Integrand size = 29, antiderivative size = 130 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {A (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac {(A b-4 a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 a^2 x^7}-\frac {b (A b-4 a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{168 a^3 x^6} \]
-1/8*A*(b*x+a)^5*((b*x+a)^2)^(1/2)/a/x^8+1/28*(A*b-4*B*a)*(b*x+a)^5*((b*x+ a)^2)^(1/2)/a^2/x^7-1/168*b*(A*b-4*B*a)*(b*x+a)^5*((b*x+a)^2)^(1/2)/a^3/x^ 6
Time = 0.82 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.48 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {(a+b x)^5 \sqrt {(a+b x)^2} \left (A b^2 x^2-2 a b x (3 A+2 B x)+3 a^2 (7 A+8 B x)\right )}{168 a^3 x^8} \]
-1/168*((a + b*x)^5*Sqrt[(a + b*x)^2]*(A*b^2*x^2 - 2*a*b*x*(3*A + 2*B*x) + 3*a^2*(7*A + 8*B*x)))/(a^3*x^8)
Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1187, 27, 87, 55, 48}
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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (A+B x)}{x^9} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5 (A+B x)}{x^9}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5 (A+B x)}{x^9}dx}{a+b x}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {(A b-4 a B) \int \frac {(a+b x)^5}{x^8}dx}{4 a}-\frac {A (a+b x)^6}{8 a x^8}\right )}{a+b x}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {(A b-4 a B) \left (-\frac {b \int \frac {(a+b x)^5}{x^7}dx}{7 a}-\frac {(a+b x)^6}{7 a x^7}\right )}{4 a}-\frac {A (a+b x)^6}{8 a x^8}\right )}{a+b x}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {\left (\frac {b (a+b x)^6}{42 a^2 x^6}-\frac {(a+b x)^6}{7 a x^7}\right ) (A b-4 a B)}{4 a}-\frac {A (a+b x)^6}{8 a x^8}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/8*(A*(a + b*x)^6)/(a*x^8) - ((A*b - 4*a *B)*(-1/7*(a + b*x)^6/(a*x^7) + (b*(a + b*x)^6)/(42*a^2*x^6)))/(4*a)))/(a + b*x)
3.7.98.3.1 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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 1.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{5} x^{6}}{2}+\left (-\frac {1}{3} A \,b^{5}-\frac {5}{3} B a \,b^{4}\right ) x^{5}+\left (-\frac {5}{4} A a \,b^{4}-\frac {5}{2} B \,a^{2} b^{3}\right ) x^{4}+\left (-2 A \,a^{2} b^{3}-2 B \,a^{3} b^{2}\right ) x^{3}+\left (-\frac {5}{3} A \,a^{3} b^{2}-\frac {5}{6} B \,a^{4} b \right ) x^{2}+\left (-\frac {5}{7} A \,a^{4} b -\frac {1}{7} a^{5} B \right ) x -\frac {A \,a^{5}}{8}\right )}{\left (b x +a \right ) x^{8}}\) | \(136\) |
| gosper | \(-\frac {\left (84 B \,b^{5} x^{6}+56 A \,b^{5} x^{5}+280 B a \,b^{4} x^{5}+210 A a \,b^{4} x^{4}+420 B \,a^{2} b^{3} x^{4}+336 A \,a^{2} b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+280 A \,a^{3} b^{2} x^{2}+140 B \,a^{4} b \,x^{2}+120 A \,a^{4} b x +24 a^{5} B x +21 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 x^{8} \left (b x +a \right )^{5}}\) | \(140\) |
| default | \(-\frac {\left (84 B \,b^{5} x^{6}+56 A \,b^{5} x^{5}+280 B a \,b^{4} x^{5}+210 A a \,b^{4} x^{4}+420 B \,a^{2} b^{3} x^{4}+336 A \,a^{2} b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+280 A \,a^{3} b^{2} x^{2}+140 B \,a^{4} b \,x^{2}+120 A \,a^{4} b x +24 a^{5} B x +21 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 x^{8} \left (b x +a \right )^{5}}\) | \(140\) |
((b*x+a)^2)^(1/2)/(b*x+a)*(-1/2*B*b^5*x^6+(-1/3*A*b^5-5/3*B*a*b^4)*x^5+(-5 /4*A*a*b^4-5/2*B*a^2*b^3)*x^4+(-2*A*a^2*b^3-2*B*a^3*b^2)*x^3+(-5/3*A*a^3*b ^2-5/6*B*a^4*b)*x^2+(-5/7*A*a^4*b-1/7*a^5*B)*x-1/8*A*a^5)/x^8
Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \]
-1/168*(84*B*b^5*x^6 + 21*A*a^5 + 56*(5*B*a*b^4 + A*b^5)*x^5 + 210*(2*B*a^ 2*b^3 + A*a*b^4)*x^4 + 336*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 140*(B*a^4*b + 2* A*a^3*b^2)*x^2 + 24*(B*a^5 + 5*A*a^4*b)*x)/x^8
\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{9}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (93) = 186\).
Time = 0.22 (sec) , antiderivative size = 495, normalized size of antiderivative = 3.81 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{7}}{6 \, a^{7}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{8}}{6 \, a^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{6}}{6 \, a^{6} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{7}}{6 \, a^{7} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{6 \, a^{5} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{6 \, a^{4} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{6 \, a^{5} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{6 \, a^{4} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{7 \, a^{2} x^{7}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{56 \, a^{3} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{8 \, a^{2} x^{8}} \]
-1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^7/a^7 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^8/a^8 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^6/(a^6*x) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^7/(a^7*x) + 1/6*(b^2*x^2 + 2*a*b* x + a^2)^(7/2)*B*b^5/(a^7*x^2) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6 /(a^8*x^2) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^4/(a^6*x^3) + 1/6*(b^ 2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^7*x^3) + 1/6*(b^2*x^2 + 2*a*b*x + a^ 2)^(7/2)*B*b^3/(a^5*x^4) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6* x^4) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^2/(a^4*x^5) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x^5) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/ 2)*B*b/(a^3*x^6) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^2/(a^4*x^6) - 1 /7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B/(a^2*x^7) + 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^7) - 1/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^8 )
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (93) = 186\).
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=\frac {{\left (4 \, B a b^{7} - A b^{8}\right )} \mathrm {sgn}\left (b x + a\right )}{168 \, a^{3}} - \frac {84 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 280 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 56 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 420 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 210 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 336 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 336 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 140 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 280 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 24 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 120 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 21 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{168 \, x^{8}} \]
1/168*(4*B*a*b^7 - A*b^8)*sgn(b*x + a)/a^3 - 1/168*(84*B*b^5*x^6*sgn(b*x + a) + 280*B*a*b^4*x^5*sgn(b*x + a) + 56*A*b^5*x^5*sgn(b*x + a) + 420*B*a^2 *b^3*x^4*sgn(b*x + a) + 210*A*a*b^4*x^4*sgn(b*x + a) + 336*B*a^3*b^2*x^3*s gn(b*x + a) + 336*A*a^2*b^3*x^3*sgn(b*x + a) + 140*B*a^4*b*x^2*sgn(b*x + a ) + 280*A*a^3*b^2*x^2*sgn(b*x + a) + 24*B*a^5*x*sgn(b*x + a) + 120*A*a^4*b *x*sgn(b*x + a) + 21*A*a^5*sgn(b*x + a))/x^8
Time = 10.12 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.18 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^7\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{3}+\frac {5\,B\,a\,b^4}{3}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,x^8\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {2\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )} \]
- (((B*a^5)/7 + (5*A*a^4*b)/7)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^7*(a + b*x)) - (((A*b^5)/3 + (5*B*a*b^4)/3)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^3 *(a + b*x)) - (A*a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(8*x^8*(a + b*x)) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*x^2*(a + b*x)) - (5*a*b^3*(A*b + 2*B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*x^4*(a + b*x)) - (5*a^3*b*(2* A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*x^6*(a + b*x)) - (2*a^2*b^2 *(A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^5*(a + b*x))