Integrand size = 24, antiderivative size = 116 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 a x^8}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 a^2 x^7}-\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{168 a^3 x^6} \]
-1/8*(b*x+a)^5*((b*x+a)^2)^(1/2)/a/x^8+1/28*b*(b*x+a)^5*((b*x+a)^2)^(1/2)/ a^2/x^7-1/168*b^2*(b*x+a)^5*((b*x+a)^2)^(1/2)/a^3/x^6
Time = 0.65 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {\sqrt {(a+b x)^2} \left (21 a^5+120 a^4 b x+280 a^3 b^2 x^2+336 a^2 b^3 x^3+210 a b^4 x^4+56 b^5 x^5\right )}{168 x^8 (a+b x)} \]
-1/168*(Sqrt[(a + b*x)^2]*(21*a^5 + 120*a^4*b*x + 280*a^3*b^2*x^2 + 336*a^ 2*b^3*x^3 + 210*a*b^4*x^4 + 56*b^5*x^5))/(x^8*(a + b*x))
Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1102, 27, 55, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx\) |
\(\Big \downarrow \) 1102 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^5}{x^9}dx}{b^5 (a+b x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^5}{x^9}dx}{a+b x}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {b \int \frac {(a+b x)^5}{x^8}dx}{4 a}-\frac {(a+b x)^6}{8 a x^8}\right )}{a+b x}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {b \left (-\frac {b \int \frac {(a+b x)^5}{x^7}dx}{7 a}-\frac {(a+b x)^6}{7 a x^7}\right )}{4 a}-\frac {(a+b x)^6}{8 a x^8}\right )}{a+b x}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {b \left (\frac {b (a+b x)^6}{42 a^2 x^6}-\frac {(a+b x)^6}{7 a x^7}\right )}{4 a}-\frac {(a+b x)^6}{8 a x^8}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/8*(a + b*x)^6/(a*x^8) - (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)
3.2.77.3.1 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[((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 = 2.73 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {1}{3} b^{5} x^{5}-\frac {5}{4} a \,b^{4} x^{4}-2 a^{2} b^{3} x^{3}-\frac {5}{3} a^{3} b^{2} x^{2}-\frac {5}{7} a^{4} b x -\frac {1}{8} a^{5}\right )}{\left (b x +a \right ) x^{8}}\) | \(73\) |
gosper | \(-\frac {\left (56 b^{5} x^{5}+210 a \,b^{4} x^{4}+336 a^{2} b^{3} x^{3}+280 a^{3} b^{2} x^{2}+120 a^{4} b x +21 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 x^{8} \left (b x +a \right )^{5}}\) | \(74\) |
default | \(-\frac {\left (56 b^{5} x^{5}+210 a \,b^{4} x^{4}+336 a^{2} b^{3} x^{3}+280 a^{3} b^{2} x^{2}+120 a^{4} b x +21 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 x^{8} \left (b x +a \right )^{5}}\) | \(74\) |
((b*x+a)^2)^(1/2)/(b*x+a)/x^8*(-1/3*b^5*x^5-5/4*a*b^4*x^4-2*a^2*b^3*x^3-5/ 3*a^3*b^2*x^2-5/7*a^4*b*x-1/8*a^5)
Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.49 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \]
-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 1 20*a^4*b*x + 21*a^5)/x^8
\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{9}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (77) = 154\).
Time = 0.20 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{8}}{6 \, a^{8}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{7}}{6 \, a^{7} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{6}}{6 \, a^{8} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{5}}{6 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{4}}{6 \, a^{6} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{6 \, a^{5} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{6 \, a^{4} x^{6}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{56 \, a^{3} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{8 \, a^{2} x^{8}} \]
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^8/a^8 + 1/6*(b^2*x^2 + 2*a*b*x + a^2 )^(5/2)*b^7/(a^7*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^6/(a^8*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^5/(a^7*x^3) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^4/(a^6*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^3/(a^5 *x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^2/(a^4*x^6) + 9/56*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b/(a^3*x^7) - 1/8*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/( a^2*x^8)
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {b^{8} \mathrm {sgn}\left (b x + a\right )}{168 \, a^{3}} - \frac {56 \, b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 336 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 280 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 120 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{168 \, x^{8}} \]
-1/168*b^8*sgn(b*x + a)/a^3 - 1/168*(56*b^5*x^5*sgn(b*x + a) + 210*a*b^4*x ^4*sgn(b*x + a) + 336*a^2*b^3*x^3*sgn(b*x + a) + 280*a^3*b^2*x^2*sgn(b*x + a) + 120*a^4*b*x*sgn(b*x + a) + 21*a^5*sgn(b*x + a))/x^8
Time = 10.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,x^8\,\left (a+b\,x\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {2\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^6\,\left (a+b\,x\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )} \]
- (a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(8*x^8*(a + b*x)) - (b^5*(a^2 + b^ 2*x^2 + 2*a*b*x)^(1/2))/(3*x^3*(a + b*x)) - (2*a^2*b^3*(a^2 + b^2*x^2 + 2* a*b*x)^(1/2))/(x^5*(a + b*x)) - (5*a^3*b^2*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) )/(3*x^6*(a + b*x)) - (5*a*b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*x^4*(a + b*x)) - (5*a^4*b*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*x^7*(a + b*x))