Integrand size = 35, antiderivative size = 452 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 (b d-a e)^5 (B d-A e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 e^7 (a+b x)} \]
2/7*(-a*e+b*d)^5*(-A*e+B*d)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/ 9*(-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^ 7/(b*x+a)+10/11*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(11/2)*((b *x+a)^2)^(1/2)/e^7/(b*x+a)-20/13*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*( e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/3*b^3*(-a*e+b*d)*(-A*b*e-2*B *a*e+3*B*b*d)*(e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/17*b^4*(-A*b* e-5*B*a*e+6*B*b*d)*(e*x+d)^(17/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/19*b^5*B *(e*x+d)^(19/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 0.44 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.09 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (46189 a^5 e^5 (-2 B d+9 A e+7 B e x)+20995 a^4 b e^4 \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-3230 a^3 b^2 e^3 \left (-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+646 a^2 b^3 e^2 \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-19 a b^4 e \left (-17 A e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+5 B \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )+b^5 \left (19 A e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+3 B \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )\right )}{2909907 e^7 (a+b x)} \]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(46189*a^5*e^5*(-2*B*d + 9*A*e + 7*B* e*x) + 20995*a^4*b*e^4*(11*A*e*(-2*d + 7*e*x) + B*(8*d^2 - 28*d*e*x + 63*e ^2*x^2)) - 3230*a^3*b^2*e^3*(-13*A*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3*B *(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3)) + 646*a^2*b^3*e^2*(1 5*A*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + B*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)) - 19*a*b^ 4*e*(-17*A*e*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 5*B*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2 - 3696*d^2*e ^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5)) + b^5*(19*A*e*(-256*d^5 + 896*d^4 *e*x - 2016*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5 ) + 3*B*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6))))/(2909907*e^7*(a + b*x))
Time = 0.46 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.64, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (A+B x) (d+e x)^{5/2}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (A+B x) (d+e x)^{5/2}dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^5 B (d+e x)^{17/2}}{e^6}+\frac {b^4 (-6 b B d+A b e+5 a B e) (d+e x)^{15/2}}{e^6}-\frac {5 b^3 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{13/2}}{e^6}+\frac {10 b^2 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{11/2}}{e^6}-\frac {5 b (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{9/2}}{e^6}+\frac {(a e-b d)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{7/2}}{e^6}+\frac {(a e-b d)^5 (A e-B d) (d+e x)^{5/2}}{e^6}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b^4 (d+e x)^{17/2} (-5 a B e-A b e+6 b B d)}{17 e^7}+\frac {2 b^3 (d+e x)^{15/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7}-\frac {20 b^2 (d+e x)^{13/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{13 e^7}+\frac {10 b (d+e x)^{11/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7}-\frac {2 (d+e x)^{9/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7}+\frac {2 (d+e x)^{7/2} (b d-a e)^5 (B d-A e)}{7 e^7}+\frac {2 b^5 B (d+e x)^{19/2}}{19 e^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(7/ 2))/(7*e^7) - (2*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(9/2) )/(9*e^7) + (10*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(11/ 2))/(11*e^7) - (20*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^( 13/2))/(13*e^7) + (2*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x) ^(15/2))/(3*e^7) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(17/2))/(1 7*e^7) + (2*b^5*B*(d + e*x)^(19/2))/(19*e^7)))/(a + b*x)
3.19.52.3.1 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 = 689, normalized size of antiderivative = 1.52
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (153153 B \,b^{5} e^{6} x^{6}+171171 A \,b^{5} e^{6} x^{5}+855855 B a \,b^{4} e^{6} x^{5}-108108 B \,b^{5} d \,e^{5} x^{5}+969969 A a \,b^{4} e^{6} x^{4}-114114 A \,b^{5} d \,e^{5} x^{4}+1939938 B \,a^{2} b^{3} e^{6} x^{4}-570570 B a \,b^{4} d \,e^{5} x^{4}+72072 B \,b^{5} d^{2} e^{4} x^{4}+2238390 A \,a^{2} b^{3} e^{6} x^{3}-596904 A a \,b^{4} d \,e^{5} x^{3}+70224 A \,b^{5} d^{2} e^{4} x^{3}+2238390 B \,a^{3} b^{2} e^{6} x^{3}-1193808 B \,a^{2} b^{3} d \,e^{5} x^{3}+351120 B a \,b^{4} d^{2} e^{4} x^{3}-44352 B \,b^{5} d^{3} e^{3} x^{3}+2645370 A \,a^{3} b^{2} e^{6} x^{2}-1220940 A \,a^{2} b^{3} d \,e^{5} x^{2}+325584 A a \,b^{4} d^{2} e^{4} x^{2}-38304 A \,b^{5} d^{3} e^{3} x^{2}+1322685 B \,a^{4} b \,e^{6} x^{2}-1220940 B \,a^{3} b^{2} d \,e^{5} x^{2}+651168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-191520 B a \,b^{4} d^{3} e^{3} x^{2}+24192 B \,b^{5} d^{4} e^{2} x^{2}+1616615 A \,a^{4} b \,e^{6} x -1175720 A \,a^{3} b^{2} d \,e^{5} x +542640 A \,a^{2} b^{3} d^{2} e^{4} x -144704 A a \,b^{4} d^{3} e^{3} x +17024 A \,b^{5} d^{4} e^{2} x +323323 B \,a^{5} e^{6} x -587860 B \,a^{4} b d \,e^{5} x +542640 B \,a^{3} b^{2} d^{2} e^{4} x -289408 B \,a^{2} b^{3} d^{3} e^{3} x +85120 B a \,b^{4} d^{4} e^{2} x -10752 B \,b^{5} d^{5} e x +415701 A \,a^{5} e^{6}-461890 A \,a^{4} b d \,e^{5}+335920 A \,a^{3} b^{2} d^{2} e^{4}-155040 A \,a^{2} b^{3} d^{3} e^{3}+41344 A a \,b^{4} d^{4} e^{2}-4864 A \,b^{5} d^{5} e -92378 B \,a^{5} d \,e^{5}+167960 B \,a^{4} b \,d^{2} e^{4}-155040 B \,a^{3} b^{2} d^{3} e^{3}+82688 B \,a^{2} b^{3} d^{4} e^{2}-24320 B a \,b^{4} d^{5} e +3072 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2909907 e^{7} \left (b x +a \right )^{5}}\) | \(689\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (153153 B \,b^{5} e^{6} x^{6}+171171 A \,b^{5} e^{6} x^{5}+855855 B a \,b^{4} e^{6} x^{5}-108108 B \,b^{5} d \,e^{5} x^{5}+969969 A a \,b^{4} e^{6} x^{4}-114114 A \,b^{5} d \,e^{5} x^{4}+1939938 B \,a^{2} b^{3} e^{6} x^{4}-570570 B a \,b^{4} d \,e^{5} x^{4}+72072 B \,b^{5} d^{2} e^{4} x^{4}+2238390 A \,a^{2} b^{3} e^{6} x^{3}-596904 A a \,b^{4} d \,e^{5} x^{3}+70224 A \,b^{5} d^{2} e^{4} x^{3}+2238390 B \,a^{3} b^{2} e^{6} x^{3}-1193808 B \,a^{2} b^{3} d \,e^{5} x^{3}+351120 B a \,b^{4} d^{2} e^{4} x^{3}-44352 B \,b^{5} d^{3} e^{3} x^{3}+2645370 A \,a^{3} b^{2} e^{6} x^{2}-1220940 A \,a^{2} b^{3} d \,e^{5} x^{2}+325584 A a \,b^{4} d^{2} e^{4} x^{2}-38304 A \,b^{5} d^{3} e^{3} x^{2}+1322685 B \,a^{4} b \,e^{6} x^{2}-1220940 B \,a^{3} b^{2} d \,e^{5} x^{2}+651168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-191520 B a \,b^{4} d^{3} e^{3} x^{2}+24192 B \,b^{5} d^{4} e^{2} x^{2}+1616615 A \,a^{4} b \,e^{6} x -1175720 A \,a^{3} b^{2} d \,e^{5} x +542640 A \,a^{2} b^{3} d^{2} e^{4} x -144704 A a \,b^{4} d^{3} e^{3} x +17024 A \,b^{5} d^{4} e^{2} x +323323 B \,a^{5} e^{6} x -587860 B \,a^{4} b d \,e^{5} x +542640 B \,a^{3} b^{2} d^{2} e^{4} x -289408 B \,a^{2} b^{3} d^{3} e^{3} x +85120 B a \,b^{4} d^{4} e^{2} x -10752 B \,b^{5} d^{5} e x +415701 A \,a^{5} e^{6}-461890 A \,a^{4} b d \,e^{5}+335920 A \,a^{3} b^{2} d^{2} e^{4}-155040 A \,a^{2} b^{3} d^{3} e^{3}+41344 A a \,b^{4} d^{4} e^{2}-4864 A \,b^{5} d^{5} e -92378 B \,a^{5} d \,e^{5}+167960 B \,a^{4} b \,d^{2} e^{4}-155040 B \,a^{3} b^{2} d^{3} e^{3}+82688 B \,a^{2} b^{3} d^{4} e^{2}-24320 B a \,b^{4} d^{5} e +3072 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2909907 e^{7} \left (b x +a \right )^{5}}\) | \(689\) |
| risch | \(\text {Expression too large to display}\) | \(1273\) |
2/2909907*(e*x+d)^(7/2)*(153153*B*b^5*e^6*x^6+171171*A*b^5*e^6*x^5+855855* B*a*b^4*e^6*x^5-108108*B*b^5*d*e^5*x^5+969969*A*a*b^4*e^6*x^4-114114*A*b^5 *d*e^5*x^4+1939938*B*a^2*b^3*e^6*x^4-570570*B*a*b^4*d*e^5*x^4+72072*B*b^5* d^2*e^4*x^4+2238390*A*a^2*b^3*e^6*x^3-596904*A*a*b^4*d*e^5*x^3+70224*A*b^5 *d^2*e^4*x^3+2238390*B*a^3*b^2*e^6*x^3-1193808*B*a^2*b^3*d*e^5*x^3+351120* B*a*b^4*d^2*e^4*x^3-44352*B*b^5*d^3*e^3*x^3+2645370*A*a^3*b^2*e^6*x^2-1220 940*A*a^2*b^3*d*e^5*x^2+325584*A*a*b^4*d^2*e^4*x^2-38304*A*b^5*d^3*e^3*x^2 +1322685*B*a^4*b*e^6*x^2-1220940*B*a^3*b^2*d*e^5*x^2+651168*B*a^2*b^3*d^2* e^4*x^2-191520*B*a*b^4*d^3*e^3*x^2+24192*B*b^5*d^4*e^2*x^2+1616615*A*a^4*b *e^6*x-1175720*A*a^3*b^2*d*e^5*x+542640*A*a^2*b^3*d^2*e^4*x-144704*A*a*b^4 *d^3*e^3*x+17024*A*b^5*d^4*e^2*x+323323*B*a^5*e^6*x-587860*B*a^4*b*d*e^5*x +542640*B*a^3*b^2*d^2*e^4*x-289408*B*a^2*b^3*d^3*e^3*x+85120*B*a*b^4*d^4*e ^2*x-10752*B*b^5*d^5*e*x+415701*A*a^5*e^6-461890*A*a^4*b*d*e^5+335920*A*a^ 3*b^2*d^2*e^4-155040*A*a^2*b^3*d^3*e^3+41344*A*a*b^4*d^4*e^2-4864*A*b^5*d^ 5*e-92378*B*a^5*d*e^5+167960*B*a^4*b*d^2*e^4-155040*B*a^3*b^2*d^3*e^3+8268 8*B*a^2*b^3*d^4*e^2-24320*B*a*b^4*d^5*e+3072*B*b^5*d^6)*((b*x+a)^2)^(5/2)/ e^7/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 993 vs. \(2 (347) = 694\).
Time = 0.32 (sec) , antiderivative size = 993, normalized size of antiderivative = 2.20 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (153153 \, B b^{5} e^{9} x^{9} + 3072 \, B b^{5} d^{9} + 415701 \, A a^{5} d^{3} e^{6} - 4864 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{8} e + 41344 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{7} e^{2} - 155040 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{6} e^{3} + 167960 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{5} e^{4} - 92378 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{4} e^{5} + 9009 \, {\left (39 \, B b^{5} d e^{8} + 19 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{9}\right )} x^{8} + 3003 \, {\left (69 \, B b^{5} d^{2} e^{7} + 133 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{8} + 323 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{9}\right )} x^{7} + 231 \, {\left (3 \, B b^{5} d^{3} e^{6} + 1045 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{7} + 10013 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{8} + 9690 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{9}\right )} x^{6} - 63 \, {\left (12 \, B b^{5} d^{4} e^{5} - 19 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{6} - 22933 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{7} - 87210 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{8} - 20995 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{9}\right )} x^{5} + 7 \, {\left (120 \, B b^{5} d^{5} e^{4} - 190 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{5} + 1615 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{6} + 513570 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{7} + 482885 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{8} + 46189 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{9}\right )} x^{4} - {\left (960 \, B b^{5} d^{6} e^{3} - 415701 \, A a^{5} e^{9} - 1520 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e^{4} + 12920 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{5} - 48450 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{6} - 2372435 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{7} - 877591 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{8}\right )} x^{3} + 3 \, {\left (384 \, B b^{5} d^{7} e^{2} + 415701 \, A a^{5} d e^{8} - 608 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{6} e^{3} + 5168 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} e^{4} - 19380 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e^{5} + 20995 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{6} + 230945 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{7}\right )} x^{2} - {\left (1536 \, B b^{5} d^{8} e - 1247103 \, A a^{5} d^{2} e^{7} - 2432 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{7} e^{2} + 20672 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{6} e^{3} - 77520 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{5} e^{4} + 83980 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} e^{5} - 46189 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} e^{6}\right )} x\right )} \sqrt {e x + d}}{2909907 \, e^{7}} \]
2/2909907*(153153*B*b^5*e^9*x^9 + 3072*B*b^5*d^9 + 415701*A*a^5*d^3*e^6 - 4864*(5*B*a*b^4 + A*b^5)*d^8*e + 41344*(2*B*a^2*b^3 + A*a*b^4)*d^7*e^2 - 1 55040*(B*a^3*b^2 + A*a^2*b^3)*d^6*e^3 + 167960*(B*a^4*b + 2*A*a^3*b^2)*d^5 *e^4 - 92378*(B*a^5 + 5*A*a^4*b)*d^4*e^5 + 9009*(39*B*b^5*d*e^8 + 19*(5*B* a*b^4 + A*b^5)*e^9)*x^8 + 3003*(69*B*b^5*d^2*e^7 + 133*(5*B*a*b^4 + A*b^5) *d*e^8 + 323*(2*B*a^2*b^3 + A*a*b^4)*e^9)*x^7 + 231*(3*B*b^5*d^3*e^6 + 104 5*(5*B*a*b^4 + A*b^5)*d^2*e^7 + 10013*(2*B*a^2*b^3 + A*a*b^4)*d*e^8 + 9690 *(B*a^3*b^2 + A*a^2*b^3)*e^9)*x^6 - 63*(12*B*b^5*d^4*e^5 - 19*(5*B*a*b^4 + A*b^5)*d^3*e^6 - 22933*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^7 - 87210*(B*a^3*b^2 + A*a^2*b^3)*d*e^8 - 20995*(B*a^4*b + 2*A*a^3*b^2)*e^9)*x^5 + 7*(120*B*b^ 5*d^5*e^4 - 190*(5*B*a*b^4 + A*b^5)*d^4*e^5 + 1615*(2*B*a^2*b^3 + A*a*b^4) *d^3*e^6 + 513570*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^7 + 482885*(B*a^4*b + 2*A* a^3*b^2)*d*e^8 + 46189*(B*a^5 + 5*A*a^4*b)*e^9)*x^4 - (960*B*b^5*d^6*e^3 - 415701*A*a^5*e^9 - 1520*(5*B*a*b^4 + A*b^5)*d^5*e^4 + 12920*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^5 - 48450*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^6 - 2372435*(B*a^ 4*b + 2*A*a^3*b^2)*d^2*e^7 - 877591*(B*a^5 + 5*A*a^4*b)*d*e^8)*x^3 + 3*(38 4*B*b^5*d^7*e^2 + 415701*A*a^5*d*e^8 - 608*(5*B*a*b^4 + A*b^5)*d^6*e^3 + 5 168*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^4 - 19380*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^ 5 + 20995*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^6 + 230945*(B*a^5 + 5*A*a^4*b)*d^2 *e^7)*x^2 - (1536*B*b^5*d^8*e - 1247103*A*a^5*d^2*e^7 - 2432*(5*B*a*b^4...
Timed out. \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1080 vs. \(2 (347) = 694\).
Time = 0.24 (sec) , antiderivative size = 1080, normalized size of antiderivative = 2.39 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d} A}{153153 \, e^{6}} + \frac {2 \, {\left (153153 \, b^{5} e^{9} x^{9} + 3072 \, b^{5} d^{9} - 24320 \, a b^{4} d^{8} e + 82688 \, a^{2} b^{3} d^{7} e^{2} - 155040 \, a^{3} b^{2} d^{6} e^{3} + 167960 \, a^{4} b d^{5} e^{4} - 92378 \, a^{5} d^{4} e^{5} + 9009 \, {\left (39 \, b^{5} d e^{8} + 95 \, a b^{4} e^{9}\right )} x^{8} + 3003 \, {\left (69 \, b^{5} d^{2} e^{7} + 665 \, a b^{4} d e^{8} + 646 \, a^{2} b^{3} e^{9}\right )} x^{7} + 231 \, {\left (3 \, b^{5} d^{3} e^{6} + 5225 \, a b^{4} d^{2} e^{7} + 20026 \, a^{2} b^{3} d e^{8} + 9690 \, a^{3} b^{2} e^{9}\right )} x^{6} - 63 \, {\left (12 \, b^{5} d^{4} e^{5} - 95 \, a b^{4} d^{3} e^{6} - 45866 \, a^{2} b^{3} d^{2} e^{7} - 87210 \, a^{3} b^{2} d e^{8} - 20995 \, a^{4} b e^{9}\right )} x^{5} + 7 \, {\left (120 \, b^{5} d^{5} e^{4} - 950 \, a b^{4} d^{4} e^{5} + 3230 \, a^{2} b^{3} d^{3} e^{6} + 513570 \, a^{3} b^{2} d^{2} e^{7} + 482885 \, a^{4} b d e^{8} + 46189 \, a^{5} e^{9}\right )} x^{4} - {\left (960 \, b^{5} d^{6} e^{3} - 7600 \, a b^{4} d^{5} e^{4} + 25840 \, a^{2} b^{3} d^{4} e^{5} - 48450 \, a^{3} b^{2} d^{3} e^{6} - 2372435 \, a^{4} b d^{2} e^{7} - 877591 \, a^{5} d e^{8}\right )} x^{3} + 3 \, {\left (384 \, b^{5} d^{7} e^{2} - 3040 \, a b^{4} d^{6} e^{3} + 10336 \, a^{2} b^{3} d^{5} e^{4} - 19380 \, a^{3} b^{2} d^{4} e^{5} + 20995 \, a^{4} b d^{3} e^{6} + 230945 \, a^{5} d^{2} e^{7}\right )} x^{2} - {\left (1536 \, b^{5} d^{8} e - 12160 \, a b^{4} d^{7} e^{2} + 41344 \, a^{2} b^{3} d^{6} e^{3} - 77520 \, a^{3} b^{2} d^{5} e^{4} + 83980 \, a^{4} b d^{4} e^{5} - 46189 \, a^{5} d^{3} e^{6}\right )} x\right )} \sqrt {e x + d} B}{2909907 \, e^{7}} \]
2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3 *d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 24310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b^4 *d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 459 0*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3 *e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^ 2*d^2*e^6 + 230945*a^4*b*d*e^7 + 21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60775*a^ 4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637 *a^5*d^2*e^6)*x)*sqrt(e*x + d)*A/e^6 + 2/2909907*(153153*b^5*e^9*x^9 + 307 2*b^5*d^9 - 24320*a*b^4*d^8*e + 82688*a^2*b^3*d^7*e^2 - 155040*a^3*b^2*d^6 *e^3 + 167960*a^4*b*d^5*e^4 - 92378*a^5*d^4*e^5 + 9009*(39*b^5*d*e^8 + 95* a*b^4*e^9)*x^8 + 3003*(69*b^5*d^2*e^7 + 665*a*b^4*d*e^8 + 646*a^2*b^3*e^9) *x^7 + 231*(3*b^5*d^3*e^6 + 5225*a*b^4*d^2*e^7 + 20026*a^2*b^3*d*e^8 + 969 0*a^3*b^2*e^9)*x^6 - 63*(12*b^5*d^4*e^5 - 95*a*b^4*d^3*e^6 - 45866*a^2*b^3 *d^2*e^7 - 87210*a^3*b^2*d*e^8 - 20995*a^4*b*e^9)*x^5 + 7*(120*b^5*d^5*e^4 - 950*a*b^4*d^4*e^5 + 3230*a^2*b^3*d^3*e^6 + 513570*a^3*b^2*d^2*e^7 + 482 885*a^4*b*d*e^8 + 46189*a^5*e^9)*x^4 - (960*b^5*d^6*e^3 - 7600*a*b^4*d^...
Leaf count of result is larger than twice the leaf count of optimal. 3831 vs. \(2 (347) = 694\).
Time = 0.38 (sec) , antiderivative size = 3831, normalized size of antiderivative = 8.48 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
2/14549535*(14549535*sqrt(e*x + d)*A*a^5*d^3*sgn(b*x + a) + 14549535*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^5*d^2*sgn(b*x + a) + 4849845*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^5*d^3*sgn(b*x + a)/e + 24249225*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^4*b*d^3*sgn(b*x + a)/e + 2909907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^5*d*sgn(b*x + a) + 4849845*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^4*b*d^3*sgn(b*x + a)/e^2 + 9699690*(3*(e*x + d)^(5/2) - 10*(e* x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^3*b^2*d^3*sgn(b*x + a)/e^2 + 29 09907*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B* a^5*d^2*sgn(b*x + a)/e + 14549535*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)* d + 15*sqrt(e*x + d)*d^2)*A*a^4*b*d^2*sgn(b*x + a)/e + 415701*(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^5*sgn(b*x + a) + 4157010*(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^3*b^2*d^3*sgn(b*x + a)/e^3 + 4157010*(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^2*b^3*d^3*sgn(b*x + a)/e^3 + 623551 5*(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^4*b*d^2*sgn(b*x + a)/e^2 + 12471030*(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*b^2*d^2*sgn(b*x + a)/e^2 + 1247103*(5*(e*x + d)^(7/2) - 21*(e*x...
Timed out. \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]