Integrand size = 35, antiderivative size = 308 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^5 (a+b x)} \]
2/9*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-2/ 11*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)*(e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/ e^5/(b*x+a)+6/13*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(13/2)*((b*x+ a)^2)^(1/2)/e^5/(b*x+a)-2/15*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(15/2)*( (b*x+a)^2)^(1/2)/e^5/(b*x+a)+2/17*b^3*B*(e*x+d)^(17/2)*((b*x+a)^2)^(1/2)/e ^5/(b*x+a)
Time = 0.14 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.80 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{9/2} \left (1105 a^3 e^3 (-2 B d+11 A e+9 B e x)+255 a^2 b e^2 \left (13 A e (-2 d+9 e x)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )-51 a b^2 e \left (-5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )+B \left (16 d^3-72 d^2 e x+198 d e^2 x^2-429 e^3 x^3\right )\right )+b^3 \left (17 A e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+B \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )\right )}{109395 e^5 (a+b x)} \]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(9/2)*(1105*a^3*e^3*(-2*B*d + 11*A*e + 9*B* e*x) + 255*a^2*b*e^2*(13*A*e*(-2*d + 9*e*x) + B*(8*d^2 - 36*d*e*x + 99*e^2 *x^2)) - 51*a*b^2*e*(-5*A*e*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + B*(16*d^3 - 72*d^2*e*x + 198*d*e^2*x^2 - 429*e^3*x^3)) + b^3*(17*A*e*(-16*d^3 + 72*d^2 *e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + B*(128*d^4 - 576*d^3*e*x + 1584*d^2* e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4))))/(109395*e^5*(a + b*x))
Time = 0.38 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.65, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1187, 27, 86, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A+B x) (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^3 (A+B x) (d+e x)^{7/2}dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^3 (A+B x) (d+e x)^{7/2}dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^3 B (d+e x)^{15/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{13/2}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{11/2}}{e^4}+\frac {(a e-b d)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{9/2}}{e^4}+\frac {(a e-b d)^3 (A e-B d) (d+e x)^{7/2}}{e^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b^2 (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5}+\frac {6 b (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5}-\frac {2 (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5}+\frac {2 (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5}+\frac {2 b^3 B (d+e x)^{17/2}}{17 e^5}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/ 2))/(9*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(11/2 ))/(11*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(13/2)) /(13*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^5) + (2*b^3*B*(d + e*x)^(17/2))/(17*e^5)))/(a + b*x)
3.19.43.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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]
Time = 0.25 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.03
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 B \,b^{3} e^{4} x^{4}+7293 A \,b^{3} e^{4} x^{3}+21879 B a \,b^{2} e^{4} x^{3}-3432 B \,b^{3} d \,e^{3} x^{3}+25245 A a \,b^{2} e^{4} x^{2}-3366 A \,b^{3} d \,e^{3} x^{2}+25245 B \,a^{2} b \,e^{4} x^{2}-10098 B a \,b^{2} d \,e^{3} x^{2}+1584 B \,b^{3} d^{2} e^{2} x^{2}+29835 A \,a^{2} b \,e^{4} x -9180 A a \,b^{2} d \,e^{3} x +1224 A \,b^{3} d^{2} e^{2} x +9945 B \,a^{3} e^{4} x -9180 B \,a^{2} b d \,e^{3} x +3672 B a \,b^{2} d^{2} e^{2} x -576 B \,b^{3} d^{3} e x +12155 A \,a^{3} e^{4}-6630 A \,a^{2} b d \,e^{3}+2040 A a \,b^{2} d^{2} e^{2}-272 A \,b^{3} d^{3} e -2210 B \,a^{3} d \,e^{3}+2040 B \,a^{2} b \,d^{2} e^{2}-816 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{109395 e^{5} \left (b x +a \right )^{3}}\) | \(317\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 B \,b^{3} e^{4} x^{4}+7293 A \,b^{3} e^{4} x^{3}+21879 B a \,b^{2} e^{4} x^{3}-3432 B \,b^{3} d \,e^{3} x^{3}+25245 A a \,b^{2} e^{4} x^{2}-3366 A \,b^{3} d \,e^{3} x^{2}+25245 B \,a^{2} b \,e^{4} x^{2}-10098 B a \,b^{2} d \,e^{3} x^{2}+1584 B \,b^{3} d^{2} e^{2} x^{2}+29835 A \,a^{2} b \,e^{4} x -9180 A a \,b^{2} d \,e^{3} x +1224 A \,b^{3} d^{2} e^{2} x +9945 B \,a^{3} e^{4} x -9180 B \,a^{2} b d \,e^{3} x +3672 B a \,b^{2} d^{2} e^{2} x -576 B \,b^{3} d^{3} e x +12155 A \,a^{3} e^{4}-6630 A \,a^{2} b d \,e^{3}+2040 A a \,b^{2} d^{2} e^{2}-272 A \,b^{3} d^{3} e -2210 B \,a^{3} d \,e^{3}+2040 B \,a^{2} b \,d^{2} e^{2}-816 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{109395 e^{5} \left (b x +a \right )^{3}}\) | \(317\) |
| risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (6435 b^{3} B \,e^{8} x^{8}+7293 A \,b^{3} e^{8} x^{7}+21879 B a \,b^{2} e^{8} x^{7}+22308 B \,b^{3} d \,e^{7} x^{7}+25245 A a \,b^{2} e^{8} x^{6}+25806 A \,b^{3} d \,e^{7} x^{6}+25245 B \,a^{2} b \,e^{8} x^{6}+77418 B a \,b^{2} d \,e^{7} x^{6}+26466 B \,b^{3} d^{2} e^{6} x^{6}+29835 A \,a^{2} b \,e^{8} x^{5}+91800 A a \,b^{2} d \,e^{7} x^{5}+31518 A \,b^{3} d^{2} e^{6} x^{5}+9945 B \,a^{3} e^{8} x^{5}+91800 B \,a^{2} b d \,e^{7} x^{5}+94554 B a \,b^{2} d^{2} e^{6} x^{5}+10908 B \,b^{3} d^{3} e^{5} x^{5}+12155 A \,a^{3} e^{8} x^{4}+112710 A \,a^{2} b d \,e^{7} x^{4}+116790 A a \,b^{2} d^{2} e^{6} x^{4}+13600 A \,b^{3} d^{3} e^{5} x^{4}+37570 B \,a^{3} d \,e^{7} x^{4}+116790 B \,a^{2} b \,d^{2} e^{6} x^{4}+40800 B a \,b^{2} d^{3} e^{5} x^{4}+35 B \,b^{3} d^{4} e^{4} x^{4}+48620 A \,a^{3} d \,e^{7} x^{3}+152490 A \,a^{2} b \,d^{2} e^{6} x^{3}+54060 A a \,b^{2} d^{3} e^{5} x^{3}+85 A \,b^{3} d^{4} e^{4} x^{3}+50830 B \,a^{3} d^{2} e^{6} x^{3}+54060 B \,a^{2} b \,d^{3} e^{5} x^{3}+255 B a \,b^{2} d^{4} e^{4} x^{3}-40 B \,b^{3} d^{5} e^{3} x^{3}+72930 A \,a^{3} d^{2} e^{6} x^{2}+79560 A \,a^{2} b \,d^{3} e^{5} x^{2}+765 A a \,b^{2} d^{4} e^{4} x^{2}-102 A \,b^{3} d^{5} e^{3} x^{2}+26520 B \,a^{3} d^{3} e^{5} x^{2}+765 B \,a^{2} b \,d^{4} e^{4} x^{2}-306 B a \,b^{2} d^{5} e^{3} x^{2}+48 B \,b^{3} d^{6} e^{2} x^{2}+48620 A \,a^{3} d^{3} e^{5} x +3315 A \,a^{2} b \,d^{4} e^{4} x -1020 A a \,b^{2} d^{5} e^{3} x +136 A \,b^{3} d^{6} e^{2} x +1105 B \,a^{3} d^{4} e^{4} x -1020 B \,a^{2} b \,d^{5} e^{3} x +408 B a \,b^{2} d^{6} e^{2} x -64 B \,b^{3} d^{7} e x +12155 A \,a^{3} d^{4} e^{4}-6630 A \,a^{2} b \,d^{5} e^{3}+2040 A a \,b^{2} d^{6} e^{2}-272 A \,b^{3} d^{7} e -2210 B \,a^{3} d^{5} e^{3}+2040 B \,a^{2} b \,d^{6} e^{2}-816 B a \,b^{2} d^{7} e +128 B \,b^{3} d^{8}\right ) \sqrt {e x +d}}{109395 \left (b x +a \right ) e^{5}}\) | \(809\) |
2/109395*(e*x+d)^(9/2)*(6435*B*b^3*e^4*x^4+7293*A*b^3*e^4*x^3+21879*B*a*b^ 2*e^4*x^3-3432*B*b^3*d*e^3*x^3+25245*A*a*b^2*e^4*x^2-3366*A*b^3*d*e^3*x^2+ 25245*B*a^2*b*e^4*x^2-10098*B*a*b^2*d*e^3*x^2+1584*B*b^3*d^2*e^2*x^2+29835 *A*a^2*b*e^4*x-9180*A*a*b^2*d*e^3*x+1224*A*b^3*d^2*e^2*x+9945*B*a^3*e^4*x- 9180*B*a^2*b*d*e^3*x+3672*B*a*b^2*d^2*e^2*x-576*B*b^3*d^3*e*x+12155*A*a^3* e^4-6630*A*a^2*b*d*e^3+2040*A*a*b^2*d^2*e^2-272*A*b^3*d^3*e-2210*B*a^3*d*e ^3+2040*B*a^2*b*d^2*e^2-816*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2) /e^5/(b*x+a)^3
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (233) = 466\).
Time = 0.28 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.06 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (6435 \, B b^{3} e^{8} x^{8} + 128 \, B b^{3} d^{8} + 12155 \, A a^{3} d^{4} e^{4} - 272 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{7} e + 2040 \, {\left (B a^{2} b + A a b^{2}\right )} d^{6} e^{2} - 2210 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5} e^{3} + 429 \, {\left (52 \, B b^{3} d e^{7} + 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{8}\right )} x^{7} + 33 \, {\left (802 \, B b^{3} d^{2} e^{6} + 782 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{7} + 765 \, {\left (B a^{2} b + A a b^{2}\right )} e^{8}\right )} x^{6} + 9 \, {\left (1212 \, B b^{3} d^{3} e^{5} + 3502 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{6} + 10200 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{7} + 1105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{8}\right )} x^{5} + 5 \, {\left (7 \, B b^{3} d^{4} e^{4} + 2431 \, A a^{3} e^{8} + 2720 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{5} + 23358 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{6} + 7514 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{7}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{5} e^{3} - 9724 \, A a^{3} d e^{7} - 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{4} - 10812 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{5} - 10166 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{6} e^{2} + 24310 \, A a^{3} d^{2} e^{6} - 34 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{3} + 255 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{4} + 8840 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{7} e - 48620 \, A a^{3} d^{3} e^{5} - 136 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e^{2} + 1020 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{3} - 1105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{4}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{5}} \]
2/109395*(6435*B*b^3*e^8*x^8 + 128*B*b^3*d^8 + 12155*A*a^3*d^4*e^4 - 272*( 3*B*a*b^2 + A*b^3)*d^7*e + 2040*(B*a^2*b + A*a*b^2)*d^6*e^2 - 2210*(B*a^3 + 3*A*a^2*b)*d^5*e^3 + 429*(52*B*b^3*d*e^7 + 17*(3*B*a*b^2 + A*b^3)*e^8)*x ^7 + 33*(802*B*b^3*d^2*e^6 + 782*(3*B*a*b^2 + A*b^3)*d*e^7 + 765*(B*a^2*b + A*a*b^2)*e^8)*x^6 + 9*(1212*B*b^3*d^3*e^5 + 3502*(3*B*a*b^2 + A*b^3)*d^2 *e^6 + 10200*(B*a^2*b + A*a*b^2)*d*e^7 + 1105*(B*a^3 + 3*A*a^2*b)*e^8)*x^5 + 5*(7*B*b^3*d^4*e^4 + 2431*A*a^3*e^8 + 2720*(3*B*a*b^2 + A*b^3)*d^3*e^5 + 23358*(B*a^2*b + A*a*b^2)*d^2*e^6 + 7514*(B*a^3 + 3*A*a^2*b)*d*e^7)*x^4 - 5*(8*B*b^3*d^5*e^3 - 9724*A*a^3*d*e^7 - 17*(3*B*a*b^2 + A*b^3)*d^4*e^4 - 10812*(B*a^2*b + A*a*b^2)*d^3*e^5 - 10166*(B*a^3 + 3*A*a^2*b)*d^2*e^6)*x^ 3 + 3*(16*B*b^3*d^6*e^2 + 24310*A*a^3*d^2*e^6 - 34*(3*B*a*b^2 + A*b^3)*d^5 *e^3 + 255*(B*a^2*b + A*a*b^2)*d^4*e^4 + 8840*(B*a^3 + 3*A*a^2*b)*d^3*e^5) *x^2 - (64*B*b^3*d^7*e - 48620*A*a^3*d^3*e^5 - 136*(3*B*a*b^2 + A*b^3)*d^6 *e^2 + 1020*(B*a^2*b + A*a*b^2)*d^5*e^3 - 1105*(B*a^3 + 3*A*a^2*b)*d^4*e^4 )*x)*sqrt(e*x + d)/e^5
Timed out. \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (233) = 466\).
Time = 0.22 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.26 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (429 \, b^{3} e^{7} x^{7} - 16 \, b^{3} d^{7} + 120 \, a b^{2} d^{6} e - 390 \, a^{2} b d^{5} e^{2} + 715 \, a^{3} d^{4} e^{3} + 33 \, {\left (46 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 9 \, {\left (206 \, b^{3} d^{2} e^{5} + 600 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 5 \, {\left (160 \, b^{3} d^{3} e^{4} + 1374 \, a b^{2} d^{2} e^{5} + 1326 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} + 5 \, {\left (b^{3} d^{4} e^{3} + 636 \, a b^{2} d^{3} e^{4} + 1794 \, a^{2} b d^{2} e^{5} + 572 \, a^{3} d e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{5} e^{2} - 15 \, a b^{2} d^{4} e^{3} - 1560 \, a^{2} b d^{3} e^{4} - 1430 \, a^{3} d^{2} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{6} e - 60 \, a b^{2} d^{5} e^{2} + 195 \, a^{2} b d^{4} e^{3} + 2860 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d} A}{6435 \, e^{4}} + \frac {2 \, {\left (6435 \, b^{3} e^{8} x^{8} + 128 \, b^{3} d^{8} - 816 \, a b^{2} d^{7} e + 2040 \, a^{2} b d^{6} e^{2} - 2210 \, a^{3} d^{5} e^{3} + 429 \, {\left (52 \, b^{3} d e^{7} + 51 \, a b^{2} e^{8}\right )} x^{7} + 33 \, {\left (802 \, b^{3} d^{2} e^{6} + 2346 \, a b^{2} d e^{7} + 765 \, a^{2} b e^{8}\right )} x^{6} + 9 \, {\left (1212 \, b^{3} d^{3} e^{5} + 10506 \, a b^{2} d^{2} e^{6} + 10200 \, a^{2} b d e^{7} + 1105 \, a^{3} e^{8}\right )} x^{5} + 5 \, {\left (7 \, b^{3} d^{4} e^{4} + 8160 \, a b^{2} d^{3} e^{5} + 23358 \, a^{2} b d^{2} e^{6} + 7514 \, a^{3} d e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{5} e^{3} - 51 \, a b^{2} d^{4} e^{4} - 10812 \, a^{2} b d^{3} e^{5} - 10166 \, a^{3} d^{2} e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{6} e^{2} - 102 \, a b^{2} d^{5} e^{3} + 255 \, a^{2} b d^{4} e^{4} + 8840 \, a^{3} d^{3} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{7} e - 408 \, a b^{2} d^{6} e^{2} + 1020 \, a^{2} b d^{5} e^{3} - 1105 \, a^{3} d^{4} e^{4}\right )} x\right )} \sqrt {e x + d} B}{109395 \, e^{5}} \]
2/6435*(429*b^3*e^7*x^7 - 16*b^3*d^7 + 120*a*b^2*d^6*e - 390*a^2*b*d^5*e^2 + 715*a^3*d^4*e^3 + 33*(46*b^3*d*e^6 + 45*a*b^2*e^7)*x^6 + 9*(206*b^3*d^2 *e^5 + 600*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 5*(160*b^3*d^3*e^4 + 1374*a* b^2*d^2*e^5 + 1326*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 + 5*(b^3*d^4*e^3 + 636*a *b^2*d^3*e^4 + 1794*a^2*b*d^2*e^5 + 572*a^3*d*e^6)*x^3 - 3*(2*b^3*d^5*e^2 - 15*a*b^2*d^4*e^3 - 1560*a^2*b*d^3*e^4 - 1430*a^3*d^2*e^5)*x^2 + (8*b^3*d ^6*e - 60*a*b^2*d^5*e^2 + 195*a^2*b*d^4*e^3 + 2860*a^3*d^3*e^4)*x)*sqrt(e* x + d)*A/e^4 + 2/109395*(6435*b^3*e^8*x^8 + 128*b^3*d^8 - 816*a*b^2*d^7*e + 2040*a^2*b*d^6*e^2 - 2210*a^3*d^5*e^3 + 429*(52*b^3*d*e^7 + 51*a*b^2*e^8 )*x^7 + 33*(802*b^3*d^2*e^6 + 2346*a*b^2*d*e^7 + 765*a^2*b*e^8)*x^6 + 9*(1 212*b^3*d^3*e^5 + 10506*a*b^2*d^2*e^6 + 10200*a^2*b*d*e^7 + 1105*a^3*e^8)* x^5 + 5*(7*b^3*d^4*e^4 + 8160*a*b^2*d^3*e^5 + 23358*a^2*b*d^2*e^6 + 7514*a ^3*d*e^7)*x^4 - 5*(8*b^3*d^5*e^3 - 51*a*b^2*d^4*e^4 - 10812*a^2*b*d^3*e^5 - 10166*a^3*d^2*e^6)*x^3 + 3*(16*b^3*d^6*e^2 - 102*a*b^2*d^5*e^3 + 255*a^2 *b*d^4*e^4 + 8840*a^3*d^3*e^5)*x^2 - (64*b^3*d^7*e - 408*a*b^2*d^6*e^2 + 1 020*a^2*b*d^5*e^3 - 1105*a^3*d^4*e^4)*x)*sqrt(e*x + d)*B/e^5
Leaf count of result is larger than twice the leaf count of optimal. 2924 vs. \(2 (233) = 466\).
Time = 0.36 (sec) , antiderivative size = 2924, normalized size of antiderivative = 9.49 \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
2/765765*(765765*sqrt(e*x + d)*A*a^3*d^4*sgn(b*x + a) + 1021020*((e*x + d) ^(3/2) - 3*sqrt(e*x + d)*d)*A*a^3*d^3*sgn(b*x + a) + 255255*((e*x + d)^(3/ 2) - 3*sqrt(e*x + d)*d)*B*a^3*d^4*sgn(b*x + a)/e + 765765*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^2*b*d^4*sgn(b*x + a)/e + 306306*(3*(e*x + d)^(5/ 2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^3*d^2*sgn(b*x + a) + 153153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)* B*a^2*b*d^4*sgn(b*x + a)/e^2 + 153153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3 /2)*d + 15*sqrt(e*x + d)*d^2)*A*a*b^2*d^4*sgn(b*x + a)/e^2 + 204204*(3*(e* x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^3*d^3*sgn( b*x + a)/e + 612612*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e* x + d)*d^2)*A*a^2*b*d^3*sgn(b*x + a)/e + 87516*(5*(e*x + d)^(7/2) - 21*(e* x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*a^3*d*sg n(b*x + a) + 65637*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d )^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*a*b^2*d^4*sgn(b*x + a)/e^3 + 21879*( 5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqr t(e*x + d)*d^3)*A*b^3*d^4*sgn(b*x + a)/e^3 + 262548*(5*(e*x + d)^(7/2) - 2 1*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*a^2 *b*d^3*sgn(b*x + a)/e^2 + 262548*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*a*b^2*d^3*sgn(b*x + a) /e^2 + 131274*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^...
Timed out. \[ \int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]