Integrand size = 27, antiderivative size = 210 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 (d+e x) \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {e \left (4 A c^2 d+3 b^2 B e-2 c (b B d+A b e+4 a B e)\right ) \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}+\frac {e (4 B c d-3 b B e+2 A c e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2}} \]
1/2*e*(2*A*c*e-3*B*b*e+4*B*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a )^(1/2))/c^(5/2)+2*(e*x+d)*(2*a*c*(A*e+B*d)-b*(A*c*d+B*a*e)-(b^2*B*e-b*c*( A*e+B*d)+2*c*(A*c*d-B*a*e))*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+e*(4*A*c ^2*d+3*b^2*B*e-2*c*(A*b*e+4*B*a*e+B*b*d))*(c*x^2+b*x+a)^(1/2)/c^2/(-4*a*c+ b^2)
Time = 1.00 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {2 \sqrt {c} \left (2 A c \left (a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)\right )+B \left (8 a^2 c e^2-b x \left (2 c^2 d^2+3 b^2 e^2+b c e (-4 d+e x)\right )+a \left (-3 b^2 e^2+2 b c e (2 d+5 e x)-4 c^2 \left (d^2+2 d e x-e^2 x^2\right )\right )\right )\right )}{\sqrt {a+x (b+c x)}}-\left (b^2-4 a c\right ) e (-4 B c d+3 b B e-2 A c e) \log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{2 c^{5/2} \left (-b^2+4 a c\right )} \]
((2*Sqrt[c]*(2*A*c*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)) + B*(8*a^2*c*e^2 - b*x*(2*c^2*d^2 + 3*b^2*e^2 + b*c *e*(-4*d + e*x)) + a*(-3*b^2*e^2 + 2*b*c*e*(2*d + 5*e*x) - 4*c^2*(d^2 + 2* d*e*x - e^2*x^2)))))/Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*e*(-4*B*c*d + 3 *b*B*e - 2*A*c*e)*Log[c^2*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/ (2*c^(5/2)*(-b^2 + 4*a*c))
Time = 0.45 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1233, 27, 1160, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int \frac {e \left (B d b^2+2 (A c d+a B e) b-4 a c (2 B d+A e)+\left (3 B e b^2+4 A c^2 d-2 c (b B d+A b e+4 a B e)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {B d b^2+2 (A c d+a B e) b-4 a c (2 B d+A e)+\left (3 B e b^2+4 A c^2 d-2 c (b B d+A b e+4 a B e)\right ) x}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {e \left (\frac {\left (b^2-4 a c\right ) (2 A c e-3 b B e+4 B c d) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c}+\frac {\sqrt {a+b x+c x^2} \left (-2 c (4 a B e+A b e+b B d)+4 A c^2 d+3 b^2 B e\right )}{c}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {e \left (\frac {\left (b^2-4 a c\right ) (2 A c e-3 b B e+4 B c d) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}+\frac {\sqrt {a+b x+c x^2} \left (-2 c (4 a B e+A b e+b B d)+4 A c^2 d+3 b^2 B e\right )}{c}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {e \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{2 c^{3/2}}+\frac {\sqrt {a+b x+c x^2} \left (-2 c (4 a B e+A b e+b B d)+4 A c^2 d+3 b^2 B e\right )}{c}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x) \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(2*(d + e*x)*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (e*(((4*A*c^2*d + 3*b^2*B*e - 2*c*(b*B*d + A*b*e + 4*a*B*e))*Sqrt[a + b* x + c*x^2])/c + ((b^2 - 4*a*c)*(4*B*c*d - 3*b*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(3/2))))/(c*(b^2 - 4*a*c))
3.25.74.3.1 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_))*((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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Time = 0.58 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.57
| method | result | size |
| risch | \(\frac {e^{2} B \sqrt {c \,x^{2}+b x +a}}{c^{2}}+\frac {\frac {4 A \,c^{2} d^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {2 B a b \,e^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (2 A \,c^{2} e^{2}-3 B \,e^{2} b c +4 B \,c^{2} d e \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (4 A \,c^{2} d e -2 B \,e^{2} a c -B \,b^{2} e^{2}+2 B \,c^{2} d^{2}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c^{2}}\) | \(330\) |
| default | \(\frac {2 A \,d^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+B \,e^{2} \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+\left (A \,e^{2}+2 B d e \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (2 A d e +B \,d^{2}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\) | \(423\) |
e^2*B/c^2*(c*x^2+b*x+a)^(1/2)+1/2/c^2*(4*A*c^2*d^2*(2*c*x+b)/(4*a*c-b^2)/( c*x^2+b*x+a)^(1/2)-2*B*a*b*e^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+( 2*A*c^2*e^2-3*B*b*c*e^2+4*B*c^2*d*e)*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1 /c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^ (3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+(4*A*c^2*d*e-2*B*a*c*e^ 2-B*b^2*e^2+2*B*c^2*d^2)*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^ 2)/(c*x^2+b*x+a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (196) = 392\).
Time = 1.51 (sec) , antiderivative size = 945, normalized size of antiderivative = 4.50 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (4 \, {\left (B a b^{2} c - 4 \, B a^{2} c^{2}\right )} d e - {\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} - 2 \, {\left (6 \, B a^{2} b + A a b^{2}\right )} c\right )} e^{2} + {\left (4 \, {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} d e - {\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \, {\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} e^{2}\right )} x^{2} + {\left (4 \, {\left (B b^{3} c - 4 \, B a b c^{2}\right )} d e - {\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c\right )} e^{2}\right )} x\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 (2 \, {\left (2 \, B a - A b\right )} c^{3} d^{2} + {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} e^{2} x^{2} - 4 \, {\left (B a b c^{2} - 2 \, A a c^{3}\right )} d e + {\left (3 \, B a b^{2} c - 2 \, {\left (4 \, B a^{2} + A a b\right )} c^{2}\right )} e^{2} + {\left (2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} - 4 \, {\left (B b^{2} c^{2} - {\left (2 \, B a + A b\right )} c^{3}\right )} d e + {\left (3 \, B b^{3} c + 4 \, A a c^{3} - 2 \, {\left (5 \, B a b + A b^{2}\right )} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}, -\frac {{\left (4 \, {\left (B a b^{2} c - 4 \, B a^{2} c^{2}\right )} d e - {\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} - 2 \, {\left (6 \, B a^{2} b + A a b^{2}\right )} c\right )} e^{2} + {\left (4 \, {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} d e - {\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \, {\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} e^{2}\right )} x^{2} + {\left (4 \, {\left (B b^{3} c - 4 \, B a b c^{2}\right )} d e - {\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c\right )} e^{2}\right )} x\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 (2 \, {\left (2 \, B a - A b\right )} c^{3} d^{2} + {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} e^{2} x^{2} - 4 \, {\left (B a b c^{2} - 2 \, A a c^{3}\right )} d e + {\left (3 \, B a b^{2} c - 2 \, {\left (4 \, B a^{2} + A a b\right )} c^{2}\right )} e^{2} + {\left (2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} - 4 \, {\left (B b^{2} c^{2} - {\left (2 \, B a + A b\right )} c^{3}\right )} d e + {\left (3 \, B b^{3} c + 4 \, A a c^{3} - 2 \, {\left (5 \, B a b + A b^{2}\right )} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}\right ] \]
[1/4*((4*(B*a*b^2*c - 4*B*a^2*c^2)*d*e - (3*B*a*b^3 + 8*A*a^2*c^2 - 2*(6*B *a^2*b + A*a*b^2)*c)*e^2 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d*e - (3*B*b^3*c + 8 *A*a*c^3 - 2*(6*B*a*b + A*b^2)*c^2)*e^2)*x^2 + (4*(B*b^3*c - 4*B*a*b*c^2)* d*e - (3*B*b^4 + 8*A*a*b*c^2 - 2*(6*B*a*b^2 + A*b^3)*c)*e^2)*x)*sqrt(c)*lo g(-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*(2*(2*B*a - A*b)*c^3*d^2 + (B*b^2*c^2 - 4*B*a*c^3)*e^2*x^2 - 4*(B*a*b*c^2 - 2*A*a*c^3)*d*e + (3*B*a*b^2*c - 2*(4*B*a^2 + A*a*b)*c^2)*e ^2 + (2*(B*b*c^3 - 2*A*c^4)*d^2 - 4*(B*b^2*c^2 - (2*B*a + A*b)*c^3)*d*e + (3*B*b^3*c + 4*A*a*c^3 - 2*(5*B*a*b + A*b^2)*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b *c^4)*x), -1/2*((4*(B*a*b^2*c - 4*B*a^2*c^2)*d*e - (3*B*a*b^3 + 8*A*a^2*c^ 2 - 2*(6*B*a^2*b + A*a*b^2)*c)*e^2 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d*e - (3*B *b^3*c + 8*A*a*c^3 - 2*(6*B*a*b + A*b^2)*c^2)*e^2)*x^2 + (4*(B*b^3*c - 4*B *a*b*c^2)*d*e - (3*B*b^4 + 8*A*a*b*c^2 - 2*(6*B*a*b^2 + A*b^3)*c)*e^2)*x)* 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*(2*(2*B*a - A*b)*c^3*d^2 + (B*b^2*c^2 - 4*B*a*c^3)*e^2*x ^2 - 4*(B*a*b*c^2 - 2*A*a*c^3)*d*e + (3*B*a*b^2*c - 2*(4*B*a^2 + A*a*b)*c^ 2)*e^2 + (2*(B*b*c^3 - 2*A*c^4)*d^2 - 4*(B*b^2*c^2 - (2*B*a + A*b)*c^3)*d* e + (3*B*b^3*c + 4*A*a*c^3 - 2*(5*B*a*b + A*b^2)*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 ...
\[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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 = 0.30 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left (\frac {{\left (B b^{2} c e^{2} - 4 \, B a c^{2} e^{2}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 8 \, B a c^{2} d e + 4 \, A b c^{2} d e + 3 \, B b^{3} e^{2} - 10 \, B a b c e^{2} - 2 \, A b^{2} c e^{2} + 4 \, A a c^{2} e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac {4 \, B a c^{2} d^{2} - 2 \, A b c^{2} d^{2} - 4 \, B a b c d e + 8 \, A a c^{2} d e + 3 \, B a b^{2} e^{2} - 8 \, B a^{2} c e^{2} - 2 \, A a b c e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt {c x^{2} + b x + a}} - \frac {{\left (4 \, B c d e - 3 \, B b e^{2} + 2 \, A c e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {5}{2}}} \]
(((B*b^2*c*e^2 - 4*B*a*c^2*e^2)*x/(b^2*c^2 - 4*a*c^3) + (2*B*b*c^2*d^2 - 4 *A*c^3*d^2 - 4*B*b^2*c*d*e + 8*B*a*c^2*d*e + 4*A*b*c^2*d*e + 3*B*b^3*e^2 - 10*B*a*b*c*e^2 - 2*A*b^2*c*e^2 + 4*A*a*c^2*e^2)/(b^2*c^2 - 4*a*c^3))*x + (4*B*a*c^2*d^2 - 2*A*b*c^2*d^2 - 4*B*a*b*c*d*e + 8*A*a*c^2*d*e + 3*B*a*b^2 *e^2 - 8*B*a^2*c*e^2 - 2*A*a*b*c*e^2)/(b^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b* x + a) - 1/2*(4*B*c*d*e - 3*B*b*e^2 + 2*A*c*e^2)*log(abs(2*(sqrt(c)*x - sq rt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(5/2)
Timed out. \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]