Integrand size = 26, antiderivative size = 341 \[ \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 c^3}+\frac {e^3 (8 B c d-5 b B e+2 A c e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{7/2}} \]
-2/3*(e*x+d)^3*(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)^(3/2)+e^3*(2*A*c*e-5*B*b*e+8*B*c*d)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/ 2))/c^(7/2)-2/3*(e*x+d)*(b*c*d^2*(10*A*b*c*e-8*A*c^2*d-B*b^2*e+4*B*b*c*d)- (16*A*c^4*d^3-5*b^4*B*e^3+4*b^2*c^2*d*e*(A*e+B*d)+2*b^3*c*e^2*(A*e+3*B*d)- 8*b*c^3*d^2*(3*A*e+B*d))*x)/b^4/c^2/(c*x^2+b*x)^(1/2)-1/3*e*(32*A*c^4*d^3- 15*b^4*B*e^3+4*b^2*c^2*d*e*(A*e+2*B*d)-16*b*c^3*d^2*(3*A*e+B*d)+2*b^3*c*e^ 2*(3*A*e+7*B*d))*(c*x^2+b*x)^(1/2)/b^4/c^3
Time = 0.92 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c} \left (b B x \left (15 b^5 e^4 x-16 c^5 d^4 x^2+8 b c^4 d^3 x (-3 d+2 e x)+b^3 c^2 e^3 x^2 (-32 d+3 e x)+4 b^4 c e^3 x (-6 d+5 e x)-6 b^2 c^3 d^2 \left (d^2-4 d e x-2 e^2 x^2\right )\right )-2 A c \left (3 b^5 e^4 x^2-16 c^5 d^4 x^3+4 b^4 c e^4 x^3+8 b c^4 d^3 x^2 (-3 d+4 e x)-6 b^2 c^3 d^2 x \left (d^2-8 d e x+2 e^2 x^2\right )+b^3 c^2 d \left (d^3+12 d^2 e x-18 d e^2 x^2-4 e^3 x^3\right )\right )\right )+3 b^4 e^3 (-8 B c d+5 b B e-2 A c e) x^{3/2} (b+c x)^{3/2} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{3 b^4 c^{7/2} (x (b+c x))^{3/2}} \]
(Sqrt[c]*(b*B*x*(15*b^5*e^4*x - 16*c^5*d^4*x^2 + 8*b*c^4*d^3*x*(-3*d + 2*e *x) + b^3*c^2*e^3*x^2*(-32*d + 3*e*x) + 4*b^4*c*e^3*x*(-6*d + 5*e*x) - 6*b ^2*c^3*d^2*(d^2 - 4*d*e*x - 2*e^2*x^2)) - 2*A*c*(3*b^5*e^4*x^2 - 16*c^5*d^ 4*x^3 + 4*b^4*c*e^4*x^3 + 8*b*c^4*d^3*x^2*(-3*d + 4*e*x) - 6*b^2*c^3*d^2*x *(d^2 - 8*d*e*x + 2*e^2*x^2) + b^3*c^2*d*(d^3 + 12*d^2*e*x - 18*d*e^2*x^2 - 4*e^3*x^3))) + 3*b^4*e^3*(-8*B*c*d + 5*b*B*e - 2*A*c*e)*x^(3/2)*(b + c*x )^(3/2)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]])/(3*b^4*c^(7/2)*(x*(b + c* x))^(3/2))
Time = 0.72 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1233, 27, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int \frac {(d+e x)^2 \left (d \left (-B e b^2+4 B c d b+10 A c e b-8 A c^2 d\right )+e \left (5 B e b^2-2 c (B d+A e) b+4 A c^2 d\right ) x\right )}{2 \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^2 \left (d \left (-B e b^2+4 B c d b+10 A c e b-8 A c^2 d\right )+e \left (5 B e b^2-2 c (B d+A e) b+4 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\frac {2 \int -\frac {e \left (b d \left (-5 B e^2 b^3+2 c e (2 B d+A e) b^2-8 c^2 d (B d+3 A e) b+16 A c^3 d^2\right )+\left (-15 B e^3 b^4+2 c e^2 (7 B d+3 A e) b^3+4 c^2 d e (2 B d+A e) b^2-16 c^3 d^2 (B d+3 A e) b+32 A c^4 d^3\right ) x\right )}{2 \sqrt {c x^2+b x}}dx}{b^2 c}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {b d \left (-5 B e^2 b^3+2 c e (2 B d+A e) b^2-8 c^2 d (B d+3 A e) b+16 A c^3 d^2\right )+\left (-15 B e^3 b^4+2 c e^2 (7 B d+3 A e) b^3+4 c^2 d e (2 B d+A e) b^2-16 c^3 d^2 (B d+3 A e) b+32 A c^4 d^3\right ) x}{\sqrt {c x^2+b x}}dx}{b^2 c}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {-\frac {e \left (\frac {\sqrt {b x+c x^2} \left (2 b^3 c e^2 (3 A e+7 B d)+4 b^2 c^2 d e (A e+2 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{c}-\frac {3 b^4 e^2 (2 A c e-5 b B e+8 B c d) \int \frac {1}{\sqrt {c x^2+b x}}dx}{2 c}\right )}{b^2 c}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {-\frac {e \left (\frac {\sqrt {b x+c x^2} \left (2 b^3 c e^2 (3 A e+7 B d)+4 b^2 c^2 d e (A e+2 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{c}-\frac {3 b^4 e^2 (2 A c e-5 b B e+8 B c d) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{c}\right )}{b^2 c}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {e \left (\frac {\sqrt {b x+c x^2} \left (2 b^3 c e^2 (3 A e+7 B d)+4 b^2 c^2 d e (A e+2 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{c}-\frac {3 b^4 e^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{c^{3/2}}\right )}{b^2 c}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
(-2*(d + e*x)^3*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3* b^2*c*(b*x + c*x^2)^(3/2)) + ((-2*(d + e*x)*(b*c*d^2*(4*b*B*c*d - 8*A*c^2* d - b^2*B*e + 10*A*b*c*e) - (16*A*c^4*d^3 - 5*b^4*B*e^3 + 4*b^2*c^2*d*e*(B *d + A*e) + 2*b^3*c*e^2*(3*B*d + A*e) - 8*b*c^3*d^2*(B*d + 3*A*e))*x))/(b^ 2*c*Sqrt[b*x + c*x^2]) - (e*(((32*A*c^4*d^3 - 15*b^4*B*e^3 + 4*b^2*c^2*d*e *(2*B*d + A*e) - 16*b*c^3*d^2*(B*d + 3*A*e) + 2*b^3*c*e^2*(7*B*d + 3*A*e)) *Sqrt[b*x + c*x^2])/c - (3*b^4*e^2*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[( Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(3/2)))/(b^2*c))/(3*b^2*c)
3.13.6.3.1 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 = 0.67 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\left (-\frac {5 B b e}{2}+\left (A e +4 B d \right ) c \right ) x \,e^{3} \left (c x +b \right ) \sqrt {x \left (c x +b \right )}\, b^{4} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+\frac {d \left (\left (-4 A \,e^{3}-6 B d \,e^{2}\right ) x^{3}+\left (-18 A d \,e^{2}-12 B \,d^{2} e \right ) x^{2}+\left (12 A \,d^{2} e +3 B \,d^{3}\right ) x +A \,d^{3}\right ) b^{3} c^{\frac {7}{2}}}{3}+\left (-2 \left (\left (2 A \,e^{2}+\frac {4}{3} B d e \right ) x^{2}+\left (-8 A d e -2 B \,d^{2}\right ) x +A \,d^{2}\right ) d^{2} b^{2} c^{\frac {9}{2}}+x \left (-8 d^{3} \left (\left (-\frac {4 A e}{3}-\frac {B d}{3}\right ) x +d A \right ) b \,c^{\frac {11}{2}}+b^{5} e^{3} \left (-\frac {10}{3} B e x +A e +4 B d \right ) c^{\frac {3}{2}}+\frac {4 \left (-\frac {3}{8} B e x +A e +4 B d \right ) x \,e^{3} b^{4} c^{\frac {5}{2}}}{3}-\frac {5 B \sqrt {c}\, b^{6} e^{4}}{2}-\frac {16 A \,c^{\frac {13}{2}} d^{4} x}{3}\right )\right ) x \right )}{\sqrt {x \left (c x +b \right )}\, c^{\frac {7}{2}} x \left (c x +b \right ) b^{4}}\) | \(300\) |
| risch | \(-\frac {\left (c x +b \right ) \left (-3 b^{4} x^{2} B \,e^{4}+24 A b \,c^{3} d^{3} e x -16 c^{4} A \,d^{4} x +6 B b \,c^{3} d^{4} x +2 A b \,c^{3} d^{4}\right )}{3 b^{4} \sqrt {x \left (c x +b \right )}\, x \,c^{3}}+\frac {-\frac {5 B \,b^{4} e^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}+2 A \,b^{3} \sqrt {c}\, e^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )+8 B \,b^{3} \sqrt {c}\, d \,e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )+\frac {2 \left (-4 A \,b^{4} c \,e^{4}+8 A \,b^{3} c^{2} d \,e^{3}-8 A \,c^{4} d^{3} e b +4 A \,d^{4} c^{5}+6 B \,b^{5} e^{4}-16 B \,b^{4} c d \,e^{3}+12 B \,b^{3} c^{2} d^{2} e^{2}-2 B \,c^{4} d^{4} b \right ) \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{c b \left (x +\frac {b}{c}\right )}+\frac {2 b \left (A \,b^{4} c \,e^{4}-4 A \,b^{3} c^{2} d \,e^{3}+6 A \,b^{2} c^{3} d^{2} e^{2}-4 A \,c^{4} d^{3} e b +A \,d^{4} c^{5}-B \,b^{5} e^{4}+4 B \,b^{4} c d \,e^{3}-6 B \,b^{3} c^{2} d^{2} e^{2}+4 B \,b^{2} c^{3} d^{3} e -B \,c^{4} d^{4} b \right ) \left (\frac {2 \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{3 b \left (x +\frac {b}{c}\right )^{2}}+\frac {4 c \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{3 b^{2} \left (x +\frac {b}{c}\right )}\right )}{c^{2}}}{2 c^{3} b^{3}}\) | \(530\) |
| default | \(A \,d^{4} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )+B \,e^{4} \left (\frac {x^{4}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {5 b \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )}{2 c}+\frac {-\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}}}}{c}\right )}{2 c}\right )+\left (A \,e^{4}+4 B d \,e^{3}\right ) \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )}{2 c}+\frac {-\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}}}}{c}\right )+\left (4 A \,d^{3} e +B \,d^{4}\right ) \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )\) | \(912\) |
-2/(x*(c*x+b))^(1/2)*(-(-5/2*B*b*e+(A*e+4*B*d)*c)*x*e^3*(c*x+b)*(x*(c*x+b) )^(1/2)*b^4*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+1/3*d*((-4*A*e^3-6*B*d*e^ 2)*x^3+(-18*A*d*e^2-12*B*d^2*e)*x^2+(12*A*d^2*e+3*B*d^3)*x+A*d^3)*b^3*c^(7 /2)+(-2*((2*A*e^2+4/3*B*d*e)*x^2+(-8*A*d*e-2*B*d^2)*x+A*d^2)*d^2*b^2*c^(9/ 2)+x*(-8*d^3*((-4/3*A*e-1/3*B*d)*x+d*A)*b*c^(11/2)+b^5*e^3*(-10/3*B*e*x+A* e+4*B*d)*c^(3/2)+4/3*(-3/8*B*e*x+A*e+4*B*d)*x*e^3*b^4*c^(5/2)-5/2*B*c^(1/2 )*b^6*e^4-16/3*A*c^(13/2)*d^4*x))*x)/c^(7/2)/x/(c*x+b)/b^4
Time = 0.32 (sec) , antiderivative size = 974, normalized size of antiderivative = 2.86 \[ \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left ({\left (8 \, B b^{4} c^{3} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{4} + 2 \, {\left (8 \, B b^{5} c^{2} d e^{3} - {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{3} + {\left (8 \, B b^{6} c d e^{3} - {\left (5 \, B b^{7} - 2 \, A b^{6} c\right )} e^{4}\right )} x^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (3 \, B b^{4} c^{3} e^{4} x^{4} - 2 \, A b^{3} c^{4} d^{4} - 4 \, {\left (4 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} - 4 \, {\left (B b^{2} c^{5} - 4 \, A b c^{6}\right )} d^{3} e - 3 \, {\left (B b^{3} c^{4} + 2 \, A b^{2} c^{5}\right )} d^{2} e^{2} + 2 \, {\left (4 \, B b^{4} c^{3} - A b^{3} c^{4}\right )} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (12 \, A b^{3} c^{4} d^{2} e^{2} - 8 \, B b^{5} c^{2} d e^{3} - 8 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 8 \, {\left (B b^{3} c^{4} - 4 \, A b^{2} c^{5}\right )} d^{3} e + {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{2} - 6 \, {\left (4 \, A b^{3} c^{4} d^{3} e + {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{6 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}}, -\frac {3 \, {\left ({\left (8 \, B b^{4} c^{3} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{4} + 2 \, {\left (8 \, B b^{5} c^{2} d e^{3} - {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{3} + {\left (8 \, B b^{6} c d e^{3} - {\left (5 \, B b^{7} - 2 \, A b^{6} c\right )} e^{4}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (3 \, B b^{4} c^{3} e^{4} x^{4} - 2 \, A b^{3} c^{4} d^{4} - 4 \, {\left (4 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} - 4 \, {\left (B b^{2} c^{5} - 4 \, A b c^{6}\right )} d^{3} e - 3 \, {\left (B b^{3} c^{4} + 2 \, A b^{2} c^{5}\right )} d^{2} e^{2} + 2 \, {\left (4 \, B b^{4} c^{3} - A b^{3} c^{4}\right )} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (12 \, A b^{3} c^{4} d^{2} e^{2} - 8 \, B b^{5} c^{2} d e^{3} - 8 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 8 \, {\left (B b^{3} c^{4} - 4 \, A b^{2} c^{5}\right )} d^{3} e + {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{2} - 6 \, {\left (4 \, A b^{3} c^{4} d^{3} e + {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}}\right ] \]
[1/6*(3*((8*B*b^4*c^3*d*e^3 - (5*B*b^5*c^2 - 2*A*b^4*c^3)*e^4)*x^4 + 2*(8* B*b^5*c^2*d*e^3 - (5*B*b^6*c - 2*A*b^5*c^2)*e^4)*x^3 + (8*B*b^6*c*d*e^3 - (5*B*b^7 - 2*A*b^6*c)*e^4)*x^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x )*sqrt(c)) + 2*(3*B*b^4*c^3*e^4*x^4 - 2*A*b^3*c^4*d^4 - 4*(4*(B*b*c^6 - 2* A*c^7)*d^4 - 4*(B*b^2*c^5 - 4*A*b*c^6)*d^3*e - 3*(B*b^3*c^4 + 2*A*b^2*c^5) *d^2*e^2 + 2*(4*B*b^4*c^3 - A*b^3*c^4)*d*e^3 - (5*B*b^5*c^2 - 2*A*b^4*c^3) *e^4)*x^3 + 3*(12*A*b^3*c^4*d^2*e^2 - 8*B*b^5*c^2*d*e^3 - 8*(B*b^2*c^5 - 2 *A*b*c^6)*d^4 + 8*(B*b^3*c^4 - 4*A*b^2*c^5)*d^3*e + (5*B*b^6*c - 2*A*b^5*c ^2)*e^4)*x^2 - 6*(4*A*b^3*c^4*d^3*e + (B*b^3*c^4 - 2*A*b^2*c^5)*d^4)*x)*sq rt(c*x^2 + b*x))/(b^4*c^6*x^4 + 2*b^5*c^5*x^3 + b^6*c^4*x^2), -1/3*(3*((8* B*b^4*c^3*d*e^3 - (5*B*b^5*c^2 - 2*A*b^4*c^3)*e^4)*x^4 + 2*(8*B*b^5*c^2*d* e^3 - (5*B*b^6*c - 2*A*b^5*c^2)*e^4)*x^3 + (8*B*b^6*c*d*e^3 - (5*B*b^7 - 2 *A*b^6*c)*e^4)*x^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (3 *B*b^4*c^3*e^4*x^4 - 2*A*b^3*c^4*d^4 - 4*(4*(B*b*c^6 - 2*A*c^7)*d^4 - 4*(B *b^2*c^5 - 4*A*b*c^6)*d^3*e - 3*(B*b^3*c^4 + 2*A*b^2*c^5)*d^2*e^2 + 2*(4*B *b^4*c^3 - A*b^3*c^4)*d*e^3 - (5*B*b^5*c^2 - 2*A*b^4*c^3)*e^4)*x^3 + 3*(12 *A*b^3*c^4*d^2*e^2 - 8*B*b^5*c^2*d*e^3 - 8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 8 *(B*b^3*c^4 - 4*A*b^2*c^5)*d^3*e + (5*B*b^6*c - 2*A*b^5*c^2)*e^4)*x^2 - 6* (4*A*b^3*c^4*d^3*e + (B*b^3*c^4 - 2*A*b^2*c^5)*d^4)*x)*sqrt(c*x^2 + b*x))/ (b^4*c^6*x^4 + 2*b^5*c^5*x^3 + b^6*c^4*x^2)]
\[ \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{4}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 795 vs. \(2 (323) = 646\).
Time = 0.20 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.33 \[ \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {5 \, B b e^{4} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )}}{6 \, c} + \frac {B e^{4} x^{4}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {4 \, A c d^{4} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d^{4} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {10 \, B b e^{4} x}{3 \, \sqrt {c x^{2} + b x} c^{3}} - \frac {5 \, B b e^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {7}{2}}} - \frac {1}{3} \, {\left (4 \, B d e^{3} + A e^{4}\right )} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} - \frac {2 \, A d^{4}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d^{4}}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {5 \, \sqrt {c x^{2} + b x} B e^{4}}{3 \, c^{3}} - \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {8 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} x}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (B d^{4} + 4 \, A d^{3} e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {4 \, {\left (4 \, B d e^{3} + A e^{4}\right )} x}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {4 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {16 \, {\left (B d^{4} + 4 \, A d^{3} e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {{\left (4 \, B d e^{3} + A e^{4}\right )} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {8 \, {\left (B d^{4} + 4 \, A d^{3} e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )}}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {2 \, {\left (4 \, B d e^{3} + A e^{4}\right )} \sqrt {c x^{2} + b x}}{3 \, b c^{2}} + \frac {4 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )}}{3 \, \sqrt {c x^{2} + b x} b c} \]
5/6*B*b*e^4*x*(3*x^2/((c*x^2 + b*x)^(3/2)*c) + b*x/((c*x^2 + b*x)^(3/2)*c^ 2) - 2*x/(sqrt(c*x^2 + b*x)*b*c) - 1/(sqrt(c*x^2 + b*x)*c^2))/c + B*e^4*x^ 4/((c*x^2 + b*x)^(3/2)*c) - 4/3*A*c*d^4*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3 *A*c^2*d^4*x/(sqrt(c*x^2 + b*x)*b^4) + 10/3*B*b*e^4*x/(sqrt(c*x^2 + b*x)*c ^3) - 5/2*B*b*e^4*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) - 1 /3*(4*B*d*e^3 + A*e^4)*x*(3*x^2/((c*x^2 + b*x)^(3/2)*c) + b*x/((c*x^2 + b* x)^(3/2)*c^2) - 2*x/(sqrt(c*x^2 + b*x)*b*c) - 1/(sqrt(c*x^2 + b*x)*c^2)) - 2/3*A*d^4/((c*x^2 + b*x)^(3/2)*b) + 16/3*A*c*d^4/(sqrt(c*x^2 + b*x)*b^3) + 5/3*sqrt(c*x^2 + b*x)*B*e^4/c^3 - 2*(3*B*d^2*e^2 + 2*A*d*e^3)*x^2/((c*x^ 2 + b*x)^(3/2)*c) + 8/3*(2*B*d^3*e + 3*A*d^2*e^2)*x/(sqrt(c*x^2 + b*x)*b^2 ) + 2/3*(B*d^4 + 4*A*d^3*e)*x/((c*x^2 + b*x)^(3/2)*b) - 4/3*(4*B*d*e^3 + A *e^4)*x/(sqrt(c*x^2 + b*x)*c^2) - 2/3*(3*B*d^2*e^2 + 2*A*d*e^3)*b*x/((c*x^ 2 + b*x)^(3/2)*c^2) - 4/3*(2*B*d^3*e + 3*A*d^2*e^2)*x/((c*x^2 + b*x)^(3/2) *c) + 4/3*(3*B*d^2*e^2 + 2*A*d*e^3)*x/(sqrt(c*x^2 + b*x)*b*c) - 16/3*(B*d^ 4 + 4*A*d^3*e)*c*x/(sqrt(c*x^2 + b*x)*b^3) + (4*B*d*e^3 + A*e^4)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 8/3*(B*d^4 + 4*A*d^3*e)/(sqr t(c*x^2 + b*x)*b^2) + 2/3*(3*B*d^2*e^2 + 2*A*d*e^3)/(sqrt(c*x^2 + b*x)*c^2 ) - 2/3*(4*B*d*e^3 + A*e^4)*sqrt(c*x^2 + b*x)/(b*c^2) + 4/3*(2*B*d^3*e + 3 *A*d^2*e^2)/(sqrt(c*x^2 + b*x)*b*c)
Time = 0.31 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {\frac {2 \, A d^{4}}{b} - {\left ({\left ({\left (\frac {3 \, B e^{4} x}{c} - \frac {4 \, {\left (4 \, B b c^{5} d^{4} - 8 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e + 16 \, A b c^{5} d^{3} e - 3 \, B b^{3} c^{3} d^{2} e^{2} - 6 \, A b^{2} c^{4} d^{2} e^{2} + 8 \, B b^{4} c^{2} d e^{3} - 2 \, A b^{3} c^{3} d e^{3} - 5 \, B b^{5} c e^{4} + 2 \, A b^{4} c^{2} e^{4}\right )}}{b^{4} c^{3}}\right )} x - \frac {3 \, {\left (8 \, B b^{2} c^{4} d^{4} - 16 \, A b c^{5} d^{4} - 8 \, B b^{3} c^{3} d^{3} e + 32 \, A b^{2} c^{4} d^{3} e - 12 \, A b^{3} c^{3} d^{2} e^{2} + 8 \, B b^{5} c d e^{3} - 5 \, B b^{6} e^{4} + 2 \, A b^{5} c e^{4}\right )}}{b^{4} c^{3}}\right )} x - \frac {6 \, {\left (B b^{3} c^{3} d^{4} - 2 \, A b^{2} c^{4} d^{4} + 4 \, A b^{3} c^{3} d^{3} e\right )}}{b^{4} c^{3}}\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} - \frac {{\left (8 \, B c d e^{3} - 5 \, B b e^{4} + 2 \, A c e^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {7}{2}}} \]
-1/3*(2*A*d^4/b - (((3*B*e^4*x/c - 4*(4*B*b*c^5*d^4 - 8*A*c^6*d^4 - 4*B*b^ 2*c^4*d^3*e + 16*A*b*c^5*d^3*e - 3*B*b^3*c^3*d^2*e^2 - 6*A*b^2*c^4*d^2*e^2 + 8*B*b^4*c^2*d*e^3 - 2*A*b^3*c^3*d*e^3 - 5*B*b^5*c*e^4 + 2*A*b^4*c^2*e^4 )/(b^4*c^3))*x - 3*(8*B*b^2*c^4*d^4 - 16*A*b*c^5*d^4 - 8*B*b^3*c^3*d^3*e + 32*A*b^2*c^4*d^3*e - 12*A*b^3*c^3*d^2*e^2 + 8*B*b^5*c*d*e^3 - 5*B*b^6*e^4 + 2*A*b^5*c*e^4)/(b^4*c^3))*x - 6*(B*b^3*c^3*d^4 - 2*A*b^2*c^4*d^4 + 4*A* b^3*c^3*d^3*e)/(b^4*c^3))*x)/(c*x^2 + b*x)^(3/2) - 1/2*(8*B*c*d*e^3 - 5*B* b*e^4 + 2*A*c*e^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b)) /c^(7/2)
Timed out. \[ \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]