Integrand size = 26, antiderivative size = 220 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b c d^2 \left (8 A c^2 d+b^2 B e-4 b c (B d+2 A e)\right )+\left (16 A c^4 d^3+2 b^3 B c d e^2-3 b^4 B e^3-8 b c^3 d^2 (B d+3 A e)+2 b^2 c^2 d e (3 B d+4 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {2 B e^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \]
-2/3*(e*x+d)^2*(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)+2*B*e^3*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(5/2)+2/3*(b*c*d ^2*(8*A*c^2*d+b^2*B*e-4*b*c*(2*A*e+B*d))+(16*A*c^4*d^3+2*b^3*B*c*d*e^2-3*b ^4*B*e^3-8*b*c^3*d^2*(3*A*e+B*d)+2*b^2*c^2*d*e*(4*A*e+3*B*d))*x)/b^4/c^2/( c*x^2+b*x)^(1/2)
Time = 0.72 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (\sqrt {c} \left (b B x \left (3 b^4 e^3 x+8 c^4 d^3 x^2+4 b^3 c e^3 x^2+6 b c^3 d^2 x (2 d-e x)+3 b^2 c^2 d \left (d^2-3 d e x-e^2 x^2\right )\right )-A c^2 \left (16 c^3 d^3 x^3+24 b c^2 d^2 x^2 (d-e x)+6 b^2 c d x \left (d^2-6 d e x+e^2 x^2\right )+b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )\right )\right )+3 b^4 B e^3 x^{3/2} (b+c x)^{3/2} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{3 b^4 c^{5/2} (x (b+c x))^{3/2}} \]
(-2*(Sqrt[c]*(b*B*x*(3*b^4*e^3*x + 8*c^4*d^3*x^2 + 4*b^3*c*e^3*x^2 + 6*b*c ^3*d^2*x*(2*d - e*x) + 3*b^2*c^2*d*(d^2 - 3*d*e*x - e^2*x^2)) - A*c^2*(16* c^3*d^3*x^3 + 24*b*c^2*d^2*x^2*(d - e*x) + 6*b^2*c*d*x*(d^2 - 6*d*e*x + e^ 2*x^2) + b^3*(-d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3))) + 3*b^4*B*e^3*x^ (3/2)*(b + c*x)^(3/2)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]]))/(3*b^4*c^( 5/2)*(x*(b + c*x))^(3/2))
Time = 0.44 (sec) , antiderivative size = 232, 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, 1224, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int -\frac {(d+e x) \left (d \left (B e b^2-4 c (B d+2 A e) b+8 A c^2 d\right )-3 b^2 B e^2 x\right )}{2 \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(d+e x) \left (d \left (B e b^2-4 c (B d+2 A e) b+8 A c^2 d\right )-3 b^2 B e^2 x\right )}{\left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1224 |
| \(\displaystyle -\frac {-\frac {3 b^2 B e^3 \int \frac {1}{\sqrt {c x^2+b x}}dx}{c}-\frac {2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle -\frac {-\frac {6 b^2 B e^3 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{c}-\frac {2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{b^2 c \sqrt {b x+c x^2}}}{3 b^2 c}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {2 \left (b c d^2 \left (-4 b c (2 A e+B d)+8 A c^2 d+b^2 B e\right )+x \left (2 b^2 c^2 d e (4 A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-3 b^4 B e^3+2 b^3 B c d e^2\right )\right )}{b^2 c \sqrt {b x+c x^2}}-\frac {6 b^2 B e^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}}{3 b^2 c}-\frac {2 (d+e x)^2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
(-2*(d + e*x)^2*(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*(b*c*d^2*(8*A*c^2*d + b^2*B*e - 4*b*c*(B *d + 2*A*e)) + (16*A*c^4*d^3 + 2*b^3*B*c*d*e^2 - 3*b^4*B*e^3 - 8*b*c^3*d^2 *(B*d + 3*A*e) + 2*b^2*c^2*d*e*(3*B*d + 4*A*e))*x))/(b^2*c*Sqrt[b*x + c*x^ 2]) - (6*b^2*B*e^3*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(3/2))/(3*b^2 *c)
3.13.7.3.1 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_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(b^2 - 4*a*c))), 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))/(c*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 1] && !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
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 = 0.54 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {2 \left (\left (-A \,e^{3}-3 B d \,e^{2}\right ) x^{3}+\left (-9 A d \,e^{2}-9 B \,d^{2} e \right ) x^{2}+\left (9 A \,d^{2} e +3 B \,d^{3}\right ) x +A \,d^{3}\right ) b^{3} c^{\frac {5}{2}}}{3}+2 \left (2 \left (\left (A \,e^{2}+B d e \right ) x^{2}+\left (-6 A d e -2 B \,d^{2}\right ) x +A \,d^{2}\right ) d \,b^{2} c^{\frac {7}{2}}+8 \left (\left (-A e -\frac {B d}{3}\right ) x +d A \right ) x \,d^{2} b \,c^{\frac {9}{2}}+\frac {16 A \,c^{\frac {11}{2}} d^{3} x^{2}}{3}+B \,e^{3} \left (-\frac {4 c^{\frac {3}{2}} x^{2}}{3}-b x \sqrt {c}+\operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right ) \sqrt {x \left (c x +b \right )}\, \left (c x +b \right )\right ) b^{4}\right ) x}{c^{\frac {5}{2}} x \left (c x +b \right ) \sqrt {x \left (c x +b \right )}\, b^{4}}\) | \(235\) |
| risch | \(-\frac {2 d^{2} \left (c x +b \right ) \left (9 A b e x -8 A c d x +3 B b d x +A b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {\frac {B \,b^{3} e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}+\frac {2 \left (A \,b^{3} c \,e^{3}-3 A b \,c^{3} d^{2} e +2 A \,c^{4} d^{3}-2 b^{4} B \,e^{3}+3 b^{3} B c d \,e^{2}-B b \,c^{3} d^{3}\right ) \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{c^{3} b \left (x +\frac {b}{c}\right )}-\frac {b \left (A \,b^{3} c \,e^{3}-3 A \,b^{2} c^{2} d \,e^{2}+3 A b \,c^{3} d^{2} e -A \,c^{4} d^{3}-b^{4} B \,e^{3}+3 b^{3} B c d \,e^{2}-3 b^{2} B \,c^{2} d^{2} e +B b \,c^{3} d^{3}\right ) \left (\frac {2 \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{3 b \left (x +\frac {b}{c}\right )^{2}}+\frac {4 c \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{3 b^{2} \left (x +\frac {b}{c}\right )}\right )}{c^{4}}}{b^{3}}\) | \(360\) |
| default | \(A \,d^{3} \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^{3} \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 (A \,e^{3}+3 B d \,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 (3 A \,d^{2} e +B \,d^{3}\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 (3 A d \,e^{2}+3 B \,d^{2} 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 )\) | \(625\) |
2*(-1/3*((-A*e^3-3*B*d*e^2)*x^3+(-9*A*d*e^2-9*B*d^2*e)*x^2+(9*A*d^2*e+3*B* d^3)*x+A*d^3)*b^3*c^(5/2)+(2*((A*e^2+B*d*e)*x^2+(-6*A*d*e-2*B*d^2)*x+A*d^2 )*d*b^2*c^(7/2)+8*((-A*e-1/3*B*d)*x+d*A)*x*d^2*b*c^(9/2)+16/3*A*c^(11/2)*d ^3*x^2+B*e^3*(-4/3*c^(3/2)*x^2-b*x*c^(1/2)+arctanh((x*(c*x+b))^(1/2)/x/c^( 1/2))*(x*(c*x+b))^(1/2)*(c*x+b))*b^4)*x)/(x*(c*x+b))^(1/2)/c^(5/2)/x/(c*x+ b)/b^4
Time = 0.30 (sec) , antiderivative size = 671, normalized size of antiderivative = 3.05 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (B b^{4} c^{2} e^{3} x^{4} + 2 \, B b^{5} c e^{3} x^{3} + B b^{6} e^{3} x^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (A b^{3} c^{3} d^{3} + {\left (8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} - 6 \, {\left (B b^{2} c^{4} - 4 \, A b c^{5}\right )} d^{2} e - 3 \, {\left (B b^{3} c^{3} + 2 \, A b^{2} c^{4}\right )} d e^{2} + {\left (4 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 4 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \, {\left (B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{2} + 3 \, {\left (3 \, A b^{3} c^{3} d^{2} e + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (B b^{4} c^{2} e^{3} x^{4} + 2 \, B b^{5} c e^{3} x^{3} + B b^{6} e^{3} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (A b^{3} c^{3} d^{3} + {\left (8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} - 6 \, {\left (B b^{2} c^{4} - 4 \, A b c^{5}\right )} d^{2} e - 3 \, {\left (B b^{3} c^{3} + 2 \, A b^{2} c^{4}\right )} d e^{2} + {\left (4 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 4 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \, {\left (B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{2} + 3 \, {\left (3 \, A b^{3} c^{3} d^{2} e + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3}\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{3 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}}\right ] \]
[1/3*(3*(B*b^4*c^2*e^3*x^4 + 2*B*b^5*c*e^3*x^3 + B*b^6*e^3*x^2)*sqrt(c)*lo g(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(A*b^3*c^3*d^3 + (8*(B*b*c^ 5 - 2*A*c^6)*d^3 - 6*(B*b^2*c^4 - 4*A*b*c^5)*d^2*e - 3*(B*b^3*c^3 + 2*A*b^ 2*c^4)*d*e^2 + (4*B*b^4*c^2 - A*b^3*c^3)*e^3)*x^3 - 3*(3*A*b^3*c^3*d*e^2 - B*b^5*c*e^3 - 4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 3*(B*b^3*c^3 - 4*A*b^2*c^4) *d^2*e)*x^2 + 3*(3*A*b^3*c^3*d^2*e + (B*b^3*c^3 - 2*A*b^2*c^4)*d^3)*x)*sqr t(c*x^2 + b*x))/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2), -2/3*(3*(B*b^ 4*c^2*e^3*x^4 + 2*B*b^5*c*e^3*x^3 + B*b^6*e^3*x^2)*sqrt(-c)*arctan(sqrt(c* x^2 + b*x)*sqrt(-c)/(c*x)) + (A*b^3*c^3*d^3 + (8*(B*b*c^5 - 2*A*c^6)*d^3 - 6*(B*b^2*c^4 - 4*A*b*c^5)*d^2*e - 3*(B*b^3*c^3 + 2*A*b^2*c^4)*d*e^2 + (4* B*b^4*c^2 - A*b^3*c^3)*e^3)*x^3 - 3*(3*A*b^3*c^3*d*e^2 - B*b^5*c*e^3 - 4*( B*b^2*c^4 - 2*A*b*c^5)*d^3 + 3*(B*b^3*c^3 - 4*A*b^2*c^4)*d^2*e)*x^2 + 3*(3 *A*b^3*c^3*d^2*e + (B*b^3*c^3 - 2*A*b^2*c^4)*d^3)*x)*sqrt(c*x^2 + b*x))/(b ^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2)]
\[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{3}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (206) = 412\).
Time = 0.21 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.50 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, B e^{3} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} - \frac {4 \, A c d^{3} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d^{3} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {4 \, B e^{3} x}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {B e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {2 \, A d^{3}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d^{3}}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} B e^{3}}{3 \, b c^{2}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, {\left (B d^{2} e + A d e^{2}\right )} x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (B d^{3} + 3 \, A d^{2} e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, {\left (B d^{2} e + A d e^{2}\right )} x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {2 \, {\left (3 \, B d e^{2} + A e^{3}\right )} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {16 \, {\left (B d^{3} + 3 \, A d^{2} e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, {\left (B d^{3} + 3 \, A d^{2} e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {3 \, B d e^{2} + A e^{3}}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {2 \, {\left (B d^{2} e + A d e^{2}\right )}}{\sqrt {c x^{2} + b x} b c} \]
-1/3*B*e^3*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)) - 4/3*A*c*d^3 *x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*A*c^2*d^3*x/(sqrt(c*x^2 + b*x)*b^4) - 4/3*B*e^3*x/(sqrt(c*x^2 + b*x)*c^2) + B*e^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 2/3*A*d^3/((c*x^2 + b*x)^(3/2)*b) + 16/3*A*c*d^3/ (sqrt(c*x^2 + b*x)*b^3) - 2/3*sqrt(c*x^2 + b*x)*B*e^3/(b*c^2) - (3*B*d*e^2 + A*e^3)*x^2/((c*x^2 + b*x)^(3/2)*c) + 4*(B*d^2*e + A*d*e^2)*x/(sqrt(c*x^ 2 + b*x)*b^2) + 2/3*(B*d^3 + 3*A*d^2*e)*x/((c*x^2 + b*x)^(3/2)*b) - 1/3*(3 *B*d*e^2 + A*e^3)*b*x/((c*x^2 + b*x)^(3/2)*c^2) - 2*(B*d^2*e + A*d*e^2)*x/ ((c*x^2 + b*x)^(3/2)*c) + 2/3*(3*B*d*e^2 + A*e^3)*x/(sqrt(c*x^2 + b*x)*b*c ) - 16/3*(B*d^3 + 3*A*d^2*e)*c*x/(sqrt(c*x^2 + b*x)*b^3) - 8/3*(B*d^3 + 3* A*d^2*e)/(sqrt(c*x^2 + b*x)*b^2) + 1/3*(3*B*d*e^2 + A*e^3)/(sqrt(c*x^2 + b *x)*c^2) + 2*(B*d^2*e + A*d*e^2)/(sqrt(c*x^2 + b*x)*b*c)
Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {B e^{3} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {5}{2}}} - \frac {2 \, {\left (\frac {A d^{3}}{b} + {\left (x {\left (\frac {{\left (8 \, B b c^{4} d^{3} - 16 \, A c^{5} d^{3} - 6 \, B b^{2} c^{3} d^{2} e + 24 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} - 6 \, A b^{2} c^{3} d e^{2} + 4 \, B b^{4} c e^{3} - A b^{3} c^{2} e^{3}\right )} x}{b^{4} c^{2}} + \frac {3 \, {\left (4 \, B b^{2} c^{3} d^{3} - 8 \, A b c^{4} d^{3} - 3 \, B b^{3} c^{2} d^{2} e + 12 \, A b^{2} c^{3} d^{2} e - 3 \, A b^{3} c^{2} d e^{2} + B b^{5} e^{3}\right )}}{b^{4} c^{2}}\right )} + \frac {3 \, {\left (B b^{3} c^{2} d^{3} - 2 \, A b^{2} c^{3} d^{3} + 3 \, A b^{3} c^{2} d^{2} e\right )}}{b^{4} c^{2}}\right )} x\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \]
-B*e^3*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(5/2) - 2 /3*(A*d^3/b + (x*((8*B*b*c^4*d^3 - 16*A*c^5*d^3 - 6*B*b^2*c^3*d^2*e + 24*A *b*c^4*d^2*e - 3*B*b^3*c^2*d*e^2 - 6*A*b^2*c^3*d*e^2 + 4*B*b^4*c*e^3 - A*b ^3*c^2*e^3)*x/(b^4*c^2) + 3*(4*B*b^2*c^3*d^3 - 8*A*b*c^4*d^3 - 3*B*b^3*c^2 *d^2*e + 12*A*b^2*c^3*d^2*e - 3*A*b^3*c^2*d*e^2 + B*b^5*e^3)/(b^4*c^2)) + 3*(B*b^3*c^2*d^3 - 2*A*b^2*c^3*d^3 + 3*A*b^3*c^2*d^2*e)/(b^4*c^2))*x)/(c*x ^2 + b*x)^(3/2)
Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]