Integrand size = 23, antiderivative size = 455 \[ \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^7}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (b^2-4 a c\right )^2 \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{15/2}} \] Output:
1/32768*(-4*a*c+b^2)*(-96*A*a^2*c^3+432*A*a*b^2*c^2-198*A*b^4*c+240*B*a^2* b*c^2-440*B*a*b^3*c+143*B*b^5)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^7-1/12288*( -96*A*a^2*c^3+432*A*a*b^2*c^2-198*A*b^4*c+240*B*a^2*b*c^2-440*B*a*b^3*c+14 3*B*b^5)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^6+1/2016*(-198*A*b*c-128*B*a*c+14 3*B*b^2)*x^2*(c*x^2+b*x+a)^(5/2)/c^3-1/144*(-18*A*c+13*B*b)*x^3*(c*x^2+b*x +a)^(5/2)/c^2+1/9*B*x^4*(c*x^2+b*x+a)^(5/2)/c+1/80640*(3003*B*b^4-4158*A*b ^3*c-7524*B*a*b^2*c+6696*A*a*b*c^2+2048*B*a^2*c^2-10*c*(504*A*a*c^2-594*A* b^2*c-748*B*a*b*c+429*B*b^3)*x)*(c*x^2+b*x+a)^(5/2)/c^5-1/65536*(-4*a*c+b^ 2)^2*(-96*A*a^2*c^3+432*A*a*b^2*c^2-198*A*b^4*c+240*B*a^2*b*c^2-440*B*a*b^ 3*c+143*B*b^5)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(15/2)
Time = 4.12 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.11 \[ \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (45045 b^8 B-2310 b^7 c (27 A+13 B x)+924 b^6 c (-475 a B+c x (45 A+26 B x))+256 c^4 \left (1024 a^4 B+560 c^4 x^7 (9 A+8 B x)+6 a^2 c^2 x^3 (105 A+64 B x)+40 a c^3 x^5 (189 A+160 B x)-a^3 c x (945 A+512 B x)\right )+72 b^5 c^2 \left (-22 c x^2 (21 A+13 B x)+7 a (1095 A+517 B x)\right )+16 b^4 c^2 \left (86499 a^2 B+22 c^2 x^3 (81 A+52 B x)-9 a c x (2247 A+1276 B x)\right )-192 b^2 c^3 \left (7641 a^3 B-40 c^3 x^5 (3 A+2 B x)+4 a c^2 x^3 (213 A+134 B x)-a^2 c x (3543 A+1970 B x)\right )-32 b^3 c^3 \left (8 c^2 x^4 (99 A+65 B x)-4 a c x^2 (1755 A+1067 B x)+9 a^2 (5103 A+2353 B x)\right )+128 b c^4 \left (24 a c^2 x^4 (39 A+25 B x)+80 c^3 x^6 (153 A+133 B x)-6 a^2 c x^2 (453 A+269 B x)+a^3 (8271 A+3701 B x)\right )\right )+315 \left (b^2-4 a c\right )^2 \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{20643840 c^{15/2}} \] Input:
Integrate[x^4*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]
Output:
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(45045*b^8*B - 2310*b^7*c*(27*A + 13*B*x) + 924*b^6*c*(-475*a*B + c*x*(45*A + 26*B*x)) + 256*c^4*(1024*a^4*B + 560* c^4*x^7*(9*A + 8*B*x) + 6*a^2*c^2*x^3*(105*A + 64*B*x) + 40*a*c^3*x^5*(189 *A + 160*B*x) - a^3*c*x*(945*A + 512*B*x)) + 72*b^5*c^2*(-22*c*x^2*(21*A + 13*B*x) + 7*a*(1095*A + 517*B*x)) + 16*b^4*c^2*(86499*a^2*B + 22*c^2*x^3* (81*A + 52*B*x) - 9*a*c*x*(2247*A + 1276*B*x)) - 192*b^2*c^3*(7641*a^3*B - 40*c^3*x^5*(3*A + 2*B*x) + 4*a*c^2*x^3*(213*A + 134*B*x) - a^2*c*x*(3543* A + 1970*B*x)) - 32*b^3*c^3*(8*c^2*x^4*(99*A + 65*B*x) - 4*a*c*x^2*(1755*A + 1067*B*x) + 9*a^2*(5103*A + 2353*B*x)) + 128*b*c^4*(24*a*c^2*x^4*(39*A + 25*B*x) + 80*c^3*x^6*(153*A + 133*B*x) - 6*a^2*c*x^2*(453*A + 269*B*x) + a^3*(8271*A + 3701*B*x))) + 315*(b^2 - 4*a*c)^2*(143*b^5*B - 198*A*b^4*c - 440*a*b^3*B*c + 432*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(20643840*c^(15/2))
Time = 0.75 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.86, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {1236, 27, 1236, 27, 1236, 27, 1225, 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.
| \(\displaystyle \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\int -\frac {1}{2} x^3 (8 a B+(13 b B-18 A c) x) \left (c x^2+b x+a\right )^{3/2}dx}{9 c}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\int x^3 (8 a B+(13 b B-18 A c) x) \left (c x^2+b x+a\right )^{3/2}dx}{18 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {\int -\frac {1}{2} x^2 \left (6 a (13 b B-18 A c)+\left (143 B b^2-198 A c b-128 a B c\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{8 c}+\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}}{18 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}-\frac {\int x^2 \left (6 a (13 b B-18 A c)+\left (143 B b^2-198 A c b-128 a B c\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{16 c}}{18 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}-\frac {\frac {\int -\frac {1}{2} x \left (4 a \left (143 B b^2-198 A c b-128 a B c\right )+3 \left (429 B b^3-594 A c b^2-748 a B c b+504 a A c^2\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{7 c}+\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{7 c}}{16 c}}{18 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{7 c}-\frac {\int x \left (4 a \left (143 B b^2-198 A c b-128 a B c\right )+3 \left (429 B b^3-594 A c b^2-748 a B c b+504 a A c^2\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}dx}{14 c}}{16 c}}{18 c}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{7 c}-\frac {\frac {21 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{20 c^2}}{14 c}}{16 c}}{18 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{7 c}-\frac {\frac {21 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{20 c^2}}{14 c}}{16 c}}{18 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{7 c}-\frac {\frac {21 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{20 c^2}}{14 c}}{16 c}}{18 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{7 c}-\frac {\frac {21 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{20 c^2}}{14 c}}{16 c}}{18 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}-\frac {\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{8 c}-\frac {\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{7 c}-\frac {\frac {21 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{8 c^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{20 c^2}}{14 c}}{16 c}}{18 c}\) |
Input:
Int[x^4*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]
Output:
(B*x^4*(a + b*x + c*x^2)^(5/2))/(9*c) - (((13*b*B - 18*A*c)*x^3*(a + b*x + c*x^2)^(5/2))/(8*c) - (((143*b^2*B - 198*A*b*c - 128*a*B*c)*x^2*(a + b*x + c*x^2)^(5/2))/(7*c) - (-1/20*((3003*b^4*B - 4158*A*b^3*c - 7524*a*b^2*B* c + 6696*a*A*b*c^2 + 2048*a^2*B*c^2 - 10*c*(429*b^3*B - 594*A*b^2*c - 748* a*b*B*c + 504*a*A*c^2)*x)*(a + b*x + c*x^2)^(5/2))/c^2 + (21*(143*b^5*B - 198*A*b^4*c - 440*a*b^3*B*c + 432*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A *c^3)*(((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)))/(8*c^2))/(14* c))/(16*c))/(18*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 = 1.38 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.43
| method | result | size |
| risch | \(\frac {\left (1146880 B \,c^{8} x^{8}+1290240 A \,c^{8} x^{7}+1361920 B b \,c^{7} x^{7}+1566720 A b \,c^{7} x^{6}+1638400 B a \,c^{7} x^{6}+15360 B \,b^{2} c^{6} x^{6}+1935360 A a \,c^{7} x^{5}+23040 A \,b^{2} c^{6} x^{5}+76800 B a b \,c^{6} x^{5}-16640 B \,b^{3} c^{5} x^{5}+119808 A a b \,c^{6} x^{4}-25344 A \,b^{3} c^{5} x^{4}+98304 B \,a^{2} c^{6} x^{4}-102912 B a \,b^{2} c^{5} x^{4}+18304 B \,b^{4} c^{4} x^{4}+161280 A \,a^{2} c^{6} x^{3}-163584 A a \,b^{2} c^{5} x^{3}+28512 A \,b^{4} c^{4} x^{3}-206592 B \,a^{2} b \,c^{5} x^{3}+136576 B a \,b^{3} c^{4} x^{3}-20592 B \,b^{5} c^{3} x^{3}-347904 A \,a^{2} b \,c^{5} x^{2}+224640 A a \,b^{3} c^{4} x^{2}-33264 A \,b^{5} c^{3} x^{2}-131072 B \,a^{3} c^{5} x^{2}+378240 B \,a^{2} b^{2} c^{4} x^{2}-183744 B a \,b^{4} c^{3} x^{2}+24024 B \,b^{6} c^{2} x^{2}-241920 A \,a^{3} c^{5} x +680256 A \,a^{2} b^{2} c^{4} x -323568 A a \,b^{4} c^{3} x +41580 A \,b^{6} c^{2} x +473728 B \,a^{3} b \,c^{4} x -677664 B \,a^{2} b^{3} c^{3} x +260568 B a \,b^{5} c^{2} x -30030 B \,b^{7} c x +1058688 A \,a^{3} b \,c^{4}-1469664 A \,a^{2} b^{3} c^{3}+551880 A a \,b^{5} c^{2}-62370 A \,b^{7} c +262144 B \,a^{4} c^{4}-1467072 B \,a^{3} b^{2} c^{3}+1383984 B \,a^{2} b^{4} c^{2}-438900 B a \,b^{6} c +45045 B \,b^{8}\right ) \sqrt {c \,x^{2}+b x +a}}{10321920 c^{7}}+\frac {\left (1536 A \,a^{4} c^{5}-7680 A \,a^{3} b^{2} c^{4}+6720 A \,a^{2} b^{4} c^{3}-2016 A a \,b^{6} c^{2}+198 A \,b^{8} c -3840 B \,a^{4} b \,c^{4}+8960 B \,a^{3} b^{3} c^{3}-6048 B \,a^{2} b^{5} c^{2}+1584 B a \,b^{7} c -143 B \,b^{9}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{65536 c^{\frac {15}{2}}}\) | \(652\) |
| default | \(\text {Expression too large to display}\) | \(1890\) |
Input:
int(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/10321920/c^7*(1146880*B*c^8*x^8+1290240*A*c^8*x^7+1361920*B*b*c^7*x^7+15 66720*A*b*c^7*x^6+1638400*B*a*c^7*x^6+15360*B*b^2*c^6*x^6+1935360*A*a*c^7* x^5+23040*A*b^2*c^6*x^5+76800*B*a*b*c^6*x^5-16640*B*b^3*c^5*x^5+119808*A*a *b*c^6*x^4-25344*A*b^3*c^5*x^4+98304*B*a^2*c^6*x^4-102912*B*a*b^2*c^5*x^4+ 18304*B*b^4*c^4*x^4+161280*A*a^2*c^6*x^3-163584*A*a*b^2*c^5*x^3+28512*A*b^ 4*c^4*x^3-206592*B*a^2*b*c^5*x^3+136576*B*a*b^3*c^4*x^3-20592*B*b^5*c^3*x^ 3-347904*A*a^2*b*c^5*x^2+224640*A*a*b^3*c^4*x^2-33264*A*b^5*c^3*x^2-131072 *B*a^3*c^5*x^2+378240*B*a^2*b^2*c^4*x^2-183744*B*a*b^4*c^3*x^2+24024*B*b^6 *c^2*x^2-241920*A*a^3*c^5*x+680256*A*a^2*b^2*c^4*x-323568*A*a*b^4*c^3*x+41 580*A*b^6*c^2*x+473728*B*a^3*b*c^4*x-677664*B*a^2*b^3*c^3*x+260568*B*a*b^5 *c^2*x-30030*B*b^7*c*x+1058688*A*a^3*b*c^4-1469664*A*a^2*b^3*c^3+551880*A* a*b^5*c^2-62370*A*b^7*c+262144*B*a^4*c^4-1467072*B*a^3*b^2*c^3+1383984*B*a ^2*b^4*c^2-438900*B*a*b^6*c+45045*B*b^8)*(c*x^2+b*x+a)^(1/2)+1/65536*(1536 *A*a^4*c^5-7680*A*a^3*b^2*c^4+6720*A*a^2*b^4*c^3-2016*A*a*b^6*c^2+198*A*b^ 8*c-3840*B*a^4*b*c^4+8960*B*a^3*b^3*c^3-6048*B*a^2*b^5*c^2+1584*B*a*b^7*c- 143*B*b^9)/c^(15/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
Time = 0.22 (sec) , antiderivative size = 1263, normalized size of antiderivative = 2.78 \[ \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
[-1/41287680*(315*(143*B*b^9 - 1536*A*a^4*c^5 + 3840*(B*a^4*b + 2*A*a^3*b^ 2)*c^4 - 2240*(4*B*a^3*b^3 + 3*A*a^2*b^4)*c^3 + 2016*(3*B*a^2*b^5 + A*a*b^ 6)*c^2 - 198*(8*B*a*b^7 + A*b^8)*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*(1146880*B*c^9 *x^8 + 45045*B*b^8*c + 71680*(19*B*b*c^8 + 18*A*c^9)*x^7 + 5120*(3*B*b^2*c ^7 + 2*(160*B*a + 153*A*b)*c^8)*x^6 + 128*(2048*B*a^4 + 8271*A*a^3*b)*c^5 - 1280*(13*B*b^3*c^6 - 1512*A*a*c^8 - 6*(10*B*a*b + 3*A*b^2)*c^7)*x^5 - 25 92*(566*B*a^3*b^2 + 567*A*a^2*b^3)*c^4 + 128*(143*B*b^4*c^5 + 24*(32*B*a^2 + 39*A*a*b)*c^7 - 6*(134*B*a*b^2 + 33*A*b^3)*c^6)*x^4 + 504*(2746*B*a^2*b ^4 + 1095*A*a*b^5)*c^3 - 16*(1287*B*b^5*c^4 - 10080*A*a^2*c^7 + 48*(269*B* a^2*b + 213*A*a*b^2)*c^6 - 22*(388*B*a*b^3 + 81*A*b^4)*c^5)*x^3 - 2310*(19 0*B*a*b^6 + 27*A*b^7)*c^2 + 8*(3003*B*b^6*c^3 - 32*(512*B*a^3 + 1359*A*a^2 *b)*c^6 + 240*(197*B*a^2*b^2 + 117*A*a*b^3)*c^5 - 198*(116*B*a*b^4 + 21*A* b^5)*c^4)*x^2 - 2*(15015*B*b^7*c^2 + 120960*A*a^3*c^6 - 32*(7402*B*a^3*b + 10629*A*a^2*b^2)*c^5 + 72*(4706*B*a^2*b^3 + 2247*A*a*b^4)*c^4 - 1386*(94* B*a*b^5 + 15*A*b^6)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^8, 1/20643840*(315*(1 43*B*b^9 - 1536*A*a^4*c^5 + 3840*(B*a^4*b + 2*A*a^3*b^2)*c^4 - 2240*(4*B*a ^3*b^3 + 3*A*a^2*b^4)*c^3 + 2016*(3*B*a^2*b^5 + A*a*b^6)*c^2 - 198*(8*B*a* b^7 + A*b^8)*c)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt (-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(1146880*B*c^9*x^8 + 45045*B*b^8*c + ...
Leaf count of result is larger than twice the leaf count of optimal. 4983 vs. \(2 (496) = 992\).
Time = 0.78 (sec) , antiderivative size = 4983, normalized size of antiderivative = 10.95 \[ \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate(x**4*(B*x+A)*(c*x**2+b*x+a)**(3/2),x)
Output:
Piecewise(((-a*(-3*a*(A*a**2 - 5*a*(2*A*a*c + A*b**2 + 2*B*a*b - 7*a*(A*c* *2 + 19*B*b*c/18)/(8*c) - 13*b*(2*A*b*c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c* *2 + 19*B*b*c/18)/(16*c))/(14*c))/(6*c) - 9*b*(2*A*a*b + B*a**2 - 6*a*(2*A *b*c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19*B*b*c/18)/(16*c))/(7*c) - 1 1*b*(2*A*a*c + A*b**2 + 2*B*a*b - 7*a*(A*c**2 + 19*B*b*c/18)/(8*c) - 13*b* (2*A*b*c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19*B*b*c/18)/(16*c))/(14*c ))/(12*c))/(10*c))/(4*c) - 5*b*(-4*a*(2*A*a*b + B*a**2 - 6*a*(2*A*b*c + 10 *B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19*B*b*c/18)/(16*c))/(7*c) - 11*b*(2*A* a*c + A*b**2 + 2*B*a*b - 7*a*(A*c**2 + 19*B*b*c/18)/(8*c) - 13*b*(2*A*b*c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19*B*b*c/18)/(16*c))/(14*c))/(12*c) )/(5*c) - 7*b*(A*a**2 - 5*a*(2*A*a*c + A*b**2 + 2*B*a*b - 7*a*(A*c**2 + 19 *B*b*c/18)/(8*c) - 13*b*(2*A*b*c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19 *B*b*c/18)/(16*c))/(14*c))/(6*c) - 9*b*(2*A*a*b + B*a**2 - 6*a*(2*A*b*c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19*B*b*c/18)/(16*c))/(7*c) - 11*b*(2* A*a*c + A*b**2 + 2*B*a*b - 7*a*(A*c**2 + 19*B*b*c/18)/(8*c) - 13*b*(2*A*b* c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19*B*b*c/18)/(16*c))/(14*c))/(12* c))/(10*c))/(8*c))/(6*c))/(2*c) - b*(-2*a*(-4*a*(2*A*a*b + B*a**2 - 6*a*(2 *A*b*c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19*B*b*c/18)/(16*c))/(7*c) - 11*b*(2*A*a*c + A*b**2 + 2*B*a*b - 7*a*(A*c**2 + 19*B*b*c/18)/(8*c) - 13* b*(2*A*b*c + 10*B*a*c/9 + B*b**2 - 15*b*(A*c**2 + 19*B*b*c/18)/(16*c))/...
Exception generated. \[ \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),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 = 0.26 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.40 \[ \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{10321920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, {\left (16 \, B c x + \frac {19 \, B b c^{8} + 18 \, A c^{9}}{c^{8}}\right )} x + \frac {3 \, B b^{2} c^{7} + 320 \, B a c^{8} + 306 \, A b c^{8}}{c^{8}}\right )} x - \frac {13 \, B b^{3} c^{6} - 60 \, B a b c^{7} - 18 \, A b^{2} c^{7} - 1512 \, A a c^{8}}{c^{8}}\right )} x + \frac {143 \, B b^{4} c^{5} - 804 \, B a b^{2} c^{6} - 198 \, A b^{3} c^{6} + 768 \, B a^{2} c^{7} + 936 \, A a b c^{7}}{c^{8}}\right )} x - \frac {1287 \, B b^{5} c^{4} - 8536 \, B a b^{3} c^{5} - 1782 \, A b^{4} c^{5} + 12912 \, B a^{2} b c^{6} + 10224 \, A a b^{2} c^{6} - 10080 \, A a^{2} c^{7}}{c^{8}}\right )} x + \frac {3003 \, B b^{6} c^{3} - 22968 \, B a b^{4} c^{4} - 4158 \, A b^{5} c^{4} + 47280 \, B a^{2} b^{2} c^{5} + 28080 \, A a b^{3} c^{5} - 16384 \, B a^{3} c^{6} - 43488 \, A a^{2} b c^{6}}{c^{8}}\right )} x - \frac {15015 \, B b^{7} c^{2} - 130284 \, B a b^{5} c^{3} - 20790 \, A b^{6} c^{3} + 338832 \, B a^{2} b^{3} c^{4} + 161784 \, A a b^{4} c^{4} - 236864 \, B a^{3} b c^{5} - 340128 \, A a^{2} b^{2} c^{5} + 120960 \, A a^{3} c^{6}}{c^{8}}\right )} x + \frac {45045 \, B b^{8} c - 438900 \, B a b^{6} c^{2} - 62370 \, A b^{7} c^{2} + 1383984 \, B a^{2} b^{4} c^{3} + 551880 \, A a b^{5} c^{3} - 1467072 \, B a^{3} b^{2} c^{4} - 1469664 \, A a^{2} b^{3} c^{4} + 262144 \, B a^{4} c^{5} + 1058688 \, A a^{3} b c^{5}}{c^{8}}\right )} + \frac {{\left (143 \, B b^{9} - 1584 \, B a b^{7} c - 198 \, A b^{8} c + 6048 \, B a^{2} b^{5} c^{2} + 2016 \, A a b^{6} c^{2} - 8960 \, B a^{3} b^{3} c^{3} - 6720 \, A a^{2} b^{4} c^{3} + 3840 \, B a^{4} b c^{4} + 7680 \, A a^{3} b^{2} c^{4} - 1536 \, A a^{4} c^{5}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{65536 \, c^{\frac {15}{2}}} \] Input:
integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
1/10321920*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*(16*B*c*x + (19*B* b*c^8 + 18*A*c^9)/c^8)*x + (3*B*b^2*c^7 + 320*B*a*c^8 + 306*A*b*c^8)/c^8)* x - (13*B*b^3*c^6 - 60*B*a*b*c^7 - 18*A*b^2*c^7 - 1512*A*a*c^8)/c^8)*x + ( 143*B*b^4*c^5 - 804*B*a*b^2*c^6 - 198*A*b^3*c^6 + 768*B*a^2*c^7 + 936*A*a* b*c^7)/c^8)*x - (1287*B*b^5*c^4 - 8536*B*a*b^3*c^5 - 1782*A*b^4*c^5 + 1291 2*B*a^2*b*c^6 + 10224*A*a*b^2*c^6 - 10080*A*a^2*c^7)/c^8)*x + (3003*B*b^6* c^3 - 22968*B*a*b^4*c^4 - 4158*A*b^5*c^4 + 47280*B*a^2*b^2*c^5 + 28080*A*a *b^3*c^5 - 16384*B*a^3*c^6 - 43488*A*a^2*b*c^6)/c^8)*x - (15015*B*b^7*c^2 - 130284*B*a*b^5*c^3 - 20790*A*b^6*c^3 + 338832*B*a^2*b^3*c^4 + 161784*A*a *b^4*c^4 - 236864*B*a^3*b*c^5 - 340128*A*a^2*b^2*c^5 + 120960*A*a^3*c^6)/c ^8)*x + (45045*B*b^8*c - 438900*B*a*b^6*c^2 - 62370*A*b^7*c^2 + 1383984*B* a^2*b^4*c^3 + 551880*A*a*b^5*c^3 - 1467072*B*a^3*b^2*c^4 - 1469664*A*a^2*b ^3*c^4 + 262144*B*a^4*c^5 + 1058688*A*a^3*b*c^5)/c^8) + 1/65536*(143*B*b^9 - 1584*B*a*b^7*c - 198*A*b^8*c + 6048*B*a^2*b^5*c^2 + 2016*A*a*b^6*c^2 - 8960*B*a^3*b^3*c^3 - 6720*A*a^2*b^4*c^3 + 3840*B*a^4*b*c^4 + 7680*A*a^3*b^ 2*c^4 - 1536*A*a^4*c^5)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt (c) + b))/c^(15/2)
Timed out. \[ \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\int x^4\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \] Input:
int(x^4*(A + B*x)*(a + b*x + c*x^2)^(3/2),x)
Output:
int(x^4*(A + B*x)*(a + b*x + c*x^2)^(3/2), x)
\[ \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx=\int x^{4} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}d x \] Input:
int(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),x)
Output:
int(x^4*(B*x+A)*(c*x^2+b*x+a)^(3/2),x)