Integrand size = 22, antiderivative size = 212 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx=-\frac {\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 A c^2+7 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}} \]
1/192*(24*A*c^2-4*C*a*c+7*C*b^2)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3-7/60*b* C*(c*x^2+b*x+a)^(5/2)/c^2+1/6*C*x*(c*x^2+b*x+a)^(5/2)/c+1/1024*(-4*a*c+b^2 )^2*(24*A*c^2-4*C*a*c+7*C*b^2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a) ^(1/2))/c^(9/2)-1/512*(-4*a*c+b^2)*(24*A*c^2-4*C*a*c+7*C*b^2)*(2*c*x+b)*(c *x^2+b*x+a)^(1/2)/c^4
Time = 1.99 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.08 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (120 A c^2 (b+2 c x) \left (-3 b^2+8 b c x+4 c \left (5 a+2 c x^2\right )\right )+C \left (-105 b^5+70 b^4 c x+8 b^3 c \left (95 a-7 c x^2\right )+48 b^2 c^2 x \left (-9 a+c x^2\right )+160 c^3 x \left (3 a^2+14 a c x^2+8 c^2 x^4\right )+16 b c^2 \left (-81 a^2+18 a c x^2+104 c^2 x^4\right )\right )\right )+15 \left (b^2-4 a c\right )^2 \left (24 A c^2+7 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{7680 c^{9/2}} \]
(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(120*A*c^2*(b + 2*c*x)*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + C*(-105*b^5 + 70*b^4*c*x + 8*b^3*c*(95*a - 7*c*x^2) + 48*b^2*c^2*x*(-9*a + c*x^2) + 160*c^3*x*(3*a^2 + 14*a*c*x^2 + 8*c^2*x^4 ) + 16*b*c^2*(-81*a^2 + 18*a*c*x^2 + 104*c^2*x^4))) + 15*(b^2 - 4*a*c)^2*( 24*A*c^2 + 7*b^2*C - 4*a*c*C)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*( b + c*x)])])/(7680*c^(9/2))
Time = 0.34 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2192, 27, 1160, 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.
| \(\displaystyle \int \left (A+C x^2\right ) \left (a+b x+c x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {\int \frac {1}{2} (12 A c-2 a C-7 b C x) \left (c x^2+b x+a\right )^{3/2}dx}{6 c}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (2 (6 A c-a C)-7 b C x) \left (c x^2+b x+a\right )^{3/2}dx}{12 c}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {\frac {\left (-4 a c C+24 A c^2+7 b^2 C\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{2 c}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{5 c}}{12 c}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\frac {\left (-4 a c C+24 A c^2+7 b^2 C\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{2 c}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{5 c}}{12 c}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\frac {\left (-4 a c C+24 A c^2+7 b^2 C\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{2 c}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{5 c}}{12 c}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {\left (-4 a c C+24 A c^2+7 b^2 C\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{2 c}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{5 c}}{12 c}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (-4 a c C+24 A c^2+7 b^2 C\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{2 c}-\frac {7 b C \left (a+b x+c x^2\right )^{5/2}}{5 c}}{12 c}+\frac {C x \left (a+b x+c x^2\right )^{5/2}}{6 c}\) |
(C*x*(a + b*x + c*x^2)^(5/2))/(6*c) + ((-7*b*C*(a + b*x + c*x^2)^(5/2))/(5 *c) + ((24*A*c^2 + 7*b^2*C - 4*a*c*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)))/(2*c))/(12*c)
3.2.79.3.1 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_)^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]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Time = 0.76 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(\frac {\left (1280 c^{5} C \,x^{5}+1664 b C \,c^{4} x^{4}+1920 A \,c^{5} x^{3}+2240 C a \,c^{4} x^{3}+48 C \,b^{2} c^{3} x^{3}+2880 A b \,c^{4} x^{2}+288 C a b \,c^{3} x^{2}-56 C \,b^{3} c^{2} x^{2}+4800 A a \,c^{4} x +240 A \,b^{2} c^{3} x +480 C \,a^{2} c^{3} x -432 C a \,b^{2} c^{2} x +70 C \,b^{4} c x +2400 A a b \,c^{3}-360 A \,b^{3} c^{2}-1296 C \,a^{2} b \,c^{2}+760 C a \,b^{3} c -105 C \,b^{5}\right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{4}}+\frac {\left (384 A \,a^{2} c^{4}-192 A a \,b^{2} c^{3}+24 A \,b^{4} c^{2}-64 C \,a^{3} c^{3}+144 C \,a^{2} b^{2} c^{2}-60 C a \,b^{4} c +7 C \,b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}\) | \(285\) |
| default | \(A \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+C \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )\) | \(370\) |
1/7680*(1280*C*c^5*x^5+1664*C*b*c^4*x^4+1920*A*c^5*x^3+2240*C*a*c^4*x^3+48 *C*b^2*c^3*x^3+2880*A*b*c^4*x^2+288*C*a*b*c^3*x^2-56*C*b^3*c^2*x^2+4800*A* a*c^4*x+240*A*b^2*c^3*x+480*C*a^2*c^3*x-432*C*a*b^2*c^2*x+70*C*b^4*c*x+240 0*A*a*b*c^3-360*A*b^3*c^2-1296*C*a^2*b*c^2+760*C*a*b^3*c-105*C*b^5)/c^4*(c *x^2+b*x+a)^(1/2)+1/1024*(384*A*a^2*c^4-192*A*a*b^2*c^3+24*A*b^4*c^2-64*C* a^3*c^3+144*C*a^2*b^2*c^2-60*C*a*b^4*c+7*C*b^6)/c^(9/2)*ln((1/2*b+c*x)/c^( 1/2)+(c*x^2+b*x+a)^(1/2))
Time = 0.31 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.85 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx=\left [\frac {15 \, {\left (7 \, C b^{6} - 60 \, C a b^{4} c + 384 \, A a^{2} c^{4} - 64 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c^{3} + 24 \, {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (1280 \, C c^{6} x^{5} + 1664 \, C b c^{5} x^{4} - 105 \, C b^{5} c + 760 \, C a b^{3} c^{2} + 2400 \, A a b c^{4} - 72 \, {\left (18 \, C a^{2} b + 5 \, A b^{3}\right )} c^{3} + 16 \, {\left (3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}\right )} x^{3} - 8 \, {\left (7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}\right )} x^{2} + 2 \, {\left (35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 2400 \, A a c^{5} + 120 \, {\left (2 \, C a^{2} + A b^{2}\right )} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{5}}, -\frac {15 \, {\left (7 \, C b^{6} - 60 \, C a b^{4} c + 384 \, A a^{2} c^{4} - 64 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c^{3} + 24 \, {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (1280 \, C c^{6} x^{5} + 1664 \, C b c^{5} x^{4} - 105 \, C b^{5} c + 760 \, C a b^{3} c^{2} + 2400 \, A a b c^{4} - 72 \, {\left (18 \, C a^{2} b + 5 \, A b^{3}\right )} c^{3} + 16 \, {\left (3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}\right )} x^{3} - 8 \, {\left (7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}\right )} x^{2} + 2 \, {\left (35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 2400 \, A a c^{5} + 120 \, {\left (2 \, C a^{2} + A b^{2}\right )} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{5}}\right ] \]
[1/30720*(15*(7*C*b^6 - 60*C*a*b^4*c + 384*A*a^2*c^4 - 64*(C*a^3 + 3*A*a*b ^2)*c^3 + 24*(6*C*a^2*b^2 + A*b^4)*c^2)*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*(1280*C*c^ 6*x^5 + 1664*C*b*c^5*x^4 - 105*C*b^5*c + 760*C*a*b^3*c^2 + 2400*A*a*b*c^4 - 72*(18*C*a^2*b + 5*A*b^3)*c^3 + 16*(3*C*b^2*c^4 + 140*C*a*c^5 + 120*A*c^ 6)*x^3 - 8*(7*C*b^3*c^3 - 36*C*a*b*c^4 - 360*A*b*c^5)*x^2 + 2*(35*C*b^4*c^ 2 - 216*C*a*b^2*c^3 + 2400*A*a*c^5 + 120*(2*C*a^2 + A*b^2)*c^4)*x)*sqrt(c* x^2 + b*x + a))/c^5, -1/15360*(15*(7*C*b^6 - 60*C*a*b^4*c + 384*A*a^2*c^4 - 64*(C*a^3 + 3*A*a*b^2)*c^3 + 24*(6*C*a^2*b^2 + A*b^4)*c^2)*sqrt(-c)*arct an(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*(1280*C*c^6*x^5 + 1664*C*b*c^5*x^4 - 105*C*b^5*c + 760*C*a*b^3*c^2 + 2400*A*a*b*c^4 - 72*(18*C*a^2*b + 5*A*b^3)*c^3 + 16*(3*C*b^2*c^4 + 140*C*a *c^5 + 120*A*c^6)*x^3 - 8*(7*C*b^3*c^3 - 36*C*a*b*c^4 - 360*A*b*c^5)*x^2 + 2*(35*C*b^4*c^2 - 216*C*a*b^2*c^3 + 2400*A*a*c^5 + 120*(2*C*a^2 + A*b^2)* c^4)*x)*sqrt(c*x^2 + b*x + a))/c^5]
Leaf count of result is larger than twice the leaf count of optimal. 775 vs. \(2 (207) = 414\).
Time = 0.51 (sec) , antiderivative size = 775, normalized size of antiderivative = 3.66 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \cdot \left (\frac {13 C b x^{4}}{60} + \frac {C c x^{5}}{6} + \frac {x^{3} \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{4 c} + \frac {x^{2} \cdot \left (2 A b c + \frac {17 C a b}{15} - \frac {7 b \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{8 c}\right )}{3 c} + \frac {x \left (2 A a c + A b^{2} + C a^{2} - \frac {3 a \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{4 c} - \frac {5 b \left (2 A b c + \frac {17 C a b}{15} - \frac {7 b \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{8 c}\right )}{6 c}\right )}{2 c} + \frac {2 A a b - \frac {2 a \left (2 A b c + \frac {17 C a b}{15} - \frac {7 b \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{8 c}\right )}{3 c} - \frac {3 b \left (2 A a c + A b^{2} + C a^{2} - \frac {3 a \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{4 c} - \frac {5 b \left (2 A b c + \frac {17 C a b}{15} - \frac {7 b \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{8 c}\right )}{6 c}\right )}{4 c}}{c}\right ) + \left (A a^{2} - \frac {a \left (2 A a c + A b^{2} + C a^{2} - \frac {3 a \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{4 c} - \frac {5 b \left (2 A b c + \frac {17 C a b}{15} - \frac {7 b \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{8 c}\right )}{6 c}\right )}{2 c} - \frac {b \left (2 A a b - \frac {2 a \left (2 A b c + \frac {17 C a b}{15} - \frac {7 b \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{8 c}\right )}{3 c} - \frac {3 b \left (2 A a c + A b^{2} + C a^{2} - \frac {3 a \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{4 c} - \frac {5 b \left (2 A b c + \frac {17 C a b}{15} - \frac {7 b \left (A c^{2} + \frac {7 C a c}{6} + \frac {C b^{2}}{40}\right )}{8 c}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (- \frac {2 C a \left (a + b x\right )^{\frac {7}{2}}}{7 b^{2}} + \frac {C \left (a + b x\right )^{\frac {9}{2}}}{9 b^{2}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (A b^{2} + C a^{2}\right )}{5 b^{2}}\right )}{b} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (A x + \frac {C x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((sqrt(a + b*x + c*x**2)*(13*C*b*x**4/60 + C*c*x**5/6 + x**3*(A*c **2 + 7*C*a*c/6 + C*b**2/40)/(4*c) + x**2*(2*A*b*c + 17*C*a*b/15 - 7*b*(A* c**2 + 7*C*a*c/6 + C*b**2/40)/(8*c))/(3*c) + x*(2*A*a*c + A*b**2 + C*a**2 - 3*a*(A*c**2 + 7*C*a*c/6 + C*b**2/40)/(4*c) - 5*b*(2*A*b*c + 17*C*a*b/15 - 7*b*(A*c**2 + 7*C*a*c/6 + C*b**2/40)/(8*c))/(6*c))/(2*c) + (2*A*a*b - 2* a*(2*A*b*c + 17*C*a*b/15 - 7*b*(A*c**2 + 7*C*a*c/6 + C*b**2/40)/(8*c))/(3* c) - 3*b*(2*A*a*c + A*b**2 + C*a**2 - 3*a*(A*c**2 + 7*C*a*c/6 + C*b**2/40) /(4*c) - 5*b*(2*A*b*c + 17*C*a*b/15 - 7*b*(A*c**2 + 7*C*a*c/6 + C*b**2/40) /(8*c))/(6*c))/(4*c))/c) + (A*a**2 - a*(2*A*a*c + A*b**2 + C*a**2 - 3*a*(A *c**2 + 7*C*a*c/6 + C*b**2/40)/(4*c) - 5*b*(2*A*b*c + 17*C*a*b/15 - 7*b*(A *c**2 + 7*C*a*c/6 + C*b**2/40)/(8*c))/(6*c))/(2*c) - b*(2*A*a*b - 2*a*(2*A *b*c + 17*C*a*b/15 - 7*b*(A*c**2 + 7*C*a*c/6 + C*b**2/40)/(8*c))/(3*c) - 3 *b*(2*A*a*c + A*b**2 + C*a**2 - 3*a*(A*c**2 + 7*C*a*c/6 + C*b**2/40)/(4*c) - 5*b*(2*A*b*c + 17*C*a*b/15 - 7*b*(A*c**2 + 7*C*a*c/6 + C*b**2/40)/(8*c) )/(6*c))/(4*c))/(2*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x)*log(b/(2*c) + x) /sqrt(c*(b/(2*c) + x)**2), True)), Ne(c, 0)), (2*(-2*C*a*(a + b*x)**(7/2)/ (7*b**2) + C*(a + b*x)**(9/2)/(9*b**2) + (a + b*x)**(5/2)*(A*b**2 + C*a**2 )/(5*b**2))/b, Ne(b, 0)), (a**(3/2)*(A*x + C*x**3/3), True))
Exception generated. \[ \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.39 \[ \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, C c x + 13 \, C b\right )} x + \frac {3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}}{c^{5}}\right )} x - \frac {7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}}{c^{5}}\right )} x + \frac {35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 240 \, C a^{2} c^{4} + 120 \, A b^{2} c^{4} + 2400 \, A a c^{5}}{c^{5}}\right )} x - \frac {105 \, C b^{5} c - 760 \, C a b^{3} c^{2} + 1296 \, C a^{2} b c^{3} + 360 \, A b^{3} c^{3} - 2400 \, A a b c^{4}}{c^{5}}\right )} - \frac {{\left (7 \, C b^{6} - 60 \, C a b^{4} c + 144 \, C a^{2} b^{2} c^{2} + 24 \, A b^{4} c^{2} - 64 \, C a^{3} c^{3} - 192 \, A a b^{2} c^{3} + 384 \, A a^{2} c^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]
1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*C*c*x + 13*C*b)*x + (3*C*b^2* c^4 + 140*C*a*c^5 + 120*A*c^6)/c^5)*x - (7*C*b^3*c^3 - 36*C*a*b*c^4 - 360* A*b*c^5)/c^5)*x + (35*C*b^4*c^2 - 216*C*a*b^2*c^3 + 240*C*a^2*c^4 + 120*A* b^2*c^4 + 2400*A*a*c^5)/c^5)*x - (105*C*b^5*c - 760*C*a*b^3*c^2 + 1296*C*a ^2*b*c^3 + 360*A*b^3*c^3 - 2400*A*a*b*c^4)/c^5) - 1/1024*(7*C*b^6 - 60*C*a *b^4*c + 144*C*a^2*b^2*c^2 + 24*A*b^4*c^2 - 64*C*a^3*c^3 - 192*A*a*b^2*c^3 + 384*A*a^2*c^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2)
Timed out. \[ \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx=\int \left (C\,x^2+A\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]