Integrand size = 26, antiderivative size = 151 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x) \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {e \left (4 A c^2 d+3 b^2 B e-2 b c (B d+A e)\right ) \sqrt {b x+c x^2}}{b^2 c^2}+\frac {e (4 B c d-3 b B e+2 A c e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \]
e*(2*A*c*e-3*B*b*e+4*B*c*d)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(5/2)-2 *(e*x+d)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)/b^2/c/(c*x^2+b*x)^( 1/2)+e*(4*A*c^2*d+3*b^2*B*e-2*b*c*(A*e+B*d))*(c*x^2+b*x)^(1/2)/b^2/c^2
Time = 0.56 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {x \left (\frac {\sqrt {c} (b+c x) \left (-2 A c \left (2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)\right )+b B x \left (2 c^2 d^2+3 b^2 e^2+b c e (-4 d+e x)\right )\right )}{b^2}+2 e (4 B c d-3 b B e+2 A c e) \sqrt {x} (b+c x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{c^{5/2} (x (b+c x))^{3/2}} \]
(x*((Sqrt[c]*(b + c*x)*(-2*A*c*(2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x )) + b*B*x*(2*c^2*d^2 + 3*b^2*e^2 + b*c*e*(-4*d + e*x))))/b^2 + 2*e*(4*B*c *d - 3*b*B*e + 2*A*c*e)*Sqrt[x]*(b + c*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[x])/ (-Sqrt[b] + Sqrt[b + c*x])]))/(c^(5/2)*(x*(b + c*x))^(3/2))
Time = 0.37 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1233, 27, 1160, 1091, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int \frac {e \left (b (b B+2 A c) d+\left (3 B e b^2-2 c (B d+A e) b+4 A c^2 d\right ) x\right )}{2 \sqrt {c x^2+b x}}dx}{b^2 c}-\frac {2 (d+e x) \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {b (b B+2 A c) d+\left (3 B e b^2-2 c (B d+A e) b+4 A c^2 d\right ) x}{\sqrt {c x^2+b x}}dx}{b^2 c}-\frac {2 (d+e x) \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {e \left (\frac {b^2 (2 A c e-3 b B e+4 B c d) \int \frac {1}{\sqrt {c x^2+b x}}dx}{2 c}+\frac {\sqrt {b x+c x^2} \left (-2 b c (A e+B d)+4 A c^2 d+3 b^2 B e\right )}{c}\right )}{b^2 c}-\frac {2 (d+e x) \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {e \left (\frac {b^2 (2 A c e-3 b B e+4 B c d) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{c}+\frac {\sqrt {b x+c x^2} \left (-2 b c (A e+B d)+4 A c^2 d+3 b^2 B e\right )}{c}\right )}{b^2 c}-\frac {2 (d+e x) \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {e \left (\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{c^{3/2}}+\frac {\sqrt {b x+c x^2} \left (-2 b c (A e+B d)+4 A c^2 d+3 b^2 B e\right )}{c}\right )}{b^2 c}-\frac {2 (d+e x) \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
(-2*(d + e*x)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2* c*Sqrt[b*x + c*x^2]) + (e*(((4*A*c^2*d + 3*b^2*B*e - 2*b*c*(B*d + A*e))*Sq rt[b*x + c*x^2])/c + (b^2*(4*B*c*d - 3*b*B*e + 2*A*c*e)*ArcTanh[(Sqrt[c]*x )/Sqrt[b*x + c*x^2]])/c^(3/2)))/(b^2*c)
3.12.100.3.1 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[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{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 = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (-\frac {\sqrt {x \left (c x +b \right )}\, \left (\left (A e +2 B d \right ) c -\frac {3 B b e}{2}\right ) e \,b^{2} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{2}+\left (-\frac {B \,e^{2} x^{2} b^{2} c}{4}+\left (A \,c^{3} d^{2}-c^{2} \left (A e +\frac {B d}{2}\right ) d b +\frac {b^{2} c e \left (A e +2 B d \right )}{2}-\frac {3 b^{3} B \,e^{2}}{4}\right ) x +\frac {A \,d^{2} b \,c^{2}}{2}\right ) \sqrt {c}\right )}{\sqrt {x \left (c x +b \right )}\, c^{\frac {5}{2}} b^{2}}\) | \(145\) |
| risch | \(-\frac {\left (c x +b \right ) \left (-B \,b^{2} e^{2} x +2 A \,c^{2} d^{2}\right )}{b^{2} \sqrt {x \left (c x +b \right )}\, c^{2}}+\frac {-\frac {3 B \,b^{2} e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}+2 A b \sqrt {c}\, e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )+4 B b \sqrt {c}\, d e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )+\frac {2 \left (-2 A \,b^{2} c \,e^{2}+4 A b \,c^{2} d e -2 A \,c^{3} d^{2}+2 b^{3} B \,e^{2}-4 B \,b^{2} c d e +2 B b \,c^{2} d^{2}\right ) \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{c b \left (x +\frac {b}{c}\right )}}{2 c^{2} b}\) | \(254\) |
| default | \(-\frac {2 A \,d^{2} \left (2 c x +b \right )}{b^{2} \sqrt {c \,x^{2}+b x}}+B \,e^{2} \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+\left (A \,e^{2}+2 B d e \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}\right )+\left (2 A d e +B \,d^{2}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )\) | \(309\) |
-4/(x*(c*x+b))^(1/2)*(-1/2*(x*(c*x+b))^(1/2)*((A*e+2*B*d)*c-3/2*B*b*e)*e*b ^2*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+(-1/4*B*e^2*x^2*b^2*c+(A*c^3*d^2-c ^2*(A*e+1/2*B*d)*d*b+1/2*b^2*c*e*(A*e+2*B*d)-3/4*b^3*B*e^2)*x+1/2*A*d^2*b* c^2)*c^(1/2))/c^(5/2)/b^2
Time = 0.37 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.95 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (4 \, B b^{2} c^{2} d e - {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} e^{2}\right )} x^{2} + {\left (4 \, B b^{3} c d e - {\left (3 \, B b^{4} - 2 \, A b^{3} c\right )} e^{2}\right )} x\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (B b^{2} c^{2} e^{2} x^{2} - 2 \, A b c^{3} d^{2} + {\left (2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} - 4 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d e + {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (b^{2} c^{4} x^{2} + b^{3} c^{3} x\right )}}, -\frac {{\left ({\left (4 \, B b^{2} c^{2} d e - {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} e^{2}\right )} x^{2} + {\left (4 \, B b^{3} c d e - {\left (3 \, B b^{4} - 2 \, A b^{3} c\right )} e^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (B b^{2} c^{2} e^{2} x^{2} - 2 \, A b c^{3} d^{2} + {\left (2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} - 4 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d e + {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{b^{2} c^{4} x^{2} + b^{3} c^{3} x}\right ] \]
[1/2*(((4*B*b^2*c^2*d*e - (3*B*b^3*c - 2*A*b^2*c^2)*e^2)*x^2 + (4*B*b^3*c* d*e - (3*B*b^4 - 2*A*b^3*c)*e^2)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(B*b^2*c^2*e^2*x^2 - 2*A*b*c^3*d^2 + (2*(B*b*c^3 - 2*A* c^4)*d^2 - 4*(B*b^2*c^2 - A*b*c^3)*d*e + (3*B*b^3*c - 2*A*b^2*c^2)*e^2)*x) *sqrt(c*x^2 + b*x))/(b^2*c^4*x^2 + b^3*c^3*x), -(((4*B*b^2*c^2*d*e - (3*B* b^3*c - 2*A*b^2*c^2)*e^2)*x^2 + (4*B*b^3*c*d*e - (3*B*b^4 - 2*A*b^3*c)*e^2 )*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (B*b^2*c^2*e^2*x^ 2 - 2*A*b*c^3*d^2 + (2*(B*b*c^3 - 2*A*c^4)*d^2 - 4*(B*b^2*c^2 - A*b*c^3)*d *e + (3*B*b^3*c - 2*A*b^2*c^2)*e^2)*x)*sqrt(c*x^2 + b*x))/(b^2*c^4*x^2 + b ^3*c^3*x)]
\[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{2}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {B e^{2} x^{2}}{\sqrt {c x^{2} + b x} c} + \frac {2 \, B d^{2} x}{\sqrt {c x^{2} + b x} b} - \frac {4 \, A c d^{2} x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {4 \, A d e x}{\sqrt {c x^{2} + b x} b} + \frac {3 \, B b e^{2} x}{\sqrt {c x^{2} + b x} c^{2}} - \frac {3 \, B b e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {5}{2}}} - \frac {2 \, A d^{2}}{\sqrt {c x^{2} + b x} b} - \frac {2 \, {\left (2 \, B d e + A e^{2}\right )} x}{\sqrt {c x^{2} + b x} c} + \frac {{\left (2 \, B d e + A e^{2}\right )} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {3}{2}}} \]
B*e^2*x^2/(sqrt(c*x^2 + b*x)*c) + 2*B*d^2*x/(sqrt(c*x^2 + b*x)*b) - 4*A*c* d^2*x/(sqrt(c*x^2 + b*x)*b^2) + 4*A*d*e*x/(sqrt(c*x^2 + b*x)*b) + 3*B*b*e^ 2*x/(sqrt(c*x^2 + b*x)*c^2) - 3/2*B*b*e^2*log(2*c*x + b + 2*sqrt(c*x^2 + b *x)*sqrt(c))/c^(5/2) - 2*A*d^2/(sqrt(c*x^2 + b*x)*b) - 2*(2*B*d*e + A*e^2) *x/(sqrt(c*x^2 + b*x)*c) + (2*B*d*e + A*e^2)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(3/2)
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {\frac {2 \, A d^{2}}{b} - {\left (\frac {B e^{2} x}{c} + \frac {2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 4 \, A b c^{2} d e + 3 \, B b^{3} e^{2} - 2 \, A b^{2} c e^{2}}{b^{2} c^{2}}\right )} x}{\sqrt {c x^{2} + b x}} - \frac {{\left (4 \, B c d e - 3 \, B b e^{2} + 2 \, A c e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {5}{2}}} \]
-(2*A*d^2/b - (B*e^2*x/c + (2*B*b*c^2*d^2 - 4*A*c^3*d^2 - 4*B*b^2*c*d*e + 4*A*b*c^2*d*e + 3*B*b^3*e^2 - 2*A*b^2*c*e^2)/(b^2*c^2))*x)/sqrt(c*x^2 + b* x) - 1/2*(4*B*c*d*e - 3*B*b*e^2 + 2*A*c*e^2)*log(abs(2*(sqrt(c)*x - sqrt(c *x^2 + b*x))*sqrt(c) + b))/c^(5/2)
Timed out. \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]