Integrand size = 35, antiderivative size = 308 \[ \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {2 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)} \] Output:
2/5*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-2/ 7*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^ 5/(b*x+a)+2/3*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(9/2)*((b*x+a)^2 )^(1/2)/e^5/(b*x+a)-2/11*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(11/2)*((b*x +a)^2)^(1/2)/e^5/(b*x+a)+2/13*b^3*B*(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^5/( b*x+a)
Time = 0.29 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.80 \[ \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (429 a^3 e^3 (-2 B d+7 A e+5 B e x)+143 a^2 b e^2 \left (9 A e (-2 d+5 e x)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-13 a b^2 e \left (-11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )+3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )+b^3 \left (13 A e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+B \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )\right )}{15015 e^5 (a+b x)} \] Input:
Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
Output:
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(429*a^3*e^3*(-2*B*d + 7*A*e + 5*B*e* x) + 143*a^2*b*e^2*(9*A*e*(-2*d + 5*e*x) + B*(8*d^2 - 20*d*e*x + 35*e^2*x^ 2)) - 13*a*b^2*e*(-11*A*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 3*B*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3)) + b^3*(13*A*e*(-16*d^3 + 40*d^2* e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + B*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2 *x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4))))/(15015*e^5*(a + b*x))
Time = 0.60 (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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A+B x) (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^3 (A+B x) (d+e x)^{3/2}dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^3 (A+B x) (d+e x)^{3/2}dx}{a+b x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^3 B (d+e x)^{11/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{9/2}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^4}+\frac {(a e-b d)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{5/2}}{e^4}+\frac {(a e-b d)^3 (A e-B d) (d+e x)^{3/2}}{e^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b^2 (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5}+\frac {2 b (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5}-\frac {2 (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac {2 (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5}+\frac {2 b^3 B (d+e x)^{13/2}}{13 e^5}\right )}{a+b x}\) |
Input:
Int[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(5/ 2))/(5*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(7/2) )/(7*e^5) + (2*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2))/(3 *e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(11/2))/(11*e^5) + (2 *b^3*B*(d + e*x)^(13/2))/(13*e^5)))/(a + b*x)
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 = 1.84 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.03
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 B \,b^{3} x^{4} e^{4}+1365 A \,b^{3} e^{4} x^{3}+4095 B a \,b^{2} e^{4} x^{3}-840 B \,b^{3} d \,e^{3} x^{3}+5005 A a \,b^{2} e^{4} x^{2}-910 A \,b^{3} d \,e^{3} x^{2}+5005 B \,a^{2} b \,e^{4} x^{2}-2730 B a \,b^{2} d \,e^{3} x^{2}+560 B \,b^{3} d^{2} e^{2} x^{2}+6435 A \,a^{2} b \,e^{4} x -2860 A a \,b^{2} d \,e^{3} x +520 A \,b^{3} d^{2} e^{2} x +2145 B \,a^{3} e^{4} x -2860 B \,a^{2} b d \,e^{3} x +1560 B a \,b^{2} d^{2} e^{2} x -320 B \,b^{3} d^{3} e x +3003 A \,a^{3} e^{4}-2574 A \,a^{2} b d \,e^{3}+1144 A a \,b^{2} d^{2} e^{2}-208 A \,b^{3} d^{3} e -858 B \,a^{3} d \,e^{3}+1144 B \,a^{2} b \,d^{2} e^{2}-624 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15015 e^{5} \left (b x +a \right )^{3}}\) | \(317\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 B \,b^{3} x^{4} e^{4}+1365 A \,b^{3} e^{4} x^{3}+4095 B a \,b^{2} e^{4} x^{3}-840 B \,b^{3} d \,e^{3} x^{3}+5005 A a \,b^{2} e^{4} x^{2}-910 A \,b^{3} d \,e^{3} x^{2}+5005 B \,a^{2} b \,e^{4} x^{2}-2730 B a \,b^{2} d \,e^{3} x^{2}+560 B \,b^{3} d^{2} e^{2} x^{2}+6435 A \,a^{2} b \,e^{4} x -2860 A a \,b^{2} d \,e^{3} x +520 A \,b^{3} d^{2} e^{2} x +2145 B \,a^{3} e^{4} x -2860 B \,a^{2} b d \,e^{3} x +1560 B a \,b^{2} d^{2} e^{2} x -320 B \,b^{3} d^{3} e x +3003 A \,a^{3} e^{4}-2574 A \,a^{2} b d \,e^{3}+1144 A a \,b^{2} d^{2} e^{2}-208 A \,b^{3} d^{3} e -858 B \,a^{3} d \,e^{3}+1144 B \,a^{2} b \,d^{2} e^{2}-624 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15015 e^{5} \left (b x +a \right )^{3}}\) | \(317\) |
| orering | \(\frac {2 \left (1155 B \,b^{3} x^{4} e^{4}+1365 A \,b^{3} e^{4} x^{3}+4095 B a \,b^{2} e^{4} x^{3}-840 B \,b^{3} d \,e^{3} x^{3}+5005 A a \,b^{2} e^{4} x^{2}-910 A \,b^{3} d \,e^{3} x^{2}+5005 B \,a^{2} b \,e^{4} x^{2}-2730 B a \,b^{2} d \,e^{3} x^{2}+560 B \,b^{3} d^{2} e^{2} x^{2}+6435 A \,a^{2} b \,e^{4} x -2860 A a \,b^{2} d \,e^{3} x +520 A \,b^{3} d^{2} e^{2} x +2145 B \,a^{3} e^{4} x -2860 B \,a^{2} b d \,e^{3} x +1560 B a \,b^{2} d^{2} e^{2} x -320 B \,b^{3} d^{3} e x +3003 A \,a^{3} e^{4}-2574 A \,a^{2} b d \,e^{3}+1144 A a \,b^{2} d^{2} e^{2}-208 A \,b^{3} d^{3} e -858 B \,a^{3} d \,e^{3}+1144 B \,a^{2} b \,d^{2} e^{2}-624 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}{15015 e^{5} \left (b x +a \right )^{3}}\) | \(326\) |
| risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (1155 B \,b^{3} x^{6} e^{6}+1365 A \,b^{3} e^{6} x^{5}+4095 B a \,b^{2} e^{6} x^{5}+1470 B \,b^{3} d \,e^{5} x^{5}+5005 A a \,b^{2} e^{6} x^{4}+1820 A \,b^{3} d \,e^{5} x^{4}+5005 B \,a^{2} b \,e^{6} x^{4}+5460 B a \,b^{2} d \,e^{5} x^{4}+35 B \,b^{3} d^{2} e^{4} x^{4}+6435 A \,a^{2} b \,e^{6} x^{3}+7150 A a \,b^{2} d \,e^{5} x^{3}+65 A \,b^{3} d^{2} e^{4} x^{3}+2145 B \,a^{3} e^{6} x^{3}+7150 B \,a^{2} b d \,e^{5} x^{3}+195 B a \,b^{2} d^{2} e^{4} x^{3}-40 B \,b^{3} d^{3} e^{3} x^{3}+3003 A \,a^{3} e^{6} x^{2}+10296 A \,a^{2} b d \,e^{5} x^{2}+429 A a \,b^{2} d^{2} e^{4} x^{2}-78 A \,b^{3} d^{3} e^{3} x^{2}+3432 B \,a^{3} d \,e^{5} x^{2}+429 B \,a^{2} b \,d^{2} e^{4} x^{2}-234 B a \,b^{2} d^{3} e^{3} x^{2}+48 B \,b^{3} d^{4} e^{2} x^{2}+6006 A \,a^{3} d \,e^{5} x +1287 A \,a^{2} b \,d^{2} e^{4} x -572 A a \,b^{2} d^{3} e^{3} x +104 A \,b^{3} d^{4} e^{2} x +429 B \,a^{3} d^{2} e^{4} x -572 B \,a^{2} b \,d^{3} e^{3} x +312 B a \,b^{2} d^{4} e^{2} x -64 B \,b^{3} d^{5} e x +3003 A \,a^{3} d^{2} e^{4}-2574 A \,a^{2} b \,d^{3} e^{3}+1144 A a \,b^{2} d^{4} e^{2}-208 A \,b^{3} d^{5} e -858 B \,a^{3} d^{3} e^{3}+1144 B \,a^{2} b \,d^{4} e^{2}-624 B a \,b^{2} d^{5} e +128 B \,b^{3} d^{6}\right ) \sqrt {e x +d}}{15015 \left (b x +a \right ) e^{5}}\) | \(561\) |
Input:
int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERB OSE)
Output:
2/15015*(e*x+d)^(5/2)*(1155*B*b^3*e^4*x^4+1365*A*b^3*e^4*x^3+4095*B*a*b^2* e^4*x^3-840*B*b^3*d*e^3*x^3+5005*A*a*b^2*e^4*x^2-910*A*b^3*d*e^3*x^2+5005* B*a^2*b*e^4*x^2-2730*B*a*b^2*d*e^3*x^2+560*B*b^3*d^2*e^2*x^2+6435*A*a^2*b* e^4*x-2860*A*a*b^2*d*e^3*x+520*A*b^3*d^2*e^2*x+2145*B*a^3*e^4*x-2860*B*a^2 *b*d*e^3*x+1560*B*a*b^2*d^2*e^2*x-320*B*b^3*d^3*e*x+3003*A*a^3*e^4-2574*A* a^2*b*d*e^3+1144*A*a*b^2*d^2*e^2-208*A*b^3*d^3*e-858*B*a^3*d*e^3+1144*B*a^ 2*b*d^2*e^2-624*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
Time = 0.08 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.45 \[ \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (1155 \, B b^{3} e^{6} x^{6} + 128 \, B b^{3} d^{6} + 3003 \, A a^{3} d^{2} e^{4} - 208 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e + 1144 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{2} - 858 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{3} + 105 \, {\left (14 \, B b^{3} d e^{5} + 13 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{6}\right )} x^{5} + 35 \, {\left (B b^{3} d^{2} e^{4} + 52 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{5} + 143 \, {\left (B a^{2} b + A a b^{2}\right )} e^{6}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{3} e^{3} - 13 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{4} - 1430 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{5} - 429 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{4} e^{2} + 1001 \, A a^{3} e^{6} - 26 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{3} + 143 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{4} + 1144 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{5} e - 6006 \, A a^{3} d e^{5} - 104 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{2} + 572 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{3} - 429 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{5}} \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm=" fricas")
Output:
2/15015*(1155*B*b^3*e^6*x^6 + 128*B*b^3*d^6 + 3003*A*a^3*d^2*e^4 - 208*(3* B*a*b^2 + A*b^3)*d^5*e + 1144*(B*a^2*b + A*a*b^2)*d^4*e^2 - 858*(B*a^3 + 3 *A*a^2*b)*d^3*e^3 + 105*(14*B*b^3*d*e^5 + 13*(3*B*a*b^2 + A*b^3)*e^6)*x^5 + 35*(B*b^3*d^2*e^4 + 52*(3*B*a*b^2 + A*b^3)*d*e^5 + 143*(B*a^2*b + A*a*b^ 2)*e^6)*x^4 - 5*(8*B*b^3*d^3*e^3 - 13*(3*B*a*b^2 + A*b^3)*d^2*e^4 - 1430*( B*a^2*b + A*a*b^2)*d*e^5 - 429*(B*a^3 + 3*A*a^2*b)*e^6)*x^3 + 3*(16*B*b^3* d^4*e^2 + 1001*A*a^3*e^6 - 26*(3*B*a*b^2 + A*b^3)*d^3*e^3 + 143*(B*a^2*b + A*a*b^2)*d^2*e^4 + 1144*(B*a^3 + 3*A*a^2*b)*d*e^5)*x^2 - (64*B*b^3*d^5*e - 6006*A*a^3*d*e^5 - 104*(3*B*a*b^2 + A*b^3)*d^4*e^2 + 572*(B*a^2*b + A*a* b^2)*d^3*e^3 - 429*(B*a^3 + 3*A*a^2*b)*d^2*e^4)*x)*sqrt(e*x + d)/e^5
\[ \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
Output:
Integral((A + B*x)*(d + e*x)**(3/2)*((a + b*x)**2)**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (233) = 466\).
Time = 0.09 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.58 \[ \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d} A}{1155 \, e^{4}} + \frac {2 \, {\left (1155 \, b^{3} e^{6} x^{6} + 128 \, b^{3} d^{6} - 624 \, a b^{2} d^{5} e + 1144 \, a^{2} b d^{4} e^{2} - 858 \, a^{3} d^{3} e^{3} + 105 \, {\left (14 \, b^{3} d e^{5} + 39 \, a b^{2} e^{6}\right )} x^{5} + 35 \, {\left (b^{3} d^{2} e^{4} + 156 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{3} e^{3} - 39 \, a b^{2} d^{2} e^{4} - 1430 \, a^{2} b d e^{5} - 429 \, a^{3} e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{4} e^{2} - 78 \, a b^{2} d^{3} e^{3} + 143 \, a^{2} b d^{2} e^{4} + 1144 \, a^{3} d e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{5} e - 312 \, a b^{2} d^{4} e^{2} + 572 \, a^{2} b d^{3} e^{3} - 429 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} B}{15015 \, e^{5}} \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm=" maxima")
Output:
2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*a^3*d^2*e^3 + 35*(4*b^3*d*e^4 + 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e^3 - 264*a^2*b*d*e^4 - 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 9 9*a^2*b*d^2*e^3 + 462*a^3*d*e^4)*x)*sqrt(e*x + d)*A/e^4 + 2/15015*(1155*b^ 3*e^6*x^6 + 128*b^3*d^6 - 624*a*b^2*d^5*e + 1144*a^2*b*d^4*e^2 - 858*a^3*d ^3*e^3 + 105*(14*b^3*d*e^5 + 39*a*b^2*e^6)*x^5 + 35*(b^3*d^2*e^4 + 156*a*b ^2*d*e^5 + 143*a^2*b*e^6)*x^4 - 5*(8*b^3*d^3*e^3 - 39*a*b^2*d^2*e^4 - 1430 *a^2*b*d*e^5 - 429*a^3*e^6)*x^3 + 3*(16*b^3*d^4*e^2 - 78*a*b^2*d^3*e^3 + 1 43*a^2*b*d^2*e^4 + 1144*a^3*d*e^5)*x^2 - (64*b^3*d^5*e - 312*a*b^2*d^4*e^2 + 572*a^2*b*d^3*e^3 - 429*a^3*d^2*e^4)*x)*sqrt(e*x + d)*B/e^5
Leaf count of result is larger than twice the leaf count of optimal. 1450 vs. \(2 (233) = 466\).
Time = 0.31 (sec) , antiderivative size = 1450, normalized size of antiderivative = 4.71 \[ \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm=" giac")
Output:
2/45045*(45045*sqrt(e*x + d)*A*a^3*d^2*sgn(b*x + a) + 30030*((e*x + d)^(3/ 2) - 3*sqrt(e*x + d)*d)*A*a^3*d*sgn(b*x + a) + 15015*((e*x + d)^(3/2) - 3* sqrt(e*x + d)*d)*B*a^3*d^2*sgn(b*x + a)/e + 45045*((e*x + d)^(3/2) - 3*sqr t(e*x + d)*d)*A*a^2*b*d^2*sgn(b*x + a)/e + 3003*(3*(e*x + d)^(5/2) - 10*(e *x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^3*sgn(b*x + a) + 9009*(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^2*sgn( b*x + a)/e^2 + 9009*(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^2*sgn(b*x + a)/e^2 + 6006*(3*(e*x + d)^(5/2) - 10*(e *x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^3*d*sgn(b*x + a)/e + 18018*(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*s gn(b*x + a)/e + 3861*(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^2*sgn(b*x + a)/e^3 + 1287* (5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sq rt(e*x + d)*d^3)*A*b^3*d^2*sgn(b*x + a)/e^3 + 7722*(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^2* b*d*sgn(b*x + a)/e^2 + 7722*(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*sgn(b*x + a)/e^2 + 1287*(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*sgn(b*x + a)/e + 3861*(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^...
Timed out. \[ \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \] Input:
int((A + B*x)*(d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)
Output:
int((A + B*x)*(d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)
Time = 0.23 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.07 \[ \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {e x +d}\, \left (1155 b^{4} e^{6} x^{6}+5460 a \,b^{3} e^{6} x^{5}+1470 b^{4} d \,e^{5} x^{5}+10010 a^{2} b^{2} e^{6} x^{4}+7280 a \,b^{3} d \,e^{5} x^{4}+35 b^{4} d^{2} e^{4} x^{4}+8580 a^{3} b \,e^{6} x^{3}+14300 a^{2} b^{2} d \,e^{5} x^{3}+260 a \,b^{3} d^{2} e^{4} x^{3}-40 b^{4} d^{3} e^{3} x^{3}+3003 a^{4} e^{6} x^{2}+13728 a^{3} b d \,e^{5} x^{2}+858 a^{2} b^{2} d^{2} e^{4} x^{2}-312 a \,b^{3} d^{3} e^{3} x^{2}+48 b^{4} d^{4} e^{2} x^{2}+6006 a^{4} d \,e^{5} x +1716 a^{3} b \,d^{2} e^{4} x -1144 a^{2} b^{2} d^{3} e^{3} x +416 a \,b^{3} d^{4} e^{2} x -64 b^{4} d^{5} e x +3003 a^{4} d^{2} e^{4}-3432 a^{3} b \,d^{3} e^{3}+2288 a^{2} b^{2} d^{4} e^{2}-832 a \,b^{3} d^{5} e +128 b^{4} d^{6}\right )}{15015 e^{5}} \] Input:
int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
Output:
(2*sqrt(d + e*x)*(3003*a**4*d**2*e**4 + 6006*a**4*d*e**5*x + 3003*a**4*e** 6*x**2 - 3432*a**3*b*d**3*e**3 + 1716*a**3*b*d**2*e**4*x + 13728*a**3*b*d* e**5*x**2 + 8580*a**3*b*e**6*x**3 + 2288*a**2*b**2*d**4*e**2 - 1144*a**2*b **2*d**3*e**3*x + 858*a**2*b**2*d**2*e**4*x**2 + 14300*a**2*b**2*d*e**5*x* *3 + 10010*a**2*b**2*e**6*x**4 - 832*a*b**3*d**5*e + 416*a*b**3*d**4*e**2* x - 312*a*b**3*d**3*e**3*x**2 + 260*a*b**3*d**2*e**4*x**3 + 7280*a*b**3*d* e**5*x**4 + 5460*a*b**3*e**6*x**5 + 128*b**4*d**6 - 64*b**4*d**5*e*x + 48* b**4*d**4*e**2*x**2 - 40*b**4*d**3*e**3*x**3 + 35*b**4*d**2*e**4*x**4 + 14 70*b**4*d*e**5*x**5 + 1155*b**4*e**6*x**6))/(15015*e**5)