e**7/12 + 7*b**5*d*e**6)/(11*b **2) + 105*a*b**4*d**2*e**5 - 19*a*(109*a**2*b**3*e**7/12 + 35*a*b**4*d*e* *6 - 21*a*(37*a*b**4*e**7/12 + 7*b**5*d*e**6)/(11*b) + 21*b**5*d**2*e**5)/ (10*b) + 35*b**5*d**3*e**4)/(9*b**2) + 175*a*b**4*d**4*e**3 - 15*a*(5*a...
Leaf count of result is larger than twice the leaf count of optimal. 2152 vs. \(2 (189) = 378\).
Time = 0.22 (sec) , antiderivative size = 2152, normalized size of antiderivative = 8.47 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
1/12*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^7*x^7/b - 19/132*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^7*x^6/b^2 + 41/220*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e ^7*x^5/b^3 - 85/396*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*e^7*x^4/b^4 + 23/9 9*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*e^7*x^3/b^5 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^7*x + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^8*e^7*x/b^7 - 8/33*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^7*x^2/b^6 + 1/4*(b^2*x^2 + 2* a*b*x + a^2)^(3/2)*a^2*d^7/b + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^9*e^7 /b^8 + 49/198*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6*e^7*x/b^7 - 247/990*(b^2 *x^2 + 2*a*b*x + a^2)^(5/2)*a^7*e^7/b^8 + 1/11*(7*b*d*e^6 + a*e^7)*(b^2*x^ 2 + 2*a*b*x + a^2)^(5/2)*x^6/b^2 - 17/110*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2 *a*b*x + a^2)^(5/2)*a*x^5/b^3 + 7/10*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2* a*b*x + a^2)^(5/2)*x^5/b^2 + 13/66*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^4/b^4 - 7/6*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x^4/b^3 + 7/9*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a*b* x + a^2)^(5/2)*x^4/b^2 - 59/264*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a ^2)^(5/2)*a^3*x^3/b^5 + 35/24*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x^3/b^4 - 91/72*(5*b*d^3*e^4 + 3*a*d^2*e^5)*(b^2*x^2 + 2*a *b*x + a^2)^(5/2)*a*x^3/b^3 + 35/8*(b*d^4*e^3 + a*d^3*e^4)*(b^2*x^2 + 2*a* b*x + a^2)^(5/2)*x^3/b^2 - 1/4*(7*b*d*e^6 + a*e^7)*(b^2*x^2 + 2*a*b*x + a^ 2)^(3/2)*a^7*x/b^7 + 7/4*(3*b*d^2*e^5 + a*d*e^6)*(b^2*x^2 + 2*a*b*x + a...
Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (189) = 378\).
Time = 0.29 (sec) , antiderivative size = 895, normalized size of antiderivative = 3.52 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
1/12*b^4*e^7*x^12*sgn(b*x + a) + 7/11*b^4*d*e^6*x^11*sgn(b*x + a) + 4/11*a *b^3*e^7*x^11*sgn(b*x + a) + 21/10*b^4*d^2*e^5*x^10*sgn(b*x + a) + 14/5*a* b^3*d*e^6*x^10*sgn(b*x + a) + 3/5*a^2*b^2*e^7*x^10*sgn(b*x + a) + 35/9*b^4 *d^3*e^4*x^9*sgn(b*x + a) + 28/3*a*b^3*d^2*e^5*x^9*sgn(b*x + a) + 14/3*a^2 *b^2*d*e^6*x^9*sgn(b*x + a) + 4/9*a^3*b*e^7*x^9*sgn(b*x + a) + 35/8*b^4*d^ 4*e^3*x^8*sgn(b*x + a) + 35/2*a*b^3*d^3*e^4*x^8*sgn(b*x + a) + 63/4*a^2*b^ 2*d^2*e^5*x^8*sgn(b*x + a) + 7/2*a^3*b*d*e^6*x^8*sgn(b*x + a) + 1/8*a^4*e^ 7*x^8*sgn(b*x + a) + 3*b^4*d^5*e^2*x^7*sgn(b*x + a) + 20*a*b^3*d^4*e^3*x^7 *sgn(b*x + a) + 30*a^2*b^2*d^3*e^4*x^7*sgn(b*x + a) + 12*a^3*b*d^2*e^5*x^7 *sgn(b*x + a) + a^4*d*e^6*x^7*sgn(b*x + a) + 7/6*b^4*d^6*e*x^6*sgn(b*x + a ) + 14*a*b^3*d^5*e^2*x^6*sgn(b*x + a) + 35*a^2*b^2*d^4*e^3*x^6*sgn(b*x + a ) + 70/3*a^3*b*d^3*e^4*x^6*sgn(b*x + a) + 7/2*a^4*d^2*e^5*x^6*sgn(b*x + a) + 1/5*b^4*d^7*x^5*sgn(b*x + a) + 28/5*a*b^3*d^6*e*x^5*sgn(b*x + a) + 126/ 5*a^2*b^2*d^5*e^2*x^5*sgn(b*x + a) + 28*a^3*b*d^4*e^3*x^5*sgn(b*x + a) + 7 *a^4*d^3*e^4*x^5*sgn(b*x + a) + a*b^3*d^7*x^4*sgn(b*x + a) + 21/2*a^2*b^2* d^6*e*x^4*sgn(b*x + a) + 21*a^3*b*d^5*e^2*x^4*sgn(b*x + a) + 35/4*a^4*d^4* e^3*x^4*sgn(b*x + a) + 2*a^2*b^2*d^7*x^3*sgn(b*x + a) + 28/3*a^3*b*d^6*e*x ^3*sgn(b*x + a) + 7*a^4*d^5*e^2*x^3*sgn(b*x + a) + 2*a^3*b*d^7*x^2*sgn(b*x + a) + 7/2*a^4*d^6*e*x^2*sgn(b*x + a) + a^4*d^7*x*sgn(b*x + a) + 1/3960*( 792*a^5*b^7*d^7 - 924*a^6*b^6*d^6*e + 792*a^7*b^5*d^5*e^2 - 495*a^8*b^4...
Timed out. \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^7\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]