Integrand size = 26, antiderivative size = 92 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (A b-(b B-2 A c) x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {8 (b B d-2 A c d+A b e) (b d+(2 c d-b e) x)}{3 b^4 \sqrt {b x+c x^2}} \]
-2/3*(A*b-(-2*A*c+B*b)*x)*(e*x+d)^2/b^2/(c*x^2+b*x)^(3/2)-8/3*(A*b*e-2*A*c *d+B*b*d)*(b*d+(-b*e+2*c*d)*x)/b^4/(c*x^2+b*x)^(1/2)
Time = 0.58 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.62 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (A \left (16 c^3 d^2 x^3+8 b c^2 d x^2 (3 d-2 e x)-b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+2 b^2 c x \left (3 d^2-12 d e x+e^2 x^2\right )\right )+b B x \left (-8 c^2 d^2 x^2+4 b c d x (-3 d+e x)+b^2 \left (-3 d^2+6 d e x+e^2 x^2\right )\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \]
(2*(A*(16*c^3*d^2*x^3 + 8*b*c^2*d*x^2*(3*d - 2*e*x) - b^3*(d^2 + 6*d*e*x - 3*e^2*x^2) + 2*b^2*c*x*(3*d^2 - 12*d*e*x + e^2*x^2)) + b*B*x*(-8*c^2*d^2* x^2 + 4*b*c*d*x*(-3*d + e*x) + b^2*(-3*d^2 + 6*d*e*x + e^2*x^2))))/(3*b^4* (x*(b + c*x))^(3/2))
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1227, 1158}
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.
\(\displaystyle \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1227 |
\(\displaystyle \frac {4 (A b e-2 A c d+b B d) \int \frac {d+e x}{\left (c x^2+b x\right )^{3/2}}dx}{3 b^2}-\frac {2 (d+e x)^2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle -\frac {8 (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (d+e x)^2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}\) |
(-2*(A*b - (b*B - 2*A*c)*x)*(d + e*x)^2)/(3*b^2*(b*x + c*x^2)^(3/2)) - (8* (b*B*d - 2*A*c*d + A*b*e)*(b*d + (2*c*d - b*e)*x))/(3*b^4*Sqrt[b*x + c*x^2 ])
3.13.8.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( (b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[m*((b*( e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
Time = 0.39 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.39
method | result | size |
risch | \(-\frac {2 d \left (c x +b \right ) \left (6 A b e x -8 A c d x +3 B b d x +A b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {2 x \left (2 A b c e x -8 A \,c^{2} d x +B \,b^{2} e x +5 B b c d x +3 A \,b^{2} e -9 A b c d +6 B \,b^{2} d \right ) \left (b e -c d \right )}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{4}}\) | \(128\) |
pseudoelliptic | \(-\frac {2 \left (\left (-B \,e^{2} x^{3}+\left (-3 A \,e^{2}-6 B d e \right ) x^{2}+\left (6 A d e +3 B \,d^{2}\right ) x +A \,d^{2}\right ) b^{3}-6 c \left (\frac {e \left (A e +2 B d \right ) x^{2}}{3}+\left (-4 A d e -2 B \,d^{2}\right ) x +A \,d^{2}\right ) x \,b^{2}-24 c^{2} x^{2} \left (\left (-\frac {2 A e}{3}-\frac {B d}{3}\right ) x +d A \right ) d b -16 A \,c^{3} d^{2} x^{3}\right )}{3 \sqrt {x \left (c x +b \right )}\, x \left (c x +b \right ) b^{4}}\) | \(153\) |
gosper | \(-\frac {2 x \left (c x +b \right ) \left (-2 A \,b^{2} c \,e^{2} x^{3}+16 A b \,c^{2} d e \,x^{3}-16 A \,c^{3} d^{2} x^{3}-B \,b^{3} e^{2} x^{3}-4 B \,b^{2} c d e \,x^{3}+8 B b \,c^{2} d^{2} x^{3}-3 A \,b^{3} e^{2} x^{2}+24 A \,b^{2} c d e \,x^{2}-24 A b \,c^{2} d^{2} x^{2}-6 B \,b^{3} d e \,x^{2}+12 B \,b^{2} c \,d^{2} x^{2}+6 A \,b^{3} d e x -6 A \,b^{2} c \,d^{2} x +3 B \,b^{3} d^{2} x +A \,d^{2} b^{3}\right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(197\) |
trager | \(-\frac {2 \left (-2 A \,b^{2} c \,e^{2} x^{3}+16 A b \,c^{2} d e \,x^{3}-16 A \,c^{3} d^{2} x^{3}-B \,b^{3} e^{2} x^{3}-4 B \,b^{2} c d e \,x^{3}+8 B b \,c^{2} d^{2} x^{3}-3 A \,b^{3} e^{2} x^{2}+24 A \,b^{2} c d e \,x^{2}-24 A b \,c^{2} d^{2} x^{2}-6 B \,b^{3} d e \,x^{2}+12 B \,b^{2} c \,d^{2} x^{2}+6 A \,b^{3} d e x -6 A \,b^{2} c \,d^{2} x +3 B \,b^{3} d^{2} x +A \,d^{2} b^{3}\right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} x^{2} \left (c x +b \right )^{2}}\) | \(201\) |
default | \(A \,d^{2} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )+B \,e^{2} \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )+\left (A \,e^{2}+2 B d e \right ) \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+\left (2 A d e +B \,d^{2}\right ) \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )\) | \(363\) |
-2/3*d*(c*x+b)*(6*A*b*e*x-8*A*c*d*x+3*B*b*d*x+A*b*d)/b^4/x/(x*(c*x+b))^(1/ 2)+2/3*x*(2*A*b*c*e*x-8*A*c^2*d*x+B*b^2*e*x+5*B*b*c*d*x+3*A*b^2*e-9*A*b*c* d+6*B*b^2*d)*(b*e-c*d)/(x*(c*x+b))^(1/2)/(c*x+b)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (84) = 168\).
Time = 0.30 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.05 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (A b^{3} d^{2} + {\left (8 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \, {\left (B b^{2} c - 4 \, A b c^{2}\right )} d e - {\left (B b^{3} + 2 \, A b^{2} c\right )} e^{2}\right )} x^{3} - 3 \, {\left (A b^{3} e^{2} - 4 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{2} + 2 \, {\left (B b^{3} - 4 \, A b^{2} c\right )} d e\right )} x^{2} + 3 \, {\left (2 \, A b^{3} d e + {\left (B b^{3} - 2 \, A b^{2} c\right )} d^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \]
-2/3*(A*b^3*d^2 + (8*(B*b*c^2 - 2*A*c^3)*d^2 - 4*(B*b^2*c - 4*A*b*c^2)*d*e - (B*b^3 + 2*A*b^2*c)*e^2)*x^3 - 3*(A*b^3*e^2 - 4*(B*b^2*c - 2*A*b*c^2)*d ^2 + 2*(B*b^3 - 4*A*b^2*c)*d*e)*x^2 + 3*(2*A*b^3*d*e + (B*b^3 - 2*A*b^2*c) *d^2)*x)*sqrt(c*x^2 + b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)
\[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{2}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (84) = 168\).
Time = 0.19 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.77 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {B e^{2} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {4 \, A c d^{2} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {B b e^{2} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} + \frac {2 \, B e^{2} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {2 \, A d^{2}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d^{2}}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {B e^{2}}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {4 \, {\left (2 \, B d e + A e^{2}\right )} x}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (B d^{2} + 2 \, A d e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {2 \, {\left (2 \, B d e + A e^{2}\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {16 \, {\left (B d^{2} + 2 \, A d e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, {\left (B d^{2} + 2 \, A d e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (2 \, B d e + A e^{2}\right )}}{3 \, \sqrt {c x^{2} + b x} b c} \]
-B*e^2*x^2/((c*x^2 + b*x)^(3/2)*c) - 4/3*A*c*d^2*x/((c*x^2 + b*x)^(3/2)*b^ 2) + 32/3*A*c^2*d^2*x/(sqrt(c*x^2 + b*x)*b^4) - 1/3*B*b*e^2*x/((c*x^2 + b* x)^(3/2)*c^2) + 2/3*B*e^2*x/(sqrt(c*x^2 + b*x)*b*c) - 2/3*A*d^2/((c*x^2 + b*x)^(3/2)*b) + 16/3*A*c*d^2/(sqrt(c*x^2 + b*x)*b^3) + 1/3*B*e^2/(sqrt(c*x ^2 + b*x)*c^2) + 4/3*(2*B*d*e + A*e^2)*x/(sqrt(c*x^2 + b*x)*b^2) + 2/3*(B* d^2 + 2*A*d*e)*x/((c*x^2 + b*x)^(3/2)*b) - 2/3*(2*B*d*e + A*e^2)*x/((c*x^2 + b*x)^(3/2)*c) - 16/3*(B*d^2 + 2*A*d*e)*c*x/(sqrt(c*x^2 + b*x)*b^3) - 8/ 3*(B*d^2 + 2*A*d*e)/(sqrt(c*x^2 + b*x)*b^2) + 2/3*(2*B*d*e + A*e^2)/(sqrt( c*x^2 + b*x)*b*c)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (84) = 168\).
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.89 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (\frac {A d^{2}}{b} + {\left (x {\left (\frac {{\left (8 \, B b c^{2} d^{2} - 16 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 16 \, A b c^{2} d e - B b^{3} e^{2} - 2 \, A b^{2} c e^{2}\right )} x}{b^{4}} + \frac {3 \, {\left (4 \, B b^{2} c d^{2} - 8 \, A b c^{2} d^{2} - 2 \, B b^{3} d e + 8 \, A b^{2} c d e - A b^{3} e^{2}\right )}}{b^{4}}\right )} + \frac {3 \, {\left (B b^{3} d^{2} - 2 \, A b^{2} c d^{2} + 2 \, A b^{3} d e\right )}}{b^{4}}\right )} x\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \]
-2/3*(A*d^2/b + (x*((8*B*b*c^2*d^2 - 16*A*c^3*d^2 - 4*B*b^2*c*d*e + 16*A*b *c^2*d*e - B*b^3*e^2 - 2*A*b^2*c*e^2)*x/b^4 + 3*(4*B*b^2*c*d^2 - 8*A*b*c^2 *d^2 - 2*B*b^3*d*e + 8*A*b^2*c*d*e - A*b^3*e^2)/b^4) + 3*(B*b^3*d^2 - 2*A* b^2*c*d^2 + 2*A*b^3*d*e)/b^4)*x)/(c*x^2 + b*x)^(3/2)
Time = 10.75 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2\,\left (-3\,B\,b^3\,d^2\,x-A\,b^3\,d^2+6\,B\,b^3\,d\,e\,x^2-6\,A\,b^3\,d\,e\,x+B\,b^3\,e^2\,x^3+3\,A\,b^3\,e^2\,x^2-12\,B\,b^2\,c\,d^2\,x^2+6\,A\,b^2\,c\,d^2\,x+4\,B\,b^2\,c\,d\,e\,x^3-24\,A\,b^2\,c\,d\,e\,x^2+2\,A\,b^2\,c\,e^2\,x^3-8\,B\,b\,c^2\,d^2\,x^3+24\,A\,b\,c^2\,d^2\,x^2-16\,A\,b\,c^2\,d\,e\,x^3+16\,A\,c^3\,d^2\,x^3\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \]
(2*(3*A*b^3*e^2*x^2 - 3*B*b^3*d^2*x - A*b^3*d^2 + 16*A*c^3*d^2*x^3 + B*b^3 *e^2*x^3 + 6*A*b^2*c*d^2*x + 6*B*b^3*d*e*x^2 + 24*A*b*c^2*d^2*x^2 - 12*B*b ^2*c*d^2*x^2 + 2*A*b^2*c*e^2*x^3 - 8*B*b*c^2*d^2*x^3 - 6*A*b^3*d*e*x - 24* A*b^2*c*d*e*x^2 - 16*A*b*c^2*d*e*x^3 + 4*B*b^2*c*d*e*x^3))/(3*b^4*(b*x + c *x^2)^(3/2))