Integrand size = 24, antiderivative size = 111 \[ \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) (b+2 c x)}{3 b^4 c \sqrt {b x+c x^2}} \]
-2/3*(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) +2/3*(8*A*c^2*d+b^2*B*e-4*b*c*(A*e+B*d))*(2*c*x+b)/b^4/c/(c*x^2+b*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b B x \left (8 c^2 d x^2+3 b^2 (d-e x)-2 b c x (-6 d+e x)\right )+A \left (-16 c^3 d x^3-6 b^2 c x (d-2 e x)+8 b c^2 x^2 (-3 d+e x)+b^3 (d+3 e x)\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \]
(-2*(b*B*x*(8*c^2*d*x^2 + 3*b^2*(d - e*x) - 2*b*c*x*(-6*d + e*x)) + A*(-16 *c^3*d*x^3 - 6*b^2*c*x*(d - 2*e*x) + 8*b*c^2*x^2*(-3*d + e*x) + b^3*(d + 3 *e*x))))/(3*b^4*(x*(b + c*x))^(3/2))
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1224, 1088}
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.
\(\displaystyle \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1224 |
\(\displaystyle -\frac {\left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right ) \int \frac {1}{\left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1088 |
\(\displaystyle \frac {2 (b+2 c x) \left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
(-2*(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*(8*A*c^2*d + b^2*B*e - 4*b*c*(B*d + A*e))*(b + 2*c*x)) /(3*b^4*c*Sqrt[b*x + c*x^2])
3.13.9.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
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])
Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-3 B e \,x^{2}+3 \left (A e +B d \right ) x +d A \right ) b^{3}-6 c x \left (\frac {B e \,x^{2}}{3}+2 \left (-A e -B d \right ) x +d A \right ) b^{2}-24 c^{2} \left (\frac {\left (-A e -B d \right ) x}{3}+d A \right ) x^{2} b -16 A \,c^{3} d \,x^{3}\right )}{3 \sqrt {x \left (c x +b \right )}\, x \left (c x +b \right ) b^{4}}\) | \(116\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (3 A b e x -8 A c d x +3 B b d x +A b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}-\frac {2 \left (5 A b \,c^{2} e x -8 A \,c^{3} d x -2 B \,b^{2} c e x +5 B b \,c^{2} d x +6 A \,b^{2} c e -9 A d b \,c^{2}-3 b^{3} B e +6 B \,b^{2} c d \right ) x}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{4}}\) | \(136\) |
gosper | \(-\frac {2 x \left (c x +b \right ) \left (8 A b \,c^{2} e \,x^{3}-16 A \,c^{3} d \,x^{3}-2 B \,b^{2} c e \,x^{3}+8 B b \,c^{2} d \,x^{3}+12 A \,b^{2} c e \,x^{2}-24 A b \,c^{2} d \,x^{2}-3 B \,b^{3} e \,x^{2}+12 B \,b^{2} c d \,x^{2}+3 A \,b^{3} e x -6 A \,b^{2} c d x +3 B \,b^{3} d x +A d \,b^{3}\right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(141\) |
trager | \(-\frac {2 \left (8 A b \,c^{2} e \,x^{3}-16 A \,c^{3} d \,x^{3}-2 B \,b^{2} c e \,x^{3}+8 B b \,c^{2} d \,x^{3}+12 A \,b^{2} c e \,x^{2}-24 A b \,c^{2} d \,x^{2}-3 B \,b^{3} e \,x^{2}+12 B \,b^{2} c d \,x^{2}+3 A \,b^{3} e x -6 A \,b^{2} c d x +3 B \,b^{3} d x +A d \,b^{3}\right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} x^{2} \left (c x +b \right )^{2}}\) | \(145\) |
default | \(d A \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )+B e \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+\left (A e +B d \right ) \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )\) | \(224\) |
-2/3*((-3*B*e*x^2+3*(A*e+B*d)*x+d*A)*b^3-6*c*x*(1/3*B*e*x^2+2*(-A*e-B*d)*x +d*A)*b^2-24*c^2*(1/3*(-A*e-B*d)*x+d*A)*x^2*b-16*A*c^3*d*x^3)/(x*(c*x+b))^ (1/2)/x/(c*x+b)/b^4
Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (A b^{3} d + 2 \, {\left (4 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c - 4 \, A b c^{2}\right )} e\right )} x^{3} + 3 \, {\left (4 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d - {\left (B b^{3} - 4 \, A b^{2} c\right )} e\right )} x^{2} + 3 \, {\left (A b^{3} e + {\left (B b^{3} - 2 \, A b^{2} c\right )} d\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \]
-2/3*(A*b^3*d + 2*(4*(B*b*c^2 - 2*A*c^3)*d - (B*b^2*c - 4*A*b*c^2)*e)*x^3 + 3*(4*(B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - 4*A*b^2*c)*e)*x^2 + 3*(A*b^3*e + (B*b^3 - 2*A*b^2*c)*d)*x)*sqrt(c*x^2 + b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)
\[ \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (103) = 206\).
Time = 0.19 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.90 \[ \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {4 \, A c d x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {4 \, B e x}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, B e x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {2 \, A d}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {2 \, B e}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {2 \, {\left (B d + A e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {16 \, {\left (B d + A e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, {\left (B d + A e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} \]
-4/3*A*c*d*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*A*c^2*d*x/(sqrt(c*x^2 + b*x) *b^4) + 4/3*B*e*x/(sqrt(c*x^2 + b*x)*b^2) - 2/3*B*e*x/((c*x^2 + b*x)^(3/2) *c) - 2/3*A*d/((c*x^2 + b*x)^(3/2)*b) + 16/3*A*c*d/(sqrt(c*x^2 + b*x)*b^3) + 2/3*B*e/(sqrt(c*x^2 + b*x)*b*c) + 2/3*(B*d + A*e)*x/((c*x^2 + b*x)^(3/2 )*b) - 16/3*(B*d + A*e)*c*x/(sqrt(c*x^2 + b*x)*b^3) - 8/3*(B*d + A*e)/(sqr t(c*x^2 + b*x)*b^2)
Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left ({\left (x {\left (\frac {2 \, {\left (4 \, B b c^{2} d - 8 \, A c^{3} d - B b^{2} c e + 4 \, A b c^{2} e\right )} x}{b^{4}} + \frac {3 \, {\left (4 \, B b^{2} c d - 8 \, A b c^{2} d - B b^{3} e + 4 \, A b^{2} c e\right )}}{b^{4}}\right )} + \frac {3 \, {\left (B b^{3} d - 2 \, A b^{2} c d + A b^{3} e\right )}}{b^{4}}\right )} x + \frac {A d}{b}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \]
-2/3*((x*(2*(4*B*b*c^2*d - 8*A*c^3*d - B*b^2*c*e + 4*A*b*c^2*e)*x/b^4 + 3* (4*B*b^2*c*d - 8*A*b*c^2*d - B*b^3*e + 4*A*b^2*c*e)/b^4) + 3*(B*b^3*d - 2* A*b^2*c*d + A*b^3*e)/b^4)*x + A*d/b)/(c*x^2 + b*x)^(3/2)
Time = 10.47 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21 \[ \int \frac {(A+B x) (d+e x)}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,\left (A\,b^3\,d+3\,A\,b^3\,e\,x+3\,B\,b^3\,d\,x-16\,A\,c^3\,d\,x^3-3\,B\,b^3\,e\,x^2-24\,A\,b\,c^2\,d\,x^2+12\,A\,b^2\,c\,e\,x^2+12\,B\,b^2\,c\,d\,x^2+8\,A\,b\,c^2\,e\,x^3+8\,B\,b\,c^2\,d\,x^3-2\,B\,b^2\,c\,e\,x^3-6\,A\,b^2\,c\,d\,x\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \]