Integrand size = 27, antiderivative size = 407 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2+15 b^2 e^2-2 c e (27 b d+8 a e)\right )+B \left (96 c^3 d^3-105 b^3 e^3+20 b c e^2 (18 b d+11 a e)-8 c^2 d e (47 b d+48 a e)\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2+35 b^2 e^2-4 c e (16 b d+9 a e)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (35 b^4 B e^3-40 b^3 c e^2 (3 B d+A e)+24 b^2 c e \left (6 B c d^2+6 A c d e-5 a B e^2\right )-32 b c^2 \left (2 B c d^3+6 A c d^2 e-9 a B d e^2-3 a A e^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}} \]
1/128*(35*b^4*B*e^3-40*b^3*c*e^2*(A*e+3*B*d)+24*b^2*c*e*(6*A*c*d*e-5*B*a*e ^2+6*B*c*d^2)-32*b*c^2*(-3*A*a*e^3+6*A*c*d^2*e-9*B*a*d*e^2+2*B*c*d^3)+16*c ^2*(4*A*c*d*(-3*a*e^2+2*c*d^2)-3*a*B*e*(-a*e^2+4*c*d^2)))*arctanh(1/2*(2*c *x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)+1/24*(8*A*c*e-7*B*b*e+6*B*c*d)* (e*x+d)^2*(c*x^2+b*x+a)^(1/2)/c^2+1/4*B*(e*x+d)^3*(c*x^2+b*x+a)^(1/2)/c+1/ 192*(8*A*c*e*(64*c^2*d^2+15*b^2*e^2-2*c*e*(8*a*e+27*b*d))+B*(96*c^3*d^3-10 5*b^3*e^3+20*b*c*e^2*(11*a*e+18*b*d)-8*c^2*d*e*(48*a*e+47*b*d))+2*c*e*(40* A*c*e*(-b*e+2*c*d)+B*(24*c^2*d^2+35*b^2*e^2-4*c*e*(9*a*e+16*b*d)))*x)*(c*x ^2+b*x+a)^(1/2)/c^4
Time = 1.17 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-8 A c e \left (-15 b^2 e^2+2 c e (27 b d+8 a e+5 b e x)-4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+B \left (-105 b^3 e^3+10 b c e^2 (36 b d+22 a e+7 b e x)-8 c^2 e \left (54 b d^2+48 a d e+30 b d e x+9 a e^2 x+7 b e^2 x^2\right )+48 c^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )\right )+3 \left (-35 b^4 B e^3+40 b^3 c e^2 (3 B d+A e)-24 b^2 c e \left (6 B c d^2+6 A c d e-5 a B e^2\right )+32 b c^2 \left (2 B c d^3+6 A c d^2 e-9 a B d e^2-3 a A e^3\right )-16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )+3 a B e \left (-4 c d^2+a e^2\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{384 c^{9/2}} \]
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-8*A*c*e*(-15*b^2*e^2 + 2*c*e*(27*b*d + 8*a*e + 5*b*e*x) - 4*c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + B*(-105*b^3*e^3 + 10*b*c*e^2*(36*b*d + 22*a*e + 7*b*e*x) - 8*c^2*e*(54*b*d^2 + 48*a*d*e + 30*b*d*e*x + 9*a*e^2*x + 7*b*e^2*x^2) + 48*c^3*(4*d^3 + 6*d^2*e*x + 4*d*e ^2*x^2 + e^3*x^3))) + 3*(-35*b^4*B*e^3 + 40*b^3*c*e^2*(3*B*d + A*e) - 24*b ^2*c*e*(6*B*c*d^2 + 6*A*c*d*e - 5*a*B*e^2) + 32*b*c^2*(2*B*c*d^3 + 6*A*c*d ^2*e - 9*a*B*d*e^2 - 3*a*A*e^3) - 16*c^2*(4*A*c*d*(2*c*d^2 - 3*a*e^2) + 3* a*B*e*(-4*c*d^2 + a*e^2)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)] ])/(384*c^(9/2))
Time = 0.91 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1236, 27, 1236, 27, 25, 1225, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^2 (b B d-8 A c d+6 a B e-(6 B c d-7 b B e+8 A c e) x)}{2 \sqrt {c x^2+b x+a}}dx}{4 c}+\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\int \frac {(d+e x)^2 (b B d-8 A c d+6 a B e-(6 B c d-7 b B e+8 A c e) x)}{\sqrt {c x^2+b x+a}}dx}{8 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (7 B d e b^2-4 \left (3 B c d^2+2 A c e d-7 a B e^2\right ) b+4 c \left (12 A c d^2-15 a B e d-8 a A e^2\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2+35 b^2 e^2-4 c e (16 b d+9 a e)\right )\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}-\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (-7 B d e b^2+12 B c d^2 b-28 a B e^2 b+8 A c d e b-48 A c^2 d^2+32 a A c e^2+60 a B c d e-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2+35 b^2 e^2-4 c e (16 b d+9 a e)\right )\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}-\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\frac {\int \frac {(d+e x) \left (-7 B d e b^2+12 B c d^2 b-28 a B e^2 b+8 A c d e b-48 A c^2 d^2+32 a A c e^2+60 a B c d e-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2+35 b^2 e^2-4 c e (16 b d+9 a e)\right )\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}-\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\frac {-\frac {3 \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (B \left (-4 c e (9 a e+16 b d)+35 b^2 e^2+24 c^2 d^2\right )+40 A c e (2 c d-b e)\right )+8 A c e \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+64 c^2 d^2\right )+B \left (-8 c^2 d e (48 a e+47 b d)+20 b c e^2 (11 a e+18 b d)-105 b^3 e^3+96 c^3 d^3\right )\right )}{4 c^2}}{6 c}-\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\frac {-\frac {3 \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (B \left (-4 c e (9 a e+16 b d)+35 b^2 e^2+24 c^2 d^2\right )+40 A c e (2 c d-b e)\right )+8 A c e \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+64 c^2 d^2\right )+B \left (-8 c^2 d e (48 a e+47 b d)+20 b c e^2 (11 a e+18 b d)-105 b^3 e^3+96 c^3 d^3\right )\right )}{4 c^2}}{6 c}-\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}-\frac {\frac {-\frac {3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right )}{8 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (B \left (-4 c e (9 a e+16 b d)+35 b^2 e^2+24 c^2 d^2\right )+40 A c e (2 c d-b e)\right )+8 A c e \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+64 c^2 d^2\right )+B \left (-8 c^2 d e (48 a e+47 b d)+20 b c e^2 (11 a e+18 b d)-105 b^3 e^3+96 c^3 d^3\right )\right )}{4 c^2}}{6 c}-\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
(B*(d + e*x)^3*Sqrt[a + b*x + c*x^2])/(4*c) - (-1/3*((6*B*c*d - 7*b*B*e + 8*A*c*e)*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/c + (-1/4*((8*A*c*e*(64*c^2*d^ 2 + 15*b^2*e^2 - 2*c*e*(27*b*d + 8*a*e)) + B*(96*c^3*d^3 - 105*b^3*e^3 + 2 0*b*c*e^2*(18*b*d + 11*a*e) - 8*c^2*d*e*(47*b*d + 48*a*e)) + 2*c*e*(40*A*c *e*(2*c*d - b*e) + B*(24*c^2*d^2 + 35*b^2*e^2 - 4*c*e*(16*b*d + 9*a*e)))*x )*Sqrt[a + b*x + c*x^2])/c^2 - (3*(35*b^4*B*e^3 - 40*b^3*c*e^2*(3*B*d + A* e) + 24*b^2*c*e*(6*B*c*d^2 + 6*A*c*d*e - 5*a*B*e^2) - 32*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3) + 16*c^2*(4*A*c*d*(2*c*d^2 - 3*a* e^2) - 3*a*B*e*(4*c*d^2 - a*e^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)))/(6*c))/(8*c)
3.25.65.3.1 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[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 = 0.69 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(-\frac {\left (-48 B \,e^{3} c^{3} x^{3}-64 A \,c^{3} e^{3} x^{2}+56 B \,e^{3} b \,c^{2} x^{2}-192 B \,c^{3} d \,e^{2} x^{2}+80 A b \,c^{2} e^{3} x -288 A \,c^{3} d \,e^{2} x +72 B \,e^{3} a \,c^{2} x -70 B \,e^{3} b^{2} c x +240 B b \,c^{2} d \,e^{2} x -288 B \,c^{3} d^{2} e x +128 A a \,c^{2} e^{3}-120 A \,b^{2} c \,e^{3}+432 A b \,c^{2} d \,e^{2}-576 A \,c^{3} d^{2} e -220 B a b c \,e^{3}+384 B a \,c^{2} d \,e^{2}+105 B \,b^{3} e^{3}-360 B \,b^{2} c d \,e^{2}+432 B b \,c^{2} d^{2} e -192 B \,c^{3} d^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{4}}+\frac {\left (96 A a b \,c^{2} e^{3}-192 A a \,c^{3} d \,e^{2}-40 A \,b^{3} c \,e^{3}+144 A \,b^{2} c^{2} d \,e^{2}-192 A b \,c^{3} d^{2} e +128 A \,c^{4} d^{3}+48 B \,a^{2} c^{2} e^{3}-120 B a \,b^{2} c \,e^{3}+288 B a b \,c^{2} d \,e^{2}-192 B a \,c^{3} d^{2} e +35 b^{4} B \,e^{3}-120 b^{3} B c d \,e^{2}+144 b^{2} B \,c^{2} d^{2} e -64 B b \,c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}\) | \(423\) |
| default | \(\frac {A \,d^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+B \,e^{3} \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )+\left (A \,e^{3}+3 B d \,e^{2}\right ) \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )+\left (3 A \,d^{2} e +B \,d^{3}\right ) \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )\) | \(743\) |
-1/192*(-48*B*c^3*e^3*x^3-64*A*c^3*e^3*x^2+56*B*b*c^2*e^3*x^2-192*B*c^3*d* e^2*x^2+80*A*b*c^2*e^3*x-288*A*c^3*d*e^2*x+72*B*a*c^2*e^3*x-70*B*b^2*c*e^3 *x+240*B*b*c^2*d*e^2*x-288*B*c^3*d^2*e*x+128*A*a*c^2*e^3-120*A*b^2*c*e^3+4 32*A*b*c^2*d*e^2-576*A*c^3*d^2*e-220*B*a*b*c*e^3+384*B*a*c^2*d*e^2+105*B*b ^3*e^3-360*B*b^2*c*d*e^2+432*B*b*c^2*d^2*e-192*B*c^3*d^3)*(c*x^2+b*x+a)^(1 /2)/c^4+1/128*(96*A*a*b*c^2*e^3-192*A*a*c^3*d*e^2-40*A*b^3*c*e^3+144*A*b^2 *c^2*d*e^2-192*A*b*c^3*d^2*e+128*A*c^4*d^3+48*B*a^2*c^2*e^3-120*B*a*b^2*c* e^3+288*B*a*b*c^2*d*e^2-192*B*a*c^3*d^2*e+35*B*b^4*e^3-120*B*b^3*c*d*e^2+1 44*B*b^2*c^2*d^2*e-64*B*b*c^3*d^3)/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b *x+a)^(1/2))
Time = 0.44 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.99 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\left [-\frac {3 \, {\left (64 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \, {\left (3 \, B b^{2} c^{2} - 4 \, {\left (B a + A b\right )} c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c + 8 \, A a c^{3} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c^{2}\right )} d e^{2} - {\left (35 \, B b^{4} + 48 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} - 40 \, {\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} e^{3}\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 (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \, {\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 24 \, {\left (15 \, B b^{2} c^{2} - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{3}\right )} d e^{2} - {\left (105 \, B b^{3} c + 128 \, A a c^{3} - 20 \, {\left (11 \, B a b + 6 \, A b^{2}\right )} c^{2}\right )} e^{3} + 8 \, {\left (24 \, B c^{4} d e^{2} - {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (144 \, B c^{4} d^{2} e - 24 \, {\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + {\left (35 \, B b^{2} c^{2} - 4 \, {\left (9 \, B a + 10 \, A b\right )} c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{5}}, \frac {3 \, {\left (64 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \, {\left (3 \, B b^{2} c^{2} - 4 \, {\left (B a + A b\right )} c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c + 8 \, A a c^{3} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c^{2}\right )} d e^{2} - {\left (35 \, B b^{4} + 48 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} - 40 \, {\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} e^{3}\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 (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \, {\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 24 \, {\left (15 \, B b^{2} c^{2} - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{3}\right )} d e^{2} - {\left (105 \, B b^{3} c + 128 \, A a c^{3} - 20 \, {\left (11 \, B a b + 6 \, A b^{2}\right )} c^{2}\right )} e^{3} + 8 \, {\left (24 \, B c^{4} d e^{2} - {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (144 \, B c^{4} d^{2} e - 24 \, {\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + {\left (35 \, B b^{2} c^{2} - 4 \, {\left (9 \, B a + 10 \, A b\right )} c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{5}}\right ] \]
[-1/768*(3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*(B*a + A*b)*c ^3)*d^2*e + 24*(5*B*b^3*c + 8*A*a*c^3 - 6*(2*B*a*b + A*b^2)*c^2)*d*e^2 - ( 35*B*b^4 + 48*(B*a^2 + 2*A*a*b)*c^2 - 40*(3*B*a*b^2 + A*b^3)*c)*e^3)*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)*sq rt(c) - 4*a*c) - 4*(48*B*c^4*e^3*x^3 + 192*B*c^4*d^3 - 144*(3*B*b*c^3 - 4* A*c^4)*d^2*e + 24*(15*B*b^2*c^2 - 2*(8*B*a + 9*A*b)*c^3)*d*e^2 - (105*B*b^ 3*c + 128*A*a*c^3 - 20*(11*B*a*b + 6*A*b^2)*c^2)*e^3 + 8*(24*B*c^4*d*e^2 - (7*B*b*c^3 - 8*A*c^4)*e^3)*x^2 + 2*(144*B*c^4*d^2*e - 24*(5*B*b*c^3 - 6*A *c^4)*d*e^2 + (35*B*b^2*c^2 - 4*(9*B*a + 10*A*b)*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/384*(3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4 *(B*a + A*b)*c^3)*d^2*e + 24*(5*B*b^3*c + 8*A*a*c^3 - 6*(2*B*a*b + A*b^2)* c^2)*d*e^2 - (35*B*b^4 + 48*(B*a^2 + 2*A*a*b)*c^2 - 40*(3*B*a*b^2 + A*b^3) *c)*e^3)*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*(48*B*c^4*e^3*x^3 + 192*B*c^4*d^3 - 144*(3*B*b* c^3 - 4*A*c^4)*d^2*e + 24*(15*B*b^2*c^2 - 2*(8*B*a + 9*A*b)*c^3)*d*e^2 - ( 105*B*b^3*c + 128*A*a*c^3 - 20*(11*B*a*b + 6*A*b^2)*c^2)*e^3 + 8*(24*B*c^4 *d*e^2 - (7*B*b*c^3 - 8*A*c^4)*e^3)*x^2 + 2*(144*B*c^4*d^2*e - 24*(5*B*b*c ^3 - 6*A*c^4)*d*e^2 + (35*B*b^2*c^2 - 4*(9*B*a + 10*A*b)*c^3)*e^3)*x)*sqrt (c*x^2 + b*x + a))/c^5]
Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (428) = 856\).
Time = 1.08 (sec) , antiderivative size = 920, normalized size of antiderivative = 2.26 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \left (\frac {B e^{3} x^{3}}{4 c} + \frac {x^{2} \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{3 c} + \frac {x \left (3 A d e^{2} - \frac {3 B a e^{3}}{4 c} + 3 B d^{2} e - \frac {5 b \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{6 c}\right )}{2 c} + \frac {3 A d^{2} e + B d^{3} - \frac {2 a \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{3 c} - \frac {3 b \left (3 A d e^{2} - \frac {3 B a e^{3}}{4 c} + 3 B d^{2} e - \frac {5 b \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{6 c}\right )}{4 c}}{c}\right ) + \left (A d^{3} - \frac {a \left (3 A d e^{2} - \frac {3 B a e^{3}}{4 c} + 3 B d^{2} e - \frac {5 b \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{6 c}\right )}{2 c} - \frac {b \left (3 A d^{2} e + B d^{3} - \frac {2 a \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{3 c} - \frac {3 b \left (3 A d e^{2} - \frac {3 B a e^{3}}{4 c} + 3 B d^{2} e - \frac {5 b \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\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 {B e^{3} \left (a + b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (A b e^{3} - 4 B a e^{3} + 3 B b d e^{2}\right )}{7 b^{4}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- 3 A a b e^{3} + 3 A b^{2} d e^{2} + 6 B a^{2} e^{3} - 9 B a b d e^{2} + 3 B b^{2} d^{2} e\right )}{5 b^{4}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \cdot \left (3 A a^{2} b e^{3} - 6 A a b^{2} d e^{2} + 3 A b^{3} d^{2} e - 4 B a^{3} e^{3} + 9 B a^{2} b d e^{2} - 6 B a b^{2} d^{2} e + B b^{3} d^{3}\right )}{3 b^{4}} + \frac {\sqrt {a + b x} \left (- A a^{3} b e^{3} + 3 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e + A b^{4} d^{3} + B a^{4} e^{3} - 3 B a^{3} b d e^{2} + 3 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3}\right )}{b^{4}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {A d^{3} x + \frac {B e^{3} x^{5}}{5} + \frac {x^{4} \left (A e^{3} + 3 B d e^{2}\right )}{4} + \frac {x^{3} \cdot \left (3 A d e^{2} + 3 B d^{2} e\right )}{3} + \frac {x^{2} \cdot \left (3 A d^{2} e + B d^{3}\right )}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Piecewise((sqrt(a + b*x + c*x**2)*(B*e**3*x**3/(4*c) + x**2*(A*e**3 - 7*B* b*e**3/(8*c) + 3*B*d*e**2)/(3*c) + x*(3*A*d*e**2 - 3*B*a*e**3/(4*c) + 3*B* d**2*e - 5*b*(A*e**3 - 7*B*b*e**3/(8*c) + 3*B*d*e**2)/(6*c))/(2*c) + (3*A* d**2*e + B*d**3 - 2*a*(A*e**3 - 7*B*b*e**3/(8*c) + 3*B*d*e**2)/(3*c) - 3*b *(3*A*d*e**2 - 3*B*a*e**3/(4*c) + 3*B*d**2*e - 5*b*(A*e**3 - 7*B*b*e**3/(8 *c) + 3*B*d*e**2)/(6*c))/(4*c))/c) + (A*d**3 - a*(3*A*d*e**2 - 3*B*a*e**3/ (4*c) + 3*B*d**2*e - 5*b*(A*e**3 - 7*B*b*e**3/(8*c) + 3*B*d*e**2)/(6*c))/( 2*c) - b*(3*A*d**2*e + B*d**3 - 2*a*(A*e**3 - 7*B*b*e**3/(8*c) + 3*B*d*e** 2)/(3*c) - 3*b*(3*A*d*e**2 - 3*B*a*e**3/(4*c) + 3*B*d**2*e - 5*b*(A*e**3 - 7*B*b*e**3/(8*c) + 3*B*d*e**2)/(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*(B*e**3*(a + b*x)**(9/2)/(9*b**4) + (a + b*x)**(7/2)*(A*b*e**3 - 4*B* a*e**3 + 3*B*b*d*e**2)/(7*b**4) + (a + b*x)**(5/2)*(-3*A*a*b*e**3 + 3*A*b* *2*d*e**2 + 6*B*a**2*e**3 - 9*B*a*b*d*e**2 + 3*B*b**2*d**2*e)/(5*b**4) + ( a + b*x)**(3/2)*(3*A*a**2*b*e**3 - 6*A*a*b**2*d*e**2 + 3*A*b**3*d**2*e - 4 *B*a**3*e**3 + 9*B*a**2*b*d*e**2 - 6*B*a*b**2*d**2*e + B*b**3*d**3)/(3*b** 4) + sqrt(a + b*x)*(-A*a**3*b*e**3 + 3*A*a**2*b**2*d*e**2 - 3*A*a*b**3*d** 2*e + A*b**4*d**3 + B*a**4*e**3 - 3*B*a**3*b*d*e**2 + 3*B*a**2*b**2*d**2*e - B*a*b**3*d**3)/b**4)/b, Ne(b, 0)), ((A*d**3*x + B*e**3*x**5/5 + x**4...
Exception generated. \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.31 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (\frac {6 \, B e^{3} x}{c} + \frac {24 \, B c^{3} d e^{2} - 7 \, B b c^{2} e^{3} + 8 \, A c^{3} e^{3}}{c^{4}}\right )} x + \frac {144 \, B c^{3} d^{2} e - 120 \, B b c^{2} d e^{2} + 144 \, A c^{3} d e^{2} + 35 \, B b^{2} c e^{3} - 36 \, B a c^{2} e^{3} - 40 \, A b c^{2} e^{3}}{c^{4}}\right )} x + \frac {192 \, B c^{3} d^{3} - 432 \, B b c^{2} d^{2} e + 576 \, A c^{3} d^{2} e + 360 \, B b^{2} c d e^{2} - 384 \, B a c^{2} d e^{2} - 432 \, A b c^{2} d e^{2} - 105 \, B b^{3} e^{3} + 220 \, B a b c e^{3} + 120 \, A b^{2} c e^{3} - 128 \, A a c^{2} e^{3}}{c^{4}}\right )} + \frac {{\left (64 \, B b c^{3} d^{3} - 128 \, A c^{4} d^{3} - 144 \, B b^{2} c^{2} d^{2} e + 192 \, B a c^{3} d^{2} e + 192 \, A b c^{3} d^{2} e + 120 \, B b^{3} c d e^{2} - 288 \, B a b c^{2} d e^{2} - 144 \, A b^{2} c^{2} d e^{2} + 192 \, A a c^{3} d e^{2} - 35 \, B b^{4} e^{3} + 120 \, B a b^{2} c e^{3} + 40 \, A b^{3} c e^{3} - 48 \, B a^{2} c^{2} e^{3} - 96 \, A a b c^{2} e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {9}{2}}} \]
1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*B*e^3*x/c + (24*B*c^3*d*e^2 - 7*B*b*c ^2*e^3 + 8*A*c^3*e^3)/c^4)*x + (144*B*c^3*d^2*e - 120*B*b*c^2*d*e^2 + 144* A*c^3*d*e^2 + 35*B*b^2*c*e^3 - 36*B*a*c^2*e^3 - 40*A*b*c^2*e^3)/c^4)*x + ( 192*B*c^3*d^3 - 432*B*b*c^2*d^2*e + 576*A*c^3*d^2*e + 360*B*b^2*c*d*e^2 - 384*B*a*c^2*d*e^2 - 432*A*b*c^2*d*e^2 - 105*B*b^3*e^3 + 220*B*a*b*c*e^3 + 120*A*b^2*c*e^3 - 128*A*a*c^2*e^3)/c^4) + 1/128*(64*B*b*c^3*d^3 - 128*A*c^ 4*d^3 - 144*B*b^2*c^2*d^2*e + 192*B*a*c^3*d^2*e + 192*A*b*c^3*d^2*e + 120* B*b^3*c*d*e^2 - 288*B*a*b*c^2*d*e^2 - 144*A*b^2*c^2*d*e^2 + 192*A*a*c^3*d* e^2 - 35*B*b^4*e^3 + 120*B*a*b^2*c*e^3 + 40*A*b^3*c*e^3 - 48*B*a^2*c^2*e^3 - 96*A*a*b*c^2*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2)
Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+b\,x+a}} \,d x \]