Integrand size = 29, antiderivative size = 304 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {5 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^6 (a+b x)}-\frac {a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
-1/9*a^5*A*((b*x+a)^2)^(1/2)/x^9/(b*x+a)-1/8*a^4*(5*A*b+B*a)*((b*x+a)^2)^( 1/2)/x^8/(b*x+a)-5/7*a^3*b*(2*A*b+B*a)*((b*x+a)^2)^(1/2)/x^7/(b*x+a)-5/3*a ^2*b^2*(A*b+B*a)*((b*x+a)^2)^(1/2)/x^6/(b*x+a)-a*b^3*(A*b+2*B*a)*((b*x+a)^ 2)^(1/2)/x^5/(b*x+a)-1/4*b^4*(A*b+5*B*a)*((b*x+a)^2)^(1/2)/x^4/(b*x+a)-1/3 *b^5*B*((b*x+a)^2)^(1/2)/x^3/(b*x+a)
Time = 0.90 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.40 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {\sqrt {(a+b x)^2} \left (\frac {(-3 A b+9 a B) \left (-\frac {a^5}{8 x^8}-\frac {5 a^4 b}{7 x^7}-\frac {5 a^3 b^2}{3 x^6}-\frac {2 a^2 b^3}{x^5}-\frac {5 a b^4}{4 x^4}-\frac {b^5}{3 x^3}\right )}{9 a}-\frac {A (a+b x)^6}{9 a x^9}\right )}{a+b x} \]
(Sqrt[(a + b*x)^2]*(((-3*A*b + 9*a*B)*(-1/8*a^5/x^8 - (5*a^4*b)/(7*x^7) - (5*a^3*b^2)/(3*x^6) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(4*x^4) - b^5/(3*x^3)))/ (9*a) - (A*(a + b*x)^6)/(9*a*x^9)))/(a + b*x)
Time = 0.30 (sec) , antiderivative size = 143, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{x^{10}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{x^{10}}dx}{a+b x}\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {A a^5}{x^{10}}+\frac {(5 A b+a B) a^4}{x^9}+\frac {5 b (2 A b+a B) a^3}{x^8}+\frac {10 b^2 (A b+a B) a^2}{x^7}+\frac {5 b^3 (A b+2 a B) a}{x^6}+\frac {b^5 B}{x^4}+\frac {b^4 (A b+5 a B)}{x^5}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {a^5 A}{9 x^9}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {5 a^3 b (a B+2 A b)}{7 x^7}-\frac {5 a^2 b^2 (a B+A b)}{3 x^6}-\frac {b^4 (5 a B+A b)}{4 x^4}-\frac {a b^3 (2 a B+A b)}{x^5}-\frac {b^5 B}{3 x^3}\right )}{a+b x}\) |
((-1/9*(a^5*A)/x^9 - (a^4*(5*A*b + a*B))/(8*x^8) - (5*a^3*b*(2*A*b + a*B)) /(7*x^7) - (5*a^2*b^2*(A*b + a*B))/(3*x^6) - (a*b^3*(A*b + 2*a*B))/x^5 - ( b^4*(A*b + 5*a*B))/(4*x^4) - (b^5*B)/(3*x^3))*Sqrt[a^2 + 2*a*b*x + b^2*x^2 ])/(a + b*x)
3.7.99.3.1 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 = 1.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{5} x^{6}}{3}+\left (-\frac {1}{4} A \,b^{5}-\frac {5}{4} B a \,b^{4}\right ) x^{5}+\left (-A a \,b^{4}-2 B \,a^{2} b^{3}\right ) x^{4}+\left (-\frac {5}{3} A \,a^{2} b^{3}-\frac {5}{3} B \,a^{3} b^{2}\right ) x^{3}+\left (-\frac {10}{7} A \,a^{3} b^{2}-\frac {5}{7} B \,a^{4} b \right ) x^{2}+\left (-\frac {5}{8} A \,a^{4} b -\frac {1}{8} a^{5} B \right ) x -\frac {A \,a^{5}}{9}\right )}{\left (b x +a \right ) x^{9}}\) | \(136\) |
| gosper | \(-\frac {\left (168 B \,b^{5} x^{6}+126 A \,b^{5} x^{5}+630 B a \,b^{4} x^{5}+504 A a \,b^{4} x^{4}+1008 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+720 A \,a^{3} b^{2} x^{2}+360 B \,a^{4} b \,x^{2}+315 A \,a^{4} b x +63 a^{5} B x +56 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 x^{9} \left (b x +a \right )^{5}}\) | \(140\) |
| default | \(-\frac {\left (168 B \,b^{5} x^{6}+126 A \,b^{5} x^{5}+630 B a \,b^{4} x^{5}+504 A a \,b^{4} x^{4}+1008 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+720 A \,a^{3} b^{2} x^{2}+360 B \,a^{4} b \,x^{2}+315 A \,a^{4} b x +63 a^{5} B x +56 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 x^{9} \left (b x +a \right )^{5}}\) | \(140\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/3*B*b^5*x^6+(-1/4*A*b^5-5/4*B*a*b^4)*x^5+(-A *a*b^4-2*B*a^2*b^3)*x^4+(-5/3*A*a^2*b^3-5/3*B*a^3*b^2)*x^3+(-10/7*A*a^3*b^ 2-5/7*B*a^4*b)*x^2+(-5/8*A*a^4*b-1/8*a^5*B)*x-1/9*A*a^5)/x^9
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.39 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {168 \, B b^{5} x^{6} + 56 \, A a^{5} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{504 \, x^{9}} \]
-1/504*(168*B*b^5*x^6 + 56*A*a^5 + 126*(5*B*a*b^4 + A*b^5)*x^5 + 504*(2*B* a^2*b^3 + A*a*b^4)*x^4 + 840*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 360*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 63*(B*a^5 + 5*A*a^4*b)*x)/x^9
\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{10}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (215) = 430\).
Time = 0.21 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.83 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{8}}{6 \, a^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{9}}{6 \, a^{9}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{7}}{6 \, a^{7} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{8}}{6 \, a^{8} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{6 \, a^{5} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{6 \, a^{4} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{6 \, a^{5} x^{6}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{56 \, a^{3} x^{7}} - \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{504 \, a^{4} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{8 \, a^{2} x^{8}} + \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{72 \, a^{3} x^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{9 \, a^{2} x^{9}} \]
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^8/a^8 - 1/6*(b^2*x^2 + 2*a*b*x + a ^2)^(5/2)*A*b^9/a^9 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^7/(a^7*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^8/(a^8*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^6/(a^8*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^7/ (a^9*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^5/(a^7*x^3) - 1/6*(b^2 *x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6/(a^8*x^3) - 1/6*(b^2*x^2 + 2*a*b*x + a^2 )^(7/2)*B*b^4/(a^6*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^7*x ^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^3/(a^5*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2 )*B*b^2/(a^4*x^6) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x^6) + 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^7) - 83/504*(b^2*x^2 + 2*a *b*x + a^2)^(7/2)*A*b^2/(a^4*x^7) - 1/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B/ (a^2*x^8) + 11/72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^8) - 1/9*(b^2 *x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^9)
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.73 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {{\left (3 \, B a b^{8} - A b^{9}\right )} \mathrm {sgn}\left (b x + a\right )}{504 \, a^{4}} - \frac {168 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 630 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1008 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 504 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 840 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 360 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 720 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 63 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 315 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 56 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{504 \, x^{9}} \]
-1/504*(3*B*a*b^8 - A*b^9)*sgn(b*x + a)/a^4 - 1/504*(168*B*b^5*x^6*sgn(b*x + a) + 630*B*a*b^4*x^5*sgn(b*x + a) + 126*A*b^5*x^5*sgn(b*x + a) + 1008*B *a^2*b^3*x^4*sgn(b*x + a) + 504*A*a*b^4*x^4*sgn(b*x + a) + 840*B*a^3*b^2*x ^3*sgn(b*x + a) + 840*A*a^2*b^3*x^3*sgn(b*x + a) + 360*B*a^4*b*x^2*sgn(b*x + a) + 720*A*a^3*b^2*x^2*sgn(b*x + a) + 63*B*a^5*x*sgn(b*x + a) + 315*A*a ^4*b*x*sgn(b*x + a) + 56*A*a^5*sgn(b*x + a))/x^9
Time = 10.15 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^8\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^4\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^6\,\left (a+b\,x\right )} \]
- (((B*a^5)/8 + (5*A*a^4*b)/8)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^8*(a + b*x)) - (((A*b^5)/4 + (5*B*a*b^4)/4)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^4 *(a + b*x)) - (A*a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(9*x^9*(a + b*x)) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(3*x^3*(a + b*x)) - (a*b^3*(A*b + 2*B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^5*(a + b*x)) - (5*a^3*b*(2*A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*x^7*(a + b*x)) - (5*a^2*b^2*(A* b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(3*x^6*(a + b*x))