Integrand size = 23, antiderivative size = 432 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^7}-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{262144 c^{15/2}} \] Output:
1/131072*(-4*a*c+b^2)^2*(240*A*a*b*c^2-220*A*b^3*c+48*B*a^2*c^2-264*B*a*b^ 2*c+143*B*b^4)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^7-1/49152*(-4*a*c+b^2)*(240 *A*a*b*c^2-220*A*b^3*c+48*B*a^2*c^2-264*B*a*b^2*c+143*B*b^4)*(2*c*x+b)*(c* x^2+b*x+a)^(3/2)/c^6+1/15360*(240*A*a*b*c^2-220*A*b^3*c+48*B*a^2*c^2-264*B *a*b^2*c+143*B*b^4)*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^5-1/180*(-20*A*c+13*B* b)*x^2*(c*x^2+b*x+a)^(7/2)/c^2+1/10*B*x^3*(c*x^2+b*x+a)^(7/2)/c-1/40320*(1 287*B*b^3-1980*A*b^2*c-1804*B*a*b*c+1280*A*a*c^2-14*c*(-220*A*b*c-108*B*a* c+143*B*b^2)*x)*(c*x^2+b*x+a)^(7/2)/c^4-1/262144*(-4*a*c+b^2)^3*(240*A*a*b *c^2-220*A*b^3*c+48*B*a^2*c^2-264*B*a*b^2*c+143*B*b^4)*arctanh(1/2*(2*c*x+ b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(15/2)
Time = 6.28 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.35 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (45045 b^9 B-2310 b^8 c (30 A+13 B x)+1848 b^7 c (-305 a B+c x (25 A+13 B x))-640 b^3 c^3 \left (6885 a^3 B-8 c^3 x^5 (5 A+3 B x)+4 a c^2 x^3 (107 A+60 B x)-3 a^2 c x (879 A+431 B x)\right )-320 b^4 c^3 \left (4 c^2 x^4 (22 A+13 B x)+207 a^2 (49 A+20 B x)-a c x^2 (1116 A+605 B x)\right )+32 b^5 c^2 \left (77742 a^2 B+22 c^2 x^3 (45 A+26 B x)-9 a c x (1715 A+869 B x)\right )+48 b^6 c^2 \left (-11 c x^2 (70 A+39 B x)+7 a (2425 A+1023 B x)\right )+512 c^5 \left (896 c^4 x^8 (10 A+9 B x)+10 a^3 c x^2 (128 A+63 B x)-5 a^4 (512 A+189 B x)+24 a^2 c^2 x^4 (800 A+651 B x)+16 a c^3 x^6 (1520 A+1323 B x)\right )+256 b^2 c^4 \left (120 a c^2 x^4 (7 A+4 B x)-15 a^2 c x^2 (266 A+139 B x)+5 a^3 (3663 A+1433 B x)+8 c^3 x^6 (3090 A+2681 B x)\right )+256 b c^4 \left (9295 a^4 B+60 a^2 c^2 x^3 (41 A+22 B x)+224 c^4 x^7 (185 A+164 B x)-10 a^3 c x (689 A+323 B x)+16 a c^3 x^5 (3765 A+3181 B x)\right )\right )+315 \left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{82575360 c^{15/2}} \] Input:
Integrate[x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
Output:
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(45045*b^9*B - 2310*b^8*c*(30*A + 13*B*x) + 1848*b^7*c*(-305*a*B + c*x*(25*A + 13*B*x)) - 640*b^3*c^3*(6885*a^3*B - 8*c^3*x^5*(5*A + 3*B*x) + 4*a*c^2*x^3*(107*A + 60*B*x) - 3*a^2*c*x*(879*A + 431*B*x)) - 320*b^4*c^3*(4*c^2*x^4*(22*A + 13*B*x) + 207*a^2*(49*A + 20 *B*x) - a*c*x^2*(1116*A + 605*B*x)) + 32*b^5*c^2*(77742*a^2*B + 22*c^2*x^3 *(45*A + 26*B*x) - 9*a*c*x*(1715*A + 869*B*x)) + 48*b^6*c^2*(-11*c*x^2*(70 *A + 39*B*x) + 7*a*(2425*A + 1023*B*x)) + 512*c^5*(896*c^4*x^8*(10*A + 9*B *x) + 10*a^3*c*x^2*(128*A + 63*B*x) - 5*a^4*(512*A + 189*B*x) + 24*a^2*c^2 *x^4*(800*A + 651*B*x) + 16*a*c^3*x^6*(1520*A + 1323*B*x)) + 256*b^2*c^4*( 120*a*c^2*x^4*(7*A + 4*B*x) - 15*a^2*c*x^2*(266*A + 139*B*x) + 5*a^3*(3663 *A + 1433*B*x) + 8*c^3*x^6*(3090*A + 2681*B*x)) + 256*b*c^4*(9295*a^4*B + 60*a^2*c^2*x^3*(41*A + 22*B*x) + 224*c^4*x^7*(185*A + 164*B*x) - 10*a^3*c* x*(689*A + 323*B*x) + 16*a*c^3*x^5*(3765*A + 3181*B*x))) + 315*(b^2 - 4*a* c)^3*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a^2*B*c ^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(82575360*c^(15/2))
Time = 0.64 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.81, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1236, 27, 1236, 27, 1225, 1087, 1087, 1087, 1092, 219}
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.
| \(\displaystyle \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\int -\frac {1}{2} x^2 (6 a B+(13 b B-20 A c) x) \left (c x^2+b x+a\right )^{5/2}dx}{10 c}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\int x^2 (6 a B+(13 b B-20 A c) x) \left (c x^2+b x+a\right )^{5/2}dx}{20 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\frac {\int -\frac {1}{2} x \left (4 a (13 b B-20 A c)+\left (143 B b^2-220 A c b-108 a B c\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}dx}{9 c}+\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{9 c}}{20 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{9 c}-\frac {\int x \left (4 a (13 b B-20 A c)+\left (143 B b^2-220 A c b-108 a B c\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}dx}{18 c}}{20 c}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{9 c}-\frac {\frac {9 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \int \left (c x^2+b x+a\right )^{5/2}dx}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{112 c^2}}{18 c}}{20 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{9 c}-\frac {\frac {9 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{112 c^2}}{18 c}}{20 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{9 c}-\frac {\frac {9 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{112 c^2}}{18 c}}{20 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{9 c}-\frac {\frac {9 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{112 c^2}}{18 c}}{20 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{9 c}-\frac {\frac {9 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{112 c^2}}{18 c}}{20 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{9 c}-\frac {\frac {9 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{112 c^2}}{18 c}}{20 c}\) |
Input:
Int[x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
Output:
(B*x^3*(a + b*x + c*x^2)^(7/2))/(10*c) - (((13*b*B - 20*A*c)*x^2*(a + b*x + c*x^2)^(7/2))/(9*c) - (-1/112*((1287*b^3*B - 1980*A*b^2*c - 1804*a*b*B*c + 1280*a*A*c^2 - 14*c*(143*b^2*B - 220*A*b*c - 108*a*B*c)*x)*(a + b*x + c *x^2)^(7/2))/c^2 + (9*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b *c^2 + 48*a^2*B*c^2)*(((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(12*c) - (5*(b ^2 - 4*a*c)*(((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8*c) - (3*(b^2 - 4*a*c )*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))))/(16*c)))/(24*c)) )/(32*c^2))/(18*c))/(20*c)
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Time = 1.25 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(-\frac {\left (-4128768 B \,c^{9} x^{9}-4587520 A \,c^{9} x^{8}-9404416 B b \,c^{8} x^{8}-10608640 A b \,c^{8} x^{7}-10838016 B a \,c^{8} x^{7}-5490688 B \,b^{2} c^{7} x^{7}-12451840 A a \,c^{8} x^{6}-6328320 A \,b^{2} c^{7} x^{6}-13029376 B a b \,c^{7} x^{6}-15360 B \,b^{3} c^{6} x^{6}-15421440 A a b \,c^{7} x^{5}-25600 A \,b^{3} c^{6} x^{5}-7999488 B \,a^{2} c^{7} x^{5}-122880 B a \,b^{2} c^{6} x^{5}+16640 B \,b^{4} c^{5} x^{5}-9830400 A \,a^{2} c^{7} x^{4}-215040 A a \,b^{2} c^{6} x^{4}+28160 A \,b^{4} c^{5} x^{4}-337920 B \,a^{2} b \,c^{6} x^{4}+153600 B a \,b^{3} c^{5} x^{4}-18304 B \,b^{5} c^{4} x^{4}-629760 A \,a^{2} b \,c^{6} x^{3}+273920 A a \,b^{3} c^{5} x^{3}-31680 A \,b^{5} c^{4} x^{3}-322560 B \,a^{3} c^{6} x^{3}+533760 B \,a^{2} b^{2} c^{5} x^{3}-193600 B a \,b^{4} c^{4} x^{3}+20592 B \,b^{6} c^{3} x^{3}-655360 A \,a^{3} c^{6} x^{2}+1021440 A \,a^{2} b^{2} c^{5} x^{2}-357120 A a \,b^{4} c^{4} x^{2}+36960 A \,b^{6} c^{3} x^{2}+826880 B \,a^{3} b \,c^{5} x^{2}-827520 B \,a^{2} b^{3} c^{4} x^{2}+250272 B a \,b^{5} c^{3} x^{2}-24024 B \,b^{7} c^{2} x^{2}+1763840 A \,a^{3} b \,c^{5} x -1687680 A \,a^{2} b^{3} c^{4} x +493920 A a \,b^{5} c^{3} x -46200 A \,b^{7} c^{2} x +483840 B \,a^{4} c^{5} x -1834240 B \,a^{3} b^{2} c^{4} x +1324800 B \,a^{2} b^{4} c^{3} x -343728 B a \,b^{6} c^{2} x +30030 B \,b^{8} c x +1310720 A \,a^{4} c^{5}-4688640 A \,a^{3} b^{2} c^{4}+3245760 A \,a^{2} b^{4} c^{3}-814800 A a \,b^{6} c^{2}+69300 A \,b^{8} c -2379520 B \,a^{4} b \,c^{4}+4406400 B \,a^{3} b^{3} c^{3}-2487744 B \,a^{2} b^{5} c^{2}+563640 B a \,b^{7} c -45045 B \,b^{9}\right ) \sqrt {c \,x^{2}+b x +a}}{41287680 c^{7}}+\frac {\left (15360 A \,a^{4} b \,c^{5}-25600 A \,a^{3} b^{3} c^{4}+13440 A \,a^{2} b^{5} c^{3}-2880 A a \,b^{7} c^{2}+220 A \,b^{9} c +3072 B \,a^{5} c^{5}-19200 B \,a^{4} b^{2} c^{4}+22400 B \,a^{3} b^{4} c^{3}-10080 B \,a^{2} b^{6} c^{2}+1980 B a \,b^{8} c -143 B \,b^{10}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{262144 c^{\frac {15}{2}}}\) | \(794\) |
| default | \(\text {Expression too large to display}\) | \(1454\) |
Input:
int(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/41287680/c^7*(-4128768*B*c^9*x^9-4587520*A*c^9*x^8-9404416*B*b*c^8*x^8- 10608640*A*b*c^8*x^7-10838016*B*a*c^8*x^7-5490688*B*b^2*c^7*x^7-12451840*A *a*c^8*x^6-6328320*A*b^2*c^7*x^6-13029376*B*a*b*c^7*x^6-15360*B*b^3*c^6*x^ 6-15421440*A*a*b*c^7*x^5-25600*A*b^3*c^6*x^5-7999488*B*a^2*c^7*x^5-122880* B*a*b^2*c^6*x^5+16640*B*b^4*c^5*x^5-9830400*A*a^2*c^7*x^4-215040*A*a*b^2*c ^6*x^4+28160*A*b^4*c^5*x^4-337920*B*a^2*b*c^6*x^4+153600*B*a*b^3*c^5*x^4-1 8304*B*b^5*c^4*x^4-629760*A*a^2*b*c^6*x^3+273920*A*a*b^3*c^5*x^3-31680*A*b ^5*c^4*x^3-322560*B*a^3*c^6*x^3+533760*B*a^2*b^2*c^5*x^3-193600*B*a*b^4*c^ 4*x^3+20592*B*b^6*c^3*x^3-655360*A*a^3*c^6*x^2+1021440*A*a^2*b^2*c^5*x^2-3 57120*A*a*b^4*c^4*x^2+36960*A*b^6*c^3*x^2+826880*B*a^3*b*c^5*x^2-827520*B* a^2*b^3*c^4*x^2+250272*B*a*b^5*c^3*x^2-24024*B*b^7*c^2*x^2+1763840*A*a^3*b *c^5*x-1687680*A*a^2*b^3*c^4*x+493920*A*a*b^5*c^3*x-46200*A*b^7*c^2*x+4838 40*B*a^4*c^5*x-1834240*B*a^3*b^2*c^4*x+1324800*B*a^2*b^4*c^3*x-343728*B*a* b^6*c^2*x+30030*B*b^8*c*x+1310720*A*a^4*c^5-4688640*A*a^3*b^2*c^4+3245760* A*a^2*b^4*c^3-814800*A*a*b^6*c^2+69300*A*b^8*c-2379520*B*a^4*b*c^4+4406400 *B*a^3*b^3*c^3-2487744*B*a^2*b^5*c^2+563640*B*a*b^7*c-45045*B*b^9)*(c*x^2+ b*x+a)^(1/2)+1/262144*(15360*A*a^4*b*c^5-25600*A*a^3*b^3*c^4+13440*A*a^2*b ^5*c^3-2880*A*a*b^7*c^2+220*A*b^9*c+3072*B*a^5*c^5-19200*B*a^4*b^2*c^4+224 00*B*a^3*b^4*c^3-10080*B*a^2*b^6*c^2+1980*B*a*b^8*c-143*B*b^10)/c^(15/2)*l n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
Time = 0.29 (sec) , antiderivative size = 1511, normalized size of antiderivative = 3.50 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
[-1/165150720*(315*(143*B*b^10 - 3072*(B*a^5 + 5*A*a^4*b)*c^5 + 6400*(3*B* a^4*b^2 + 4*A*a^3*b^3)*c^4 - 4480*(5*B*a^3*b^4 + 3*A*a^2*b^5)*c^3 + 1440*( 7*B*a^2*b^6 + 2*A*a*b^7)*c^2 - 220*(9*B*a*b^8 + A*b^9)*c)*sqrt(c)*log(-8*c ^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a *c) - 4*(4128768*B*c^10*x^9 + 45045*B*b^9*c - 1310720*A*a^4*c^6 + 229376*( 41*B*b*c^9 + 20*A*c^10)*x^8 + 14336*(383*B*b^2*c^8 + 4*(189*B*a + 185*A*b) *c^9)*x^7 + 1024*(15*B*b^3*c^7 + 12160*A*a*c^9 + 4*(3181*B*a*b + 1545*A*b^ 2)*c^8)*x^6 + 14080*(169*B*a^4*b + 333*A*a^3*b^2)*c^5 - 256*(65*B*b^4*c^6 - 48*(651*B*a^2 + 1255*A*a*b)*c^8 - 20*(24*B*a*b^2 + 5*A*b^3)*c^7)*x^5 - 2 880*(1530*B*a^3*b^3 + 1127*A*a^2*b^4)*c^4 + 128*(143*B*b^5*c^5 + 76800*A*a ^2*c^8 + 240*(11*B*a^2*b + 7*A*a*b^2)*c^7 - 20*(60*B*a*b^3 + 11*A*b^4)*c^6 )*x^4 + 336*(7404*B*a^2*b^5 + 2425*A*a*b^6)*c^3 - 16*(1287*B*b^6*c^4 - 960 *(21*B*a^3 + 41*A*a^2*b)*c^7 + 80*(417*B*a^2*b^2 + 214*A*a*b^3)*c^6 - 220* (55*B*a*b^4 + 9*A*b^5)*c^5)*x^3 - 4620*(122*B*a*b^7 + 15*A*b^8)*c^2 + 8*(3 003*B*b^7*c^3 + 81920*A*a^3*c^7 - 6080*(17*B*a^3*b + 21*A*a^2*b^2)*c^6 + 2 40*(431*B*a^2*b^3 + 186*A*a*b^4)*c^5 - 132*(237*B*a*b^5 + 35*A*b^6)*c^4)*x ^2 - 2*(15015*B*b^8*c^2 + 1280*(189*B*a^4 + 689*A*a^3*b)*c^6 - 320*(2866*B *a^3*b^2 + 2637*A*a^2*b^3)*c^5 + 720*(920*B*a^2*b^4 + 343*A*a*b^5)*c^4 - 9 24*(186*B*a*b^6 + 25*A*b^7)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^8, 1/82575360 *(315*(143*B*b^10 - 3072*(B*a^5 + 5*A*a^4*b)*c^5 + 6400*(3*B*a^4*b^2 + ...
Leaf count of result is larger than twice the leaf count of optimal. 10654 vs. \(2 (461) = 922\).
Time = 0.91 (sec) , antiderivative size = 10654, normalized size of antiderivative = 24.66 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)
Output:
Piecewise(((-a*(-3*a*(3*A*a**2*b + B*a**3 - 5*a*(6*A*a*b*c + A*b**3 + 3*B* a**2*c + 3*B*a*b**2 - 7*a*(3*A*b*c**2 + 21*B*a*c**2/10 + 3*B*b**2*c - 17*b *(A*c**3 + 41*B*b*c**2/20)/(18*c))/(8*c) - 13*b*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 8*a*(A*c**3 + 41*B*b*c**2/20)/(9*c) - 15*b*(3*A*b*c* *2 + 21*B*a*c**2/10 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B*b*c**2/20)/(18*c))/ (16*c))/(14*c))/(6*c) - 9*b*(3*A*a**2*c + 3*A*a*b**2 + 3*B*a**2*b - 6*a*(3 *A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 8*a*(A*c**3 + 41*B*b*c**2/20 )/(9*c) - 15*b*(3*A*b*c**2 + 21*B*a*c**2/10 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B*b*c**2/20)/(18*c))/(16*c))/(7*c) - 11*b*(6*A*a*b*c + A*b**3 + 3*B*a** 2*c + 3*B*a*b**2 - 7*a*(3*A*b*c**2 + 21*B*a*c**2/10 + 3*B*b**2*c - 17*b*(A *c**3 + 41*B*b*c**2/20)/(18*c))/(8*c) - 13*b*(3*A*a*c**2 + 3*A*b**2*c + 6* B*a*b*c + B*b**3 - 8*a*(A*c**3 + 41*B*b*c**2/20)/(9*c) - 15*b*(3*A*b*c**2 + 21*B*a*c**2/10 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B*b*c**2/20)/(18*c))/(16 *c))/(14*c))/(12*c))/(10*c))/(4*c) - 5*b*(A*a**3 - 4*a*(3*A*a**2*c + 3*A*a *b**2 + 3*B*a**2*b - 6*a*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 8 *a*(A*c**3 + 41*B*b*c**2/20)/(9*c) - 15*b*(3*A*b*c**2 + 21*B*a*c**2/10 + 3 *B*b**2*c - 17*b*(A*c**3 + 41*B*b*c**2/20)/(18*c))/(16*c))/(7*c) - 11*b*(6 *A*a*b*c + A*b**3 + 3*B*a**2*c + 3*B*a*b**2 - 7*a*(3*A*b*c**2 + 21*B*a*c** 2/10 + 3*B*b**2*c - 17*b*(A*c**3 + 41*B*b*c**2/20)/(18*c))/(8*c) - 13*b*(3 *A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 8*a*(A*c**3 + 41*B*b*c**2...
Exception generated. \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.28 (sec) , antiderivative size = 767, normalized size of antiderivative = 1.78 \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:
integrate(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
1/41287680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*(18*B*c^2*x + ( 41*B*b*c^10 + 20*A*c^11)/c^9)*x + (383*B*b^2*c^9 + 756*B*a*c^10 + 740*A*b* c^10)/c^9)*x + (15*B*b^3*c^8 + 12724*B*a*b*c^9 + 6180*A*b^2*c^9 + 12160*A* a*c^10)/c^9)*x - (65*B*b^4*c^7 - 480*B*a*b^2*c^8 - 100*A*b^3*c^8 - 31248*B *a^2*c^9 - 60240*A*a*b*c^9)/c^9)*x + (143*B*b^5*c^6 - 1200*B*a*b^3*c^7 - 2 20*A*b^4*c^7 + 2640*B*a^2*b*c^8 + 1680*A*a*b^2*c^8 + 76800*A*a^2*c^9)/c^9) *x - (1287*B*b^6*c^5 - 12100*B*a*b^4*c^6 - 1980*A*b^5*c^6 + 33360*B*a^2*b^ 2*c^7 + 17120*A*a*b^3*c^7 - 20160*B*a^3*c^8 - 39360*A*a^2*b*c^8)/c^9)*x + (3003*B*b^7*c^4 - 31284*B*a*b^5*c^5 - 4620*A*b^6*c^5 + 103440*B*a^2*b^3*c^ 6 + 44640*A*a*b^4*c^6 - 103360*B*a^3*b*c^7 - 127680*A*a^2*b^2*c^7 + 81920* A*a^3*c^8)/c^9)*x - (15015*B*b^8*c^3 - 171864*B*a*b^6*c^4 - 23100*A*b^7*c^ 4 + 662400*B*a^2*b^4*c^5 + 246960*A*a*b^5*c^5 - 917120*B*a^3*b^2*c^6 - 843 840*A*a^2*b^3*c^6 + 241920*B*a^4*c^7 + 881920*A*a^3*b*c^7)/c^9)*x + (45045 *B*b^9*c^2 - 563640*B*a*b^7*c^3 - 69300*A*b^8*c^3 + 2487744*B*a^2*b^5*c^4 + 814800*A*a*b^6*c^4 - 4406400*B*a^3*b^3*c^5 - 3245760*A*a^2*b^4*c^5 + 237 9520*B*a^4*b*c^6 + 4688640*A*a^3*b^2*c^6 - 1310720*A*a^4*c^7)/c^9) + 1/262 144*(143*B*b^10 - 1980*B*a*b^8*c - 220*A*b^9*c + 10080*B*a^2*b^6*c^2 + 288 0*A*a*b^7*c^2 - 22400*B*a^3*b^4*c^3 - 13440*A*a^2*b^5*c^3 + 19200*B*a^4*b^ 2*c^4 + 25600*A*a^3*b^3*c^4 - 3072*B*a^5*c^5 - 15360*A*a^4*b*c^5)*log(abs( 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(15/2)
Timed out. \[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int x^3\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \] Input:
int(x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2),x)
Output:
int(x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2), x)
\[ \int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int x^{3} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}d x \] Input:
int(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)
Output:
int(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)