Integrand size = 28, antiderivative size = 114 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {(b d-a e)^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^3}+\frac {2 e (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^3}+\frac {e^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^3} \]
1/4*(-a*e+b*d)^2*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/b^3+2/5*e*(-a*e+b*d)* (b^2*x^2+2*a*b*x+a^2)^(5/2)/b^3+1/6*e^2*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2 )/b^3
Time = 1.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.11 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (20 a^3 \left (3 d^2+3 d e x+e^2 x^2\right )+15 a^2 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+6 a b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )}{60 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(20*a^3*(3*d^2 + 3*d*e*x + e^2*x^2) + 15*a^2*b*x*(6*d ^2 + 8*d*e*x + 3*e^2*x^2) + 6*a*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2)))/(60*(a + b*x))
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1102, 27, 49, 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.
\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 1102 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^3 (d+e x)^2dx}{b^3 (a+b x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^3 (d+e x)^2dx}{a+b x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {e^2 (a+b x)^5}{b^2}+\frac {2 e (b d-a e) (a+b x)^4}{b^2}+\frac {(b d-a e)^2 (a+b x)^3}{b^2}\right )dx}{a+b x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {2 e (a+b x)^5 (b d-a e)}{5 b^3}+\frac {(a+b x)^4 (b d-a e)^2}{4 b^3}+\frac {e^2 (a+b x)^6}{6 b^3}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^2*(a + b*x)^4)/(4*b^3) + (2*e *(b*d - a*e)*(a + b*x)^5)/(5*b^3) + (e^2*(a + b*x)^6)/(6*b^3)))/(a + b*x)
3.16.57.3.1 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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 2.95 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.30
method | result | size |
gosper | \(\frac {x \left (10 e^{2} b^{3} x^{5}+36 x^{4} a \,b^{2} e^{2}+24 x^{4} b^{3} d e +45 x^{3} e^{2} a^{2} b +90 x^{3} a d e \,b^{2}+15 x^{3} b^{3} d^{2}+20 x^{2} a^{3} e^{2}+120 x^{2} a^{2} b d e +60 x^{2} a \,b^{2} d^{2}+60 x d e \,a^{3}+90 x \,a^{2} b \,d^{2}+60 a^{3} d^{2}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (b x +a \right )^{3}}\) | \(148\) |
default | \(\frac {x \left (10 e^{2} b^{3} x^{5}+36 x^{4} a \,b^{2} e^{2}+24 x^{4} b^{3} d e +45 x^{3} e^{2} a^{2} b +90 x^{3} a d e \,b^{2}+15 x^{3} b^{3} d^{2}+20 x^{2} a^{3} e^{2}+120 x^{2} a^{2} b d e +60 x^{2} a \,b^{2} d^{2}+60 x d e \,a^{3}+90 x \,a^{2} b \,d^{2}+60 a^{3} d^{2}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (b x +a \right )^{3}}\) | \(148\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{2} b^{3} x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a \,b^{2} e^{2}+2 b^{3} d e \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 e^{2} a^{2} b +6 a d e \,b^{2}+b^{3} d^{2}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{3} e^{2}+6 a^{2} b d e +3 a \,b^{2} d^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 d e \,a^{3}+3 a^{2} b \,d^{2}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{3} d^{2} x}{b x +a}\) | \(221\) |
1/60*x*(10*b^3*e^2*x^5+36*a*b^2*e^2*x^4+24*b^3*d*e*x^4+45*a^2*b*e^2*x^3+90 *a*b^2*d*e*x^3+15*b^3*d^2*x^3+20*a^3*e^2*x^2+120*a^2*b*d*e*x^2+60*a*b^2*d^ 2*x^2+60*a^3*d*e*x+90*a^2*b*d^2*x+60*a^3*d^2)*((b*x+a)^2)^(3/2)/(b*x+a)^3
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.09 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{6} \, b^{3} e^{2} x^{6} + a^{3} d^{2} x + \frac {1}{5} \, {\left (2 \, b^{3} d e + 3 \, a b^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{2} + 6 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} d^{2} + 6 \, a^{2} b d e + a^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \]
1/6*b^3*e^2*x^6 + a^3*d^2*x + 1/5*(2*b^3*d*e + 3*a*b^2*e^2)*x^5 + 1/4*(b^3 *d^2 + 6*a*b^2*d*e + 3*a^2*b*e^2)*x^4 + 1/3*(3*a*b^2*d^2 + 6*a^2*b*d*e + a ^3*e^2)*x^3 + 1/2*(3*a^2*b*d^2 + 2*a^3*d*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 1695 vs. \(2 (107) = 214\).
Time = 0.88 (sec) , antiderivative size = 1695, normalized size of antiderivative = 14.87 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**2*e**2*x**5/6 + x**4*(13*a *b**3*e**2/6 + 2*b**4*d*e)/(5*b**2) + x**3*(31*a**2*b**2*e**2/6 + 8*a*b**3 *d*e - 9*a*(13*a*b**3*e**2/6 + 2*b**4*d*e)/(5*b) + b**4*d**2)/(4*b**2) + x **2*(4*a**3*b*e**2 + 12*a**2*b**2*d*e - 4*a**2*(13*a*b**3*e**2/6 + 2*b**4* d*e)/(5*b**2) + 4*a*b**3*d**2 - 7*a*(31*a**2*b**2*e**2/6 + 8*a*b**3*d*e - 9*a*(13*a*b**3*e**2/6 + 2*b**4*d*e)/(5*b) + b**4*d**2)/(4*b))/(3*b**2) + x *(a**4*e**2 + 8*a**3*b*d*e + 6*a**2*b**2*d**2 - 3*a**2*(31*a**2*b**2*e**2/ 6 + 8*a*b**3*d*e - 9*a*(13*a*b**3*e**2/6 + 2*b**4*d*e)/(5*b) + b**4*d**2)/ (4*b**2) - 5*a*(4*a**3*b*e**2 + 12*a**2*b**2*d*e - 4*a**2*(13*a*b**3*e**2/ 6 + 2*b**4*d*e)/(5*b**2) + 4*a*b**3*d**2 - 7*a*(31*a**2*b**2*e**2/6 + 8*a* b**3*d*e - 9*a*(13*a*b**3*e**2/6 + 2*b**4*d*e)/(5*b) + b**4*d**2)/(4*b))/( 3*b))/(2*b**2) + (2*a**4*d*e + 4*a**3*b*d**2 - 2*a**2*(4*a**3*b*e**2 + 12* a**2*b**2*d*e - 4*a**2*(13*a*b**3*e**2/6 + 2*b**4*d*e)/(5*b**2) + 4*a*b**3 *d**2 - 7*a*(31*a**2*b**2*e**2/6 + 8*a*b**3*d*e - 9*a*(13*a*b**3*e**2/6 + 2*b**4*d*e)/(5*b) + b**4*d**2)/(4*b))/(3*b**2) - 3*a*(a**4*e**2 + 8*a**3*b *d*e + 6*a**2*b**2*d**2 - 3*a**2*(31*a**2*b**2*e**2/6 + 8*a*b**3*d*e - 9*a *(13*a*b**3*e**2/6 + 2*b**4*d*e)/(5*b) + b**4*d**2)/(4*b**2) - 5*a*(4*a**3 *b*e**2 + 12*a**2*b**2*d*e - 4*a**2*(13*a*b**3*e**2/6 + 2*b**4*d*e)/(5*b** 2) + 4*a*b**3*d**2 - 7*a*(31*a**2*b**2*e**2/6 + 8*a*b**3*d*e - 9*a*(13*a*b **3*e**2/6 + 2*b**4*d*e)/(5*b) + b**4*d**2)/(4*b))/(3*b))/(2*b))/b**2) ...
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (102) = 204\).
Time = 0.20 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.15 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d e x}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} e^{2} x}{4 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{2}}{4 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d e}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} e^{2}}{4 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e^{2} x}{6 \, b^{2}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d e}{5 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{2}}{30 \, b^{3}} \]
1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*d^2*x - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^ (3/2)*a*d*e*x/b + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*e^2*x/b^2 + 1/4* (b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*d^2/b - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(3 /2)*a^2*d*e/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*e^2/b^3 + 1/6*(b ^2*x^2 + 2*a*b*x + a^2)^(5/2)*e^2*x/b^2 + 2/5*(b^2*x^2 + 2*a*b*x + a^2)^(5 /2)*d*e/b^2 - 7/30*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e^2/b^3
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (102) = 204\).
Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.11 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {1}{6} \, b^{3} e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, b^{3} d e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a b^{2} e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a b^{2} d e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, a^{2} b e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + a b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b d e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{3} e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} d e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} d^{2} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (15 \, a^{4} b^{2} d^{2} - 6 \, a^{5} b d e + a^{6} e^{2}\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, b^{3}} \]
1/6*b^3*e^2*x^6*sgn(b*x + a) + 2/5*b^3*d*e*x^5*sgn(b*x + a) + 3/5*a*b^2*e^ 2*x^5*sgn(b*x + a) + 1/4*b^3*d^2*x^4*sgn(b*x + a) + 3/2*a*b^2*d*e*x^4*sgn( b*x + a) + 3/4*a^2*b*e^2*x^4*sgn(b*x + a) + a*b^2*d^2*x^3*sgn(b*x + a) + 2 *a^2*b*d*e*x^3*sgn(b*x + a) + 1/3*a^3*e^2*x^3*sgn(b*x + a) + 3/2*a^2*b*d^2 *x^2*sgn(b*x + a) + a^3*d*e*x^2*sgn(b*x + a) + a^3*d^2*x*sgn(b*x + a) + 1/ 60*(15*a^4*b^2*d^2 - 6*a^5*b*d*e + a^6*e^2)*sgn(b*x + a)/b^3
Timed out. \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]