Integrand size = 31, antiderivative size = 124 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac {2 (b d-a e) (3 b B d-2 A b e-a B e)}{e^4 \sqrt {d+e x}}-\frac {2 b (3 b B d-A b e-2 a B e) \sqrt {d+e x}}{e^4}+\frac {2 b^2 B (d+e x)^{3/2}}{3 e^4} \]
2/3*(-a*e+b*d)^2*(-A*e+B*d)/e^4/(e*x+d)^(3/2)+2/3*b^2*B*(e*x+d)^(3/2)/e^4- 2*(-a*e+b*d)*(-2*A*b*e-B*a*e+3*B*b*d)/e^4/(e*x+d)^(1/2)-2*b*(-A*b*e-2*B*a* e+3*B*b*d)*(e*x+d)^(1/2)/e^4
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (a^2 e^2 (2 B d+A e+3 B e x)-2 a b e \left (-A e (2 d+3 e x)+B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+b^2 \left (-A e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )\right )}{3 e^4 (d+e x)^{3/2}} \]
(-2*(a^2*e^2*(2*B*d + A*e + 3*B*e*x) - 2*a*b*e*(-(A*e*(2*d + 3*e*x)) + B*( 8*d^2 + 12*d*e*x + 3*e^2*x^2)) + b^2*(-(A*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) ) + B*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3))))/(3*e^4*(d + e*x)^(3 /2))
Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 86, 2009}
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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right ) (A+B x)}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^2 (a+b x)^2 (A+B x)}{(d+e x)^{5/2}}dx}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b (2 a B e+A b e-3 b B d)}{e^3 \sqrt {d+e x}}+\frac {(a e-b d) (a B e+2 A b e-3 b B d)}{e^3 (d+e x)^{3/2}}+\frac {(a e-b d)^2 (A e-B d)}{e^3 (d+e x)^{5/2}}+\frac {b^2 B \sqrt {d+e x}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b \sqrt {d+e x} (-2 a B e-A b e+3 b B d)}{e^4}-\frac {2 (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 \sqrt {d+e x}}+\frac {2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 b^2 B (d+e x)^{3/2}}{3 e^4}\) |
(2*(b*d - a*e)^2*(B*d - A*e))/(3*e^4*(d + e*x)^(3/2)) - (2*(b*d - a*e)*(3* b*B*d - 2*A*b*e - a*B*e))/(e^4*Sqrt[d + e*x]) - (2*b*(3*b*B*d - A*b*e - 2* a*B*e)*Sqrt[d + e*x])/e^4 + (2*b^2*B*(d + e*x)^(3/2))/(3*e^4)
3.18.90.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(\frac {2 b \left (B b e x +3 A b e +6 B a e -8 B b d \right ) \sqrt {e x +d}}{3 e^{4}}-\frac {2 \left (6 A b \,e^{2} x +3 B a \,e^{2} x -9 B b d e x +A a \,e^{2}+5 A b d e +2 B a d e -8 B b \,d^{2}\right ) \left (a e -b d \right )}{3 e^{4} \left (e x +d \right )^{\frac {3}{2}}}\) | \(105\) |
| pseudoelliptic | \(-\frac {2 \left (\left (\left (-B \,x^{3}-3 A \,x^{2}\right ) b^{2}+6 a x \left (-B x +A \right ) b +a^{2} \left (3 B x +A \right )\right ) e^{3}+4 \left (\left (\frac {3}{2} B \,x^{2}-3 A x \right ) b^{2}+a \left (-6 B x +A \right ) b +\frac {B \,a^{2}}{2}\right ) d \,e^{2}-8 b \left (\left (-3 B x +A \right ) b +2 B a \right ) d^{2} e +16 B \,b^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(123\) |
| gosper | \(-\frac {2 \left (-B \,b^{2} x^{3} e^{3}-3 A \,b^{2} e^{3} x^{2}-6 B \,x^{2} a b \,e^{3}+6 B \,x^{2} b^{2} d \,e^{2}+6 A x a b \,e^{3}-12 A x \,b^{2} d \,e^{2}+3 B x \,a^{2} e^{3}-24 B x a b d \,e^{2}+24 B x \,b^{2} d^{2} e +A \,a^{2} e^{3}+4 A a b d \,e^{2}-8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}-16 B a b \,d^{2} e +16 B \,b^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(168\) |
| trager | \(-\frac {2 \left (-B \,b^{2} x^{3} e^{3}-3 A \,b^{2} e^{3} x^{2}-6 B \,x^{2} a b \,e^{3}+6 B \,x^{2} b^{2} d \,e^{2}+6 A x a b \,e^{3}-12 A x \,b^{2} d \,e^{2}+3 B x \,a^{2} e^{3}-24 B x a b d \,e^{2}+24 B x \,b^{2} d^{2} e +A \,a^{2} e^{3}+4 A a b d \,e^{2}-8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}-16 B a b \,d^{2} e +16 B \,b^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(168\) |
| derivativedivides | \(\frac {\frac {2 B \,b^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} e \sqrt {e x +d}+4 B a b e \sqrt {e x +d}-6 B \,b^{2} d \sqrt {e x +d}-\frac {2 \left (A \,a^{2} e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -B \,b^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +a^{2} B \,e^{2}-4 B a b d e +3 B \,b^{2} d^{2}\right )}{\sqrt {e x +d}}}{e^{4}}\) | \(174\) |
| default | \(\frac {\frac {2 B \,b^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} e \sqrt {e x +d}+4 B a b e \sqrt {e x +d}-6 B \,b^{2} d \sqrt {e x +d}-\frac {2 \left (A \,a^{2} e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -B \,b^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +a^{2} B \,e^{2}-4 B a b d e +3 B \,b^{2} d^{2}\right )}{\sqrt {e x +d}}}{e^{4}}\) | \(174\) |
2/3*b*(B*b*e*x+3*A*b*e+6*B*a*e-8*B*b*d)*(e*x+d)^(1/2)/e^4-2/3*(6*A*b*e^2*x +3*B*a*e^2*x-9*B*b*d*e*x+A*a*e^2+5*A*b*d*e+2*B*a*d*e-8*B*b*d^2)*(a*e-b*d)/ e^4/(e*x+d)^(3/2)
Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (B b^{2} e^{3} x^{3} - 16 \, B b^{2} d^{3} - A a^{2} e^{3} + 8 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e - 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \, {\left (2 \, B b^{2} d e^{2} - {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} - 3 \, {\left (8 \, B b^{2} d^{2} e - 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
2/3*(B*b^2*e^3*x^3 - 16*B*b^2*d^3 - A*a^2*e^3 + 8*(2*B*a*b + A*b^2)*d^2*e - 2*(B*a^2 + 2*A*a*b)*d*e^2 - 3*(2*B*b^2*d*e^2 - (2*B*a*b + A*b^2)*e^3)*x^ 2 - 3*(8*B*b^2*d^2*e - 4*(2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a*b)*e^3)* x)*sqrt(e*x + d)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (124) = 248\).
Time = 0.38 (sec) , antiderivative size = 709, normalized size of antiderivative = 5.72 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{5/2}} \, dx=\begin {cases} - \frac {2 A a^{2} e^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {8 A a b d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 A a b e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {16 A b^{2} d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {24 A b^{2} d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {6 A b^{2} e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {4 B a^{2} d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 B a^{2} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {32 B a b d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 B a b d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {12 B a b e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 B b^{2} d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 B b^{2} d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 B b^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 B b^{2} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-2*A*a**2*e**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x) ) - 8*A*a*b*d*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 12* A*a*b*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 16*A*b**2 *d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 24*A*b**2*d*e* *2*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 6*A*b**2*e**3*x** 2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 4*B*a**2*d*e**2/(3*d *e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 6*B*a**2*e**3*x/(3*d*e**4* sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 32*B*a*b*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 48*B*a*b*d*e**2*x/(3*d*e**4*sqrt(d + e *x) + 3*e**5*x*sqrt(d + e*x)) + 12*B*a*b*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 32*B*b**2*d**3/(3*d*e**4*sqrt(d + e*x) + 3*e* *5*x*sqrt(d + e*x)) - 48*B*b**2*d**2*e*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5* x*sqrt(d + e*x)) - 12*B*b**2*d*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5* x*sqrt(d + e*x)) + 2*B*b**2*e**3*x**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*s qrt(d + e*x)), Ne(e, 0)), ((A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2 *x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4)/d**(5/2), True))
Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} B b^{2} - 3 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} \sqrt {e x + d}}{e^{3}} + \frac {B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \]
2/3*(((e*x + d)^(3/2)*B*b^2 - 3*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*sqrt(e*x + d))/e^3 + (B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b + A*b^2)*d^2*e + (B*a^2 + 2 *A*a*b)*d*e^2 - 3*(3*B*b^2*d^2 - 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a* b)*e^2)*(e*x + d))/((e*x + d)^(3/2)*e^3))/e
Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.64 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (9 \, {\left (e x + d\right )} B b^{2} d^{2} - B b^{2} d^{3} - 12 \, {\left (e x + d\right )} B a b d e - 6 \, {\left (e x + d\right )} A b^{2} d e + 2 \, B a b d^{2} e + A b^{2} d^{2} e + 3 \, {\left (e x + d\right )} B a^{2} e^{2} + 6 \, {\left (e x + d\right )} A a b e^{2} - B a^{2} d e^{2} - 2 \, A a b d e^{2} + A a^{2} e^{3}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{4}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{2} e^{8} - 9 \, \sqrt {e x + d} B b^{2} d e^{8} + 6 \, \sqrt {e x + d} B a b e^{9} + 3 \, \sqrt {e x + d} A b^{2} e^{9}\right )}}{3 \, e^{12}} \]
-2/3*(9*(e*x + d)*B*b^2*d^2 - B*b^2*d^3 - 12*(e*x + d)*B*a*b*d*e - 6*(e*x + d)*A*b^2*d*e + 2*B*a*b*d^2*e + A*b^2*d^2*e + 3*(e*x + d)*B*a^2*e^2 + 6*( e*x + d)*A*a*b*e^2 - B*a^2*d*e^2 - 2*A*a*b*d*e^2 + A*a^2*e^3)/((e*x + d)^( 3/2)*e^4) + 2/3*((e*x + d)^(3/2)*B*b^2*e^8 - 9*sqrt(e*x + d)*B*b^2*d*e^8 + 6*sqrt(e*x + d)*B*a*b*e^9 + 3*sqrt(e*x + d)*A*b^2*e^9)/e^12
Time = 10.85 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.52 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{5/2}} \, dx=\frac {2\,B\,b^2\,d^3-2\,A\,a^2\,e^3+2\,B\,b^2\,{\left (d+e\,x\right )}^3+6\,A\,b^2\,e\,{\left (d+e\,x\right )}^2-6\,B\,a^2\,e^2\,\left (d+e\,x\right )-18\,B\,b^2\,d\,{\left (d+e\,x\right )}^2-18\,B\,b^2\,d^2\,\left (d+e\,x\right )-2\,A\,b^2\,d^2\,e+2\,B\,a^2\,d\,e^2-12\,A\,a\,b\,e^2\,\left (d+e\,x\right )+12\,B\,a\,b\,e\,{\left (d+e\,x\right )}^2+12\,A\,b^2\,d\,e\,\left (d+e\,x\right )+4\,A\,a\,b\,d\,e^2-4\,B\,a\,b\,d^2\,e+24\,B\,a\,b\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \]
(2*B*b^2*d^3 - 2*A*a^2*e^3 + 2*B*b^2*(d + e*x)^3 + 6*A*b^2*e*(d + e*x)^2 - 6*B*a^2*e^2*(d + e*x) - 18*B*b^2*d*(d + e*x)^2 - 18*B*b^2*d^2*(d + e*x) - 2*A*b^2*d^2*e + 2*B*a^2*d*e^2 - 12*A*a*b*e^2*(d + e*x) + 12*B*a*b*e*(d + e*x)^2 + 12*A*b^2*d*e*(d + e*x) + 4*A*a*b*d*e^2 - 4*B*a*b*d^2*e + 24*B*a*b *d*e*(d + e*x))/(3*e^4*(d + e*x)^(3/2))