Integrand size = 29, antiderivative size = 306 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]
-1/11*a^5*A*((b*x+a)^2)^(1/2)/x^11/(b*x+a)-1/10*a^4*(5*A*b+B*a)*((b*x+a)^2 )^(1/2)/x^10/(b*x+a)-5/9*a^3*b*(2*A*b+B*a)*((b*x+a)^2)^(1/2)/x^9/(b*x+a)-5 /4*a^2*b^2*(A*b+B*a)*((b*x+a)^2)^(1/2)/x^8/(b*x+a)-5/7*a*b^3*(A*b+2*B*a)*( (b*x+a)^2)^(1/2)/x^7/(b*x+a)-1/6*b^4*(A*b+5*B*a)*((b*x+a)^2)^(1/2)/x^6/(b* x+a)-1/5*b^5*B*((b*x+a)^2)^(1/2)/x^5/(b*x+a)
Time = 1.02 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.33 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx=\frac {\sqrt {(a+b x)^2} \left (-1260 A (a+b x)^6+(5 A b-11 a B) x \left (126 a^5+700 a^4 b x+1575 a^3 b^2 x^2+1800 a^2 b^3 x^3+1050 a b^4 x^4+252 b^5 x^5\right )\right )}{13860 a x^{11} (a+b x)} \]
(Sqrt[(a + b*x)^2]*(-1260*A*(a + b*x)^6 + (5*A*b - 11*a*B)*x*(126*a^5 + 70 0*a^4*b*x + 1575*a^3*b^2*x^2 + 1800*a^2*b^3*x^3 + 1050*a*b^4*x^4 + 252*b^5 *x^5)))/(13860*a*x^11*(a + b*x))
Time = 0.30 (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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{x^{12}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{x^{12}}dx}{a+b x}\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {A a^5}{x^{12}}+\frac {(5 A b+a B) a^4}{x^{11}}+\frac {5 b (2 A b+a B) a^3}{x^{10}}+\frac {10 b^2 (A b+a B) a^2}{x^9}+\frac {5 b^3 (A b+2 a B) a}{x^8}+\frac {b^5 B}{x^6}+\frac {b^4 (A b+5 a B)}{x^7}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {a^5 A}{11 x^{11}}-\frac {a^4 (a B+5 A b)}{10 x^{10}}-\frac {5 a^3 b (a B+2 A b)}{9 x^9}-\frac {5 a^2 b^2 (a B+A b)}{4 x^8}-\frac {b^4 (5 a B+A b)}{6 x^6}-\frac {5 a b^3 (2 a B+A b)}{7 x^7}-\frac {b^5 B}{5 x^5}\right )}{a+b x}\) |
((-1/11*(a^5*A)/x^11 - (a^4*(5*A*b + a*B))/(10*x^10) - (5*a^3*b*(2*A*b + a *B))/(9*x^9) - (5*a^2*b^2*(A*b + a*B))/(4*x^8) - (5*a*b^3*(A*b + 2*a*B))/( 7*x^7) - (b^4*(A*b + 5*a*B))/(6*x^6) - (b^5*B)/(5*x^5))*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x)
3.8.1.3.1 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 = 2.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{5} x^{6}}{5}+\left (-\frac {1}{6} A \,b^{5}-\frac {5}{6} B a \,b^{4}\right ) x^{5}+\left (-\frac {5}{7} A a \,b^{4}-\frac {10}{7} B \,a^{2} b^{3}\right ) x^{4}+\left (-\frac {5}{4} A \,a^{2} b^{3}-\frac {5}{4} B \,a^{3} b^{2}\right ) x^{3}+\left (-\frac {10}{9} A \,a^{3} b^{2}-\frac {5}{9} B \,a^{4} b \right ) x^{2}+\left (-\frac {1}{2} A \,a^{4} b -\frac {1}{10} a^{5} B \right ) x -\frac {A \,a^{5}}{11}\right )}{\left (b x +a \right ) x^{11}}\) | \(136\) |
| gosper | \(-\frac {\left (2772 B \,b^{5} x^{6}+2310 A \,b^{5} x^{5}+11550 B a \,b^{4} x^{5}+9900 A a \,b^{4} x^{4}+19800 B \,a^{2} b^{3} x^{4}+17325 A \,a^{2} b^{3} x^{3}+17325 B \,a^{3} b^{2} x^{3}+15400 A \,a^{3} b^{2} x^{2}+7700 B \,a^{4} b \,x^{2}+6930 A \,a^{4} b x +1386 a^{5} B x +1260 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 x^{11} \left (b x +a \right )^{5}}\) | \(140\) |
| default | \(-\frac {\left (2772 B \,b^{5} x^{6}+2310 A \,b^{5} x^{5}+11550 B a \,b^{4} x^{5}+9900 A a \,b^{4} x^{4}+19800 B \,a^{2} b^{3} x^{4}+17325 A \,a^{2} b^{3} x^{3}+17325 B \,a^{3} b^{2} x^{3}+15400 A \,a^{3} b^{2} x^{2}+7700 B \,a^{4} b \,x^{2}+6930 A \,a^{4} b x +1386 a^{5} B x +1260 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 x^{11} \left (b x +a \right )^{5}}\) | \(140\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/5*B*b^5*x^6+(-1/6*A*b^5-5/6*B*a*b^4)*x^5+(-5 /7*A*a*b^4-10/7*B*a^2*b^3)*x^4+(-5/4*A*a^2*b^3-5/4*B*a^3*b^2)*x^3+(-10/9*A *a^3*b^2-5/9*B*a^4*b)*x^2+(-1/2*A*a^4*b-1/10*a^5*B)*x-1/11*A*a^5)/x^11
Time = 0.38 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.39 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \]
-1/13860*(2772*B*b^5*x^6 + 1260*A*a^5 + 2310*(5*B*a*b^4 + A*b^5)*x^5 + 990 0*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 17325*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 7700*( B*a^4*b + 2*A*a^3*b^2)*x^2 + 1386*(B*a^5 + 5*A*a^4*b)*x)/x^11
\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{12}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (215) = 430\).
Time = 0.22 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{10}}{6 \, a^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{11}}{6 \, a^{11}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{9}}{6 \, a^{9} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{10}}{6 \, a^{10} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{8}}{6 \, a^{10} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{9}}{6 \, a^{11} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{7}}{6 \, a^{9} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{8}}{6 \, a^{10} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{6}} + \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{1260 \, a^{5} x^{7}} - \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{2772 \, a^{6} x^{7}} - \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{180 \, a^{4} x^{8}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{396 \, a^{5} x^{8}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{90 \, a^{3} x^{9}} - \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{198 \, a^{4} x^{9}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{10 \, a^{2} x^{10}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{22 \, a^{3} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{11 \, a^{2} x^{11}} \]
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^10/a^10 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^11/a^11 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^9/(a^9*x ) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^10/(a^10*x) - 1/6*(b^2*x^2 + 2 *a*b*x + a^2)^(7/2)*B*b^8/(a^10*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2) *A*b^9/(a^11*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^7/(a^9*x^3) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^8/(a^10*x^3) - 1/6*(b^2*x^2 + 2*a* b*x + a^2)^(7/2)*B*b^6/(a^8*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b ^7/(a^9*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^5/(a^7*x^5) - 1/6*( b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6/(a^8*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^4/(a^6*x^6) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^ 7*x^6) + 209/1260*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^3/(a^5*x^7) - 461/27 72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6*x^7) - 29/180*(b^2*x^2 + 2*a *b*x + a^2)^(7/2)*B*b^2/(a^4*x^8) + 65/396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2) *A*b^3/(a^5*x^8) + 13/90*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^9) - 3 1/198*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^2/(a^4*x^9) - 1/10*(b^2*x^2 + 2* a*b*x + a^2)^(7/2)*B/(a^2*x^10) + 3/22*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b /(a^3*x^10) - 1/11*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^11)
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.72 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {{\left (11 \, B a b^{10} - 5 \, A b^{11}\right )} \mathrm {sgn}\left (b x + a\right )}{13860 \, a^{6}} - \frac {2772 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 11550 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 2310 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 19800 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 9900 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 17325 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 17325 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 7700 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 15400 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1386 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 6930 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 1260 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{13860 \, x^{11}} \]
-1/13860*(11*B*a*b^10 - 5*A*b^11)*sgn(b*x + a)/a^6 - 1/13860*(2772*B*b^5*x ^6*sgn(b*x + a) + 11550*B*a*b^4*x^5*sgn(b*x + a) + 2310*A*b^5*x^5*sgn(b*x + a) + 19800*B*a^2*b^3*x^4*sgn(b*x + a) + 9900*A*a*b^4*x^4*sgn(b*x + a) + 17325*B*a^3*b^2*x^3*sgn(b*x + a) + 17325*A*a^2*b^3*x^3*sgn(b*x + a) + 7700 *B*a^4*b*x^2*sgn(b*x + a) + 15400*A*a^3*b^2*x^2*sgn(b*x + a) + 1386*B*a^5* x*sgn(b*x + a) + 6930*A*a^4*b*x*sgn(b*x + a) + 1260*A*a^5*sgn(b*x + a))/x^ 11
Time = 10.09 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {\left (\frac {B\,a^5}{10}+\frac {A\,b\,a^4}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{10}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^6\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^8\,\left (a+b\,x\right )} \]
- (((B*a^5)/10 + (A*a^4*b)/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^10*(a + b*x)) - (((A*b^5)/6 + (5*B*a*b^4)/6)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^6 *(a + b*x)) - (A*a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(11*x^11*(a + b*x)) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(5*x^5*(a + b*x)) - (5*a*b^3*(A* b + 2*B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*x^7*(a + b*x)) - (5*a^3*b*( 2*A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(9*x^9*(a + b*x)) - (5*a^2*b ^2*(A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*x^8*(a + b*x))