Integrand size = 20, antiderivative size = 203 \[ \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 \left (b^2-4 a c\right )^2 (b B-2 A c) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}+\frac {5 \left (b^2-4 a c\right ) (b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {5 \left (b^2-4 a c\right )^3 (b B-2 A c) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}} \]
5/384*(-4*a*c+b^2)*(-2*A*c+B*b)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3-1/24*(-2 *A*c+B*b)*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^2+1/7*B*(c*x^2+b*x+a)^(7/2)/c+5/ 2048*(-4*a*c+b^2)^3*(-2*A*c+B*b)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+ a)^(1/2))/c^(9/2)-5/1024*(-4*a*c+b^2)^2*(-2*A*c+B*b)*(2*c*x+b)*(c*x^2+b*x+ a)^(1/2)/c^4
Time = 2.77 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.53 \[ \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^6 B+70 b^5 c (3 A+B x)+28 b^4 c (40 a B-c x (5 A+2 B x))+16 b^3 c^2 \left (c x^2 (7 A+3 B x)-14 a (10 A+3 B x)\right )+64 c^3 \left (48 a^3 B+8 c^3 x^5 (7 A+6 B x)+3 a^2 c x (77 A+48 B x)+2 a c^2 x^3 (91 A+72 B x)\right )+16 b^2 c^2 \left (-231 a^2 B+6 a c x (14 A+5 B x)+2 c^2 x^3 (189 A+148 B x)\right )+32 b c^3 \left (3 a^2 (77 A+19 B x)+8 c^2 x^4 (35 A+29 B x)+2 a c x^2 (273 A+197 B x)\right )\right )+105 \left (b^2-4 a c\right )^3 (b B-2 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{21504 c^{9/2}} \]
(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^6*B + 70*b^5*c*(3*A + B*x) + 28*b^4 *c*(40*a*B - c*x*(5*A + 2*B*x)) + 16*b^3*c^2*(c*x^2*(7*A + 3*B*x) - 14*a*( 10*A + 3*B*x)) + 64*c^3*(48*a^3*B + 8*c^3*x^5*(7*A + 6*B*x) + 3*a^2*c*x*(7 7*A + 48*B*x) + 2*a*c^2*x^3*(91*A + 72*B*x)) + 16*b^2*c^2*(-231*a^2*B + 6* a*c*x*(14*A + 5*B*x) + 2*c^2*x^3*(189*A + 148*B*x)) + 32*b*c^3*(3*a^2*(77* A + 19*B*x) + 8*c^2*x^4*(35*A + 29*B*x) + 2*a*c*x^2*(273*A + 197*B*x))) + 105*(b^2 - 4*a*c)^3*(b*B - 2*A*c)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(21504*c^(9/2))
Time = 0.32 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1160, 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.
| \(\displaystyle \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \int \left (c x^2+b x+a\right )^{5/2}dx}{2 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{2 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{2 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {(b B-2 A c) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\) |
(B*(a + b*x + c*x^2)^(7/2))/(7*c) - ((b*B - 2*A*c)*(((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)))/(2*c)
3.10.39.3.1 Defintions of rubi rules used
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_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Time = 0.42 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.55
| method | result | size |
| default | \(A \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+B \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\) | \(314\) |
| risch | \(\frac {\left (3072 B \,c^{6} x^{6}+3584 A \,c^{6} x^{5}+7424 B b \,c^{5} x^{5}+8960 A b \,c^{5} x^{4}+9216 B a \,c^{5} x^{4}+4736 B \,b^{2} c^{4} x^{4}+11648 A a \,c^{5} x^{3}+6048 A \,b^{2} c^{4} x^{3}+12608 B a b \,c^{4} x^{3}+48 B \,b^{3} c^{3} x^{3}+17472 A a b \,c^{4} x^{2}+112 A \,b^{3} c^{3} x^{2}+9216 B \,a^{2} c^{4} x^{2}+480 B a \,b^{2} c^{3} x^{2}-56 B \,b^{4} c^{2} x^{2}+14784 A \,a^{2} c^{4} x +1344 A a \,b^{2} c^{3} x -140 A \,b^{4} c^{2} x +1824 B \,a^{2} b \,c^{3} x -672 B a \,b^{3} c^{2} x +70 B \,b^{5} c x +7392 a^{2} A b \,c^{3}-2240 A a \,b^{3} c^{2}+210 A \,b^{5} c +3072 B \,a^{3} c^{3}-3696 B \,a^{2} b^{2} c^{2}+1120 B a \,b^{4} c -105 B \,b^{6}\right ) \sqrt {c \,x^{2}+b x +a}}{21504 c^{4}}+\frac {5 \left (128 A \,a^{3} c^{4}-96 A \,a^{2} b^{2} c^{3}+24 A a \,b^{4} c^{2}-2 A \,b^{6} c -64 B \,a^{3} b \,c^{3}+48 B \,a^{2} b^{3} c^{2}-12 B a \,b^{5} c +B \,b^{7}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {9}{2}}}\) | \(412\) |
A*(1/12*(2*c*x+b)/c*(c*x^2+b*x+a)^(5/2)+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)/ c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1 /2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))) +B*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)/c*(c*x^2+b*x+a)^(5/2 )+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2) /c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+ c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (177) = 354\).
Time = 0.35 (sec) , antiderivative size = 843, normalized size of antiderivative = 4.15 \[ \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\left [-\frac {105 \, {\left (B b^{7} + 128 \, A a^{3} c^{4} - 32 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 24 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (3072 \, B c^{7} x^{6} - 105 \, B b^{6} c + 256 \, {\left (29 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 96 \, {\left (32 \, B a^{3} + 77 \, A a^{2} b\right )} c^{4} + 128 \, {\left (37 \, B b^{2} c^{5} + 2 \, {\left (36 \, B a + 35 \, A b\right )} c^{6}\right )} x^{4} - 112 \, {\left (33 \, B a^{2} b^{2} + 20 \, A a b^{3}\right )} c^{3} + 16 \, {\left (3 \, B b^{3} c^{4} + 728 \, A a c^{6} + 2 \, {\left (394 \, B a b + 189 \, A b^{2}\right )} c^{5}\right )} x^{3} + 70 \, {\left (16 \, B a b^{4} + 3 \, A b^{5}\right )} c^{2} - 8 \, {\left (7 \, B b^{4} c^{3} - 24 \, {\left (48 \, B a^{2} + 91 \, A a b\right )} c^{5} - 2 \, {\left (30 \, B a b^{2} + 7 \, A b^{3}\right )} c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{5} c^{2} + 7392 \, A a^{2} c^{5} + 48 \, {\left (19 \, B a^{2} b + 14 \, A a b^{2}\right )} c^{4} - 14 \, {\left (24 \, B a b^{3} + 5 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{86016 \, c^{5}}, -\frac {105 \, {\left (B b^{7} + 128 \, A a^{3} c^{4} - 32 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 24 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (3072 \, B c^{7} x^{6} - 105 \, B b^{6} c + 256 \, {\left (29 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 96 \, {\left (32 \, B a^{3} + 77 \, A a^{2} b\right )} c^{4} + 128 \, {\left (37 \, B b^{2} c^{5} + 2 \, {\left (36 \, B a + 35 \, A b\right )} c^{6}\right )} x^{4} - 112 \, {\left (33 \, B a^{2} b^{2} + 20 \, A a b^{3}\right )} c^{3} + 16 \, {\left (3 \, B b^{3} c^{4} + 728 \, A a c^{6} + 2 \, {\left (394 \, B a b + 189 \, A b^{2}\right )} c^{5}\right )} x^{3} + 70 \, {\left (16 \, B a b^{4} + 3 \, A b^{5}\right )} c^{2} - 8 \, {\left (7 \, B b^{4} c^{3} - 24 \, {\left (48 \, B a^{2} + 91 \, A a b\right )} c^{5} - 2 \, {\left (30 \, B a b^{2} + 7 \, A b^{3}\right )} c^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{5} c^{2} + 7392 \, A a^{2} c^{5} + 48 \, {\left (19 \, B a^{2} b + 14 \, A a b^{2}\right )} c^{4} - 14 \, {\left (24 \, B a b^{3} + 5 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{43008 \, c^{5}}\right ] \]
[-1/86016*(105*(B*b^7 + 128*A*a^3*c^4 - 32*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 24*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 2*(6*B*a*b^5 + A*b^6)*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*(3072*B*c^7*x^6 - 105*B*b^6*c + 256*(29*B*b*c^6 + 14*A*c^7)*x^5 + 96*(32*B*a^3 + 77*A*a^2*b)*c^4 + 128*(37*B*b^2*c^5 + 2*(36*B*a + 35*A*b)* c^6)*x^4 - 112*(33*B*a^2*b^2 + 20*A*a*b^3)*c^3 + 16*(3*B*b^3*c^4 + 728*A*a *c^6 + 2*(394*B*a*b + 189*A*b^2)*c^5)*x^3 + 70*(16*B*a*b^4 + 3*A*b^5)*c^2 - 8*(7*B*b^4*c^3 - 24*(48*B*a^2 + 91*A*a*b)*c^5 - 2*(30*B*a*b^2 + 7*A*b^3) *c^4)*x^2 + 2*(35*B*b^5*c^2 + 7392*A*a^2*c^5 + 48*(19*B*a^2*b + 14*A*a*b^2 )*c^4 - 14*(24*B*a*b^3 + 5*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^5, -1/4 3008*(105*(B*b^7 + 128*A*a^3*c^4 - 32*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 24*( 2*B*a^2*b^3 + A*a*b^4)*c^2 - 2*(6*B*a*b^5 + A*b^6)*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*(3 072*B*c^7*x^6 - 105*B*b^6*c + 256*(29*B*b*c^6 + 14*A*c^7)*x^5 + 96*(32*B*a ^3 + 77*A*a^2*b)*c^4 + 128*(37*B*b^2*c^5 + 2*(36*B*a + 35*A*b)*c^6)*x^4 - 112*(33*B*a^2*b^2 + 20*A*a*b^3)*c^3 + 16*(3*B*b^3*c^4 + 728*A*a*c^6 + 2*(3 94*B*a*b + 189*A*b^2)*c^5)*x^3 + 70*(16*B*a*b^4 + 3*A*b^5)*c^2 - 8*(7*B*b^ 4*c^3 - 24*(48*B*a^2 + 91*A*a*b)*c^5 - 2*(30*B*a*b^2 + 7*A*b^3)*c^4)*x^2 + 2*(35*B*b^5*c^2 + 7392*A*a^2*c^5 + 48*(19*B*a^2*b + 14*A*a*b^2)*c^4 - 14* (24*B*a*b^3 + 5*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^5]
Leaf count of result is larger than twice the leaf count of optimal. 2428 vs. \(2 (197) = 394\).
Time = 0.65 (sec) , antiderivative size = 2428, normalized size of antiderivative = 11.96 \[ \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a + b*x + c*x**2)*(B*c**2*x**6/7 + x**5*(A*c**3 + 29*B*b*c **2/14)/(6*c) + x**4*(3*A*b*c**2 + 15*B*a*c**2/7 + 3*B*b**2*c - 11*b*(A*c* *3 + 29*B*b*c**2/14)/(12*c))/(5*c) + x**3*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a *b*c + B*b**3 - 5*a*(A*c**3 + 29*B*b*c**2/14)/(6*c) - 9*b*(3*A*b*c**2 + 15 *B*a*c**2/7 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/(10*c))/ (4*c) + x**2*(6*A*a*b*c + A*b**3 + 3*B*a**2*c + 3*B*a*b**2 - 4*a*(3*A*b*c* *2 + 15*B*a*c**2/7 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/( 5*c) - 7*b*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 5*a*(A*c**3 + 2 9*B*b*c**2/14)/(6*c) - 9*b*(3*A*b*c**2 + 15*B*a*c**2/7 + 3*B*b**2*c - 11*b *(A*c**3 + 29*B*b*c**2/14)/(12*c))/(10*c))/(8*c))/(3*c) + x*(3*A*a**2*c + 3*A*a*b**2 + 3*B*a**2*b - 3*a*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b** 3 - 5*a*(A*c**3 + 29*B*b*c**2/14)/(6*c) - 9*b*(3*A*b*c**2 + 15*B*a*c**2/7 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/(10*c))/(4*c) - 5*b* (6*A*a*b*c + A*b**3 + 3*B*a**2*c + 3*B*a*b**2 - 4*a*(3*A*b*c**2 + 15*B*a*c **2/7 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/(5*c) - 7*b*(3 *A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 5*a*(A*c**3 + 29*B*b*c**2/14 )/(6*c) - 9*b*(3*A*b*c**2 + 15*B*a*c**2/7 + 3*B*b**2*c - 11*b*(A*c**3 + 29 *B*b*c**2/14)/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) + (3*A*a**2*b + B*a**3 - 2*a*(6*A*a*b*c + A*b**3 + 3*B*a**2*c + 3*B*a*b**2 - 4*a*(3*A*b*c**2 + 15* B*a*c**2/7 + 3*B*b**2*c - 11*b*(A*c**3 + 29*B*b*c**2/14)/(12*c))/(5*c) ...
Exception generated. \[ \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (177) = 354\).
Time = 0.30 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.08 \[ \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{21504} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, B c^{2} x + \frac {29 \, B b c^{7} + 14 \, A c^{8}}{c^{6}}\right )} x + \frac {37 \, B b^{2} c^{6} + 72 \, B a c^{7} + 70 \, A b c^{7}}{c^{6}}\right )} x + \frac {3 \, B b^{3} c^{5} + 788 \, B a b c^{6} + 378 \, A b^{2} c^{6} + 728 \, A a c^{7}}{c^{6}}\right )} x - \frac {7 \, B b^{4} c^{4} - 60 \, B a b^{2} c^{5} - 14 \, A b^{3} c^{5} - 1152 \, B a^{2} c^{6} - 2184 \, A a b c^{6}}{c^{6}}\right )} x + \frac {35 \, B b^{5} c^{3} - 336 \, B a b^{3} c^{4} - 70 \, A b^{4} c^{4} + 912 \, B a^{2} b c^{5} + 672 \, A a b^{2} c^{5} + 7392 \, A a^{2} c^{6}}{c^{6}}\right )} x - \frac {105 \, B b^{6} c^{2} - 1120 \, B a b^{4} c^{3} - 210 \, A b^{5} c^{3} + 3696 \, B a^{2} b^{2} c^{4} + 2240 \, A a b^{3} c^{4} - 3072 \, B a^{3} c^{5} - 7392 \, A a^{2} b c^{5}}{c^{6}}\right )} - \frac {5 \, {\left (B b^{7} - 12 \, B a b^{5} c - 2 \, A b^{6} c + 48 \, B a^{2} b^{3} c^{2} + 24 \, A a b^{4} c^{2} - 64 \, B a^{3} b c^{3} - 96 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {9}{2}}} \]
1/21504*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*B*c^2*x + (29*B*b*c^7 + 1 4*A*c^8)/c^6)*x + (37*B*b^2*c^6 + 72*B*a*c^7 + 70*A*b*c^7)/c^6)*x + (3*B*b ^3*c^5 + 788*B*a*b*c^6 + 378*A*b^2*c^6 + 728*A*a*c^7)/c^6)*x - (7*B*b^4*c^ 4 - 60*B*a*b^2*c^5 - 14*A*b^3*c^5 - 1152*B*a^2*c^6 - 2184*A*a*b*c^6)/c^6)* x + (35*B*b^5*c^3 - 336*B*a*b^3*c^4 - 70*A*b^4*c^4 + 912*B*a^2*b*c^5 + 672 *A*a*b^2*c^5 + 7392*A*a^2*c^6)/c^6)*x - (105*B*b^6*c^2 - 1120*B*a*b^4*c^3 - 210*A*b^5*c^3 + 3696*B*a^2*b^2*c^4 + 2240*A*a*b^3*c^4 - 3072*B*a^3*c^5 - 7392*A*a^2*b*c^5)/c^6) - 5/2048*(B*b^7 - 12*B*a*b^5*c - 2*A*b^6*c + 48*B* a^2*b^3*c^2 + 24*A*a*b^4*c^2 - 64*B*a^3*b*c^3 - 96*A*a^2*b^2*c^3 + 128*A*a ^3*c^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2 )
Timed out. \[ \int (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]