Integrand size = 29, antiderivative size = 306 \[ \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {a^5 A x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {a^4 (5 A b+a B) x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {5 a^3 b (2 A b+a B) x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac {10 a^2 b^2 (A b+a B) x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {a b^3 (A b+2 a B) x^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^4 (A b+5 a B) x^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {b^5 B x^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 (a+b x)} \]
1/6*a^5*A*x^6*((b*x+a)^2)^(1/2)/(b*x+a)+1/7*a^4*(5*A*b+B*a)*x^7*((b*x+a)^2 )^(1/2)/(b*x+a)+5/8*a^3*b*(2*A*b+B*a)*x^8*((b*x+a)^2)^(1/2)/(b*x+a)+10/9*a ^2*b^2*(A*b+B*a)*x^9*((b*x+a)^2)^(1/2)/(b*x+a)+1/2*a*b^3*(A*b+2*B*a)*x^10* ((b*x+a)^2)^(1/2)/(b*x+a)+1/11*b^4*(A*b+5*B*a)*x^11*((b*x+a)^2)^(1/2)/(b*x +a)+1/12*b^5*B*x^12*((b*x+a)^2)^(1/2)/(b*x+a)
Time = 1.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.41 \[ \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x^6 \sqrt {(a+b x)^2} \left (132 a^5 (7 A+6 B x)+495 a^4 b x (8 A+7 B x)+770 a^3 b^2 x^2 (9 A+8 B x)+616 a^2 b^3 x^3 (10 A+9 B x)+252 a b^4 x^4 (11 A+10 B x)+42 b^5 x^5 (12 A+11 B x)\right )}{5544 (a+b x)} \]
(x^6*Sqrt[(a + b*x)^2]*(132*a^5*(7*A + 6*B*x) + 495*a^4*b*x*(8*A + 7*B*x) + 770*a^3*b^2*x^2*(9*A + 8*B*x) + 616*a^2*b^3*x^3*(10*A + 9*B*x) + 252*a*b ^4*x^4*(11*A + 10*B*x) + 42*b^5*x^5*(12*A + 11*B*x)))/(5544*(a + b*x))
Time = 0.32 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.47, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1187, 27, 85, 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.
| \(\displaystyle \int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 x^5 (a+b x)^5 (A+B x)dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^5 (a+b x)^5 (A+B x)dx}{a+b x}\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^5 B x^{11}+b^4 (A b+5 a B) x^{10}+5 a b^3 (A b+2 a B) x^9+10 a^2 b^2 (A b+a B) x^8+5 a^3 b (2 A b+a B) x^7+a^4 (5 A b+a B) x^6+a^5 A x^5\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 x^7 (a B+5 A b)+\frac {5}{8} a^3 b x^8 (a B+2 A b)+\frac {10}{9} a^2 b^2 x^9 (a B+A b)+\frac {1}{11} b^4 x^{11} (5 a B+A b)+\frac {1}{2} a b^3 x^{10} (2 a B+A b)+\frac {1}{12} b^5 B x^{12}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((a^5*A*x^6)/6 + (a^4*(5*A*b + a*B)*x^7)/7 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + (10*a^2*b^2*(A*b + a*B)*x^9)/9 + (a*b^3* (A*b + 2*a*B)*x^10)/2 + (b^4*(A*b + 5*a*B)*x^11)/11 + (b^5*B*x^12)/12))/(a + b*x)
3.7.84.3.1 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[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.46
| method | result | size |
| gosper | \(\frac {x^{6} \left (462 B \,b^{5} x^{6}+504 A \,b^{5} x^{5}+2520 B a \,b^{4} x^{5}+2772 A a \,b^{4} x^{4}+5544 B \,a^{2} b^{3} x^{4}+6160 A \,a^{2} b^{3} x^{3}+6160 B \,a^{3} b^{2} x^{3}+6930 A \,a^{3} b^{2} x^{2}+3465 B \,a^{4} b \,x^{2}+3960 A \,a^{4} b x +792 a^{5} B x +924 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 \left (b x +a \right )^{5}}\) | \(140\) |
| default | \(\frac {x^{6} \left (462 B \,b^{5} x^{6}+504 A \,b^{5} x^{5}+2520 B a \,b^{4} x^{5}+2772 A a \,b^{4} x^{4}+5544 B \,a^{2} b^{3} x^{4}+6160 A \,a^{2} b^{3} x^{3}+6160 B \,a^{3} b^{2} x^{3}+6930 A \,a^{3} b^{2} x^{2}+3465 B \,a^{4} b \,x^{2}+3960 A \,a^{4} b x +792 a^{5} B x +924 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 \left (b x +a \right )^{5}}\) | \(140\) |
| risch | \(\frac {b^{5} B \,x^{12} \sqrt {\left (b x +a \right )^{2}}}{12 b x +12 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A \,b^{5}+5 B a \,b^{4}\right ) x^{11}}{11 b x +11 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A a \,b^{4}+10 B \,a^{2} b^{3}\right ) x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,a^{2} b^{3}+10 B \,a^{3} b^{2}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 A \,a^{3} b^{2}+5 B \,a^{4} b \right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 A \,a^{4} b +a^{5} B \right ) x^{7}}{7 b x +7 a}+\frac {a^{5} A \,x^{6} \sqrt {\left (b x +a \right )^{2}}}{6 b x +6 a}\) | \(236\) |
1/5544*x^6*(462*B*b^5*x^6+504*A*b^5*x^5+2520*B*a*b^4*x^5+2772*A*a*b^4*x^4+ 5544*B*a^2*b^3*x^4+6160*A*a^2*b^3*x^3+6160*B*a^3*b^2*x^3+6930*A*a^3*b^2*x^ 2+3465*B*a^4*b*x^2+3960*A*a^4*b*x+792*B*a^5*x+924*A*a^5)*((b*x+a)^2)^(5/2) /(b*x+a)^5
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.39 \[ \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]
1/12*B*b^5*x^12 + 1/6*A*a^5*x^6 + 1/11*(5*B*a*b^4 + A*b^5)*x^11 + 1/2*(2*B *a^2*b^3 + A*a*b^4)*x^10 + 10/9*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 5/8*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 1/7*(B*a^5 + 5*A*a^4*b)*x^7
Leaf count of result is larger than twice the leaf count of optimal. 24941 vs. \(2 (231) = 462\).
Time = 1.17 (sec) , antiderivative size = 24941, normalized size of antiderivative = 81.51 \[ \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(B*b**4*x**11/12 + x**10*(A*b* *6 + 49*B*a*b**5/12)/(11*b**2) + x**9*(6*A*a*b**5 + 169*B*a**2*b**4/12 - 2 1*a*(A*b**6 + 49*B*a*b**5/12)/(11*b))/(10*b**2) + x**8*(15*A*a**2*b**4 + 2 0*B*a**3*b**3 - 10*a**2*(A*b**6 + 49*B*a*b**5/12)/(11*b**2) - 19*a*(6*A*a* b**5 + 169*B*a**2*b**4/12 - 21*a*(A*b**6 + 49*B*a*b**5/12)/(11*b))/(10*b)) /(9*b**2) + x**7*(20*A*a**3*b**3 + 15*B*a**4*b**2 - 9*a**2*(6*A*a*b**5 + 1 69*B*a**2*b**4/12 - 21*a*(A*b**6 + 49*B*a*b**5/12)/(11*b))/(10*b**2) - 17* a*(15*A*a**2*b**4 + 20*B*a**3*b**3 - 10*a**2*(A*b**6 + 49*B*a*b**5/12)/(11 *b**2) - 19*a*(6*A*a*b**5 + 169*B*a**2*b**4/12 - 21*a*(A*b**6 + 49*B*a*b** 5/12)/(11*b))/(10*b))/(9*b))/(8*b**2) + x**6*(15*A*a**4*b**2 + 6*B*a**5*b - 8*a**2*(15*A*a**2*b**4 + 20*B*a**3*b**3 - 10*a**2*(A*b**6 + 49*B*a*b**5/ 12)/(11*b**2) - 19*a*(6*A*a*b**5 + 169*B*a**2*b**4/12 - 21*a*(A*b**6 + 49* B*a*b**5/12)/(11*b))/(10*b))/(9*b**2) - 15*a*(20*A*a**3*b**3 + 15*B*a**4*b **2 - 9*a**2*(6*A*a*b**5 + 169*B*a**2*b**4/12 - 21*a*(A*b**6 + 49*B*a*b**5 /12)/(11*b))/(10*b**2) - 17*a*(15*A*a**2*b**4 + 20*B*a**3*b**3 - 10*a**2*( A*b**6 + 49*B*a*b**5/12)/(11*b**2) - 19*a*(6*A*a*b**5 + 169*B*a**2*b**4/12 - 21*a*(A*b**6 + 49*B*a*b**5/12)/(11*b))/(10*b))/(9*b))/(8*b))/(7*b**2) + x**5*(6*A*a**5*b + B*a**6 - 7*a**2*(20*A*a**3*b**3 + 15*B*a**4*b**2 - 9*a **2*(6*A*a*b**5 + 169*B*a**2*b**4/12 - 21*a*(A*b**6 + 49*B*a*b**5/12)/(11* b))/(10*b**2) - 17*a*(15*A*a**2*b**4 + 20*B*a**3*b**3 - 10*a**2*(A*b**6...
Time = 0.20 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.38 \[ \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B x^{5}}{12 \, b^{2}} - \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a x^{4}}{132 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A x^{4}}{11 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} x^{3}}{33 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a x^{3}}{22 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{6} x}{6 \, b^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{5} x}{6 \, b^{5}} - \frac {16 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{3} x^{2}}{99 \, b^{5}} + \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{2} x^{2}}{198 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{7}}{6 \, b^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{6}}{6 \, b^{6}} + \frac {131 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{4} x}{792 \, b^{6}} - \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{3} x}{396 \, b^{5}} - \frac {923 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{5}}{5544 \, b^{7}} + \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{4}}{2772 \, b^{6}} \]
1/12*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*x^5/b^2 - 17/132*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*x^4/b^3 + 1/11*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*x^4/b^2 + 5/33*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^2*x^3/b^4 - 3/22*(b^2*x^2 + 2* a*b*x + a^2)^(7/2)*A*a*x^3/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^6 *x/b^6 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a^5*x/b^5 - 16/99*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^3*x^2/b^5 + 31/198*(b^2*x^2 + 2*a*b*x + a^2)^(7 /2)*A*a^2*x^2/b^4 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^7/b^7 - 1/6*(b ^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a^6/b^6 + 131/792*(b^2*x^2 + 2*a*b*x + a^2 )^(7/2)*B*a^4*x/b^6 - 65/396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*a^3*x/b^5 - 923/5544*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^5/b^7 + 461/2772*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*a^4/b^6
Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.72 \[ \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{12} \, B b^{5} x^{12} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{11} \, B a b^{4} x^{11} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{11} \, A b^{5} x^{11} \mathrm {sgn}\left (b x + a\right ) + B a^{2} b^{3} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a b^{4} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, B a^{3} b^{2} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, A a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, B a^{4} b x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, A a^{3} b^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, B a^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, A a^{4} b x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A a^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (B a^{12} - 2 \, A a^{11} b\right )} \mathrm {sgn}\left (b x + a\right )}{5544 \, b^{7}} \]
1/12*B*b^5*x^12*sgn(b*x + a) + 5/11*B*a*b^4*x^11*sgn(b*x + a) + 1/11*A*b^5 *x^11*sgn(b*x + a) + B*a^2*b^3*x^10*sgn(b*x + a) + 1/2*A*a*b^4*x^10*sgn(b* x + a) + 10/9*B*a^3*b^2*x^9*sgn(b*x + a) + 10/9*A*a^2*b^3*x^9*sgn(b*x + a) + 5/8*B*a^4*b*x^8*sgn(b*x + a) + 5/4*A*a^3*b^2*x^8*sgn(b*x + a) + 1/7*B*a ^5*x^7*sgn(b*x + a) + 5/7*A*a^4*b*x^7*sgn(b*x + a) + 1/6*A*a^5*x^6*sgn(b*x + a) + 1/5544*(B*a^12 - 2*A*a^11*b)*sgn(b*x + a)/b^7
Timed out. \[ \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int x^5\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]