Integrand size = 26, antiderivative size = 72 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}}+\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}} \]
-1/12*(b*x^2+a)*(b^2*x^4+2*a*b*x^2+a^2)^(5/2)/a/x^14+1/84*(b^2*x^4+2*a*b*x ^2+a^2)^(7/2)/a^2/x^14
Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {\sqrt {\left (a+b x^2\right )^2} \left (6 a^5+35 a^4 b x^2+84 a^3 b^2 x^4+105 a^2 b^3 x^6+70 a b^4 x^8+21 b^5 x^{10}\right )}{84 x^{14} \left (a+b x^2\right )} \]
-1/84*(Sqrt[(a + b*x^2)^2]*(6*a^5 + 35*a^4*b*x^2 + 84*a^3*b^2*x^4 + 105*a^ 2*b^3*x^6 + 70*a*b^4*x^8 + 21*b^5*x^10))/(x^14*(a + b*x^2))
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1384, 27, 243, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {b^5 \left (b x^2+a\right )^5}{x^{15}}dx}{b^5 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (b x^2+a\right )^5}{x^{15}}dx}{a+b x^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (b x^2+a\right )^5}{x^{16}}dx^2}{2 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (-\frac {b \int \frac {\left (b x^2+a\right )^5}{x^{14}}dx^2}{7 a}-\frac {\left (a+b x^2\right )^6}{7 a x^{14}}\right )}{2 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (\frac {b \left (a+b x^2\right )^6}{42 a^2 x^{12}}-\frac {\left (a+b x^2\right )^6}{7 a x^{14}}\right )}{2 \left (a+b x^2\right )}\) |
(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-1/7*(a + b*x^2)^6/(a*x^14) + (b*(a + b* x^2)^6)/(42*a^2*x^12)))/(2*(a + b*x^2))
3.6.99.3.1 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[((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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.47 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(-\frac {\left (21 x^{10} b^{5}+70 a \,x^{8} b^{4}+105 a^{2} x^{6} b^{3}+84 a^{3} x^{4} b^{2}+35 x^{2} a^{4} b +6 a^{5}\right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{84 x^{14}}\) | \(68\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {1}{14} a^{5}-\frac {5}{12} x^{2} a^{4} b -a^{3} x^{4} b^{2}-\frac {5}{4} a^{2} x^{6} b^{3}-\frac {5}{6} a \,x^{8} b^{4}-\frac {1}{4} x^{10} b^{5}\right )}{\left (b \,x^{2}+a \right ) x^{14}}\) | \(79\) |
gosper | \(-\frac {\left (21 x^{10} b^{5}+70 a \,x^{8} b^{4}+105 a^{2} x^{6} b^{3}+84 a^{3} x^{4} b^{2}+35 x^{2} a^{4} b +6 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{84 x^{14} \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
default | \(-\frac {\left (21 x^{10} b^{5}+70 a \,x^{8} b^{4}+105 a^{2} x^{6} b^{3}+84 a^{3} x^{4} b^{2}+35 x^{2} a^{4} b +6 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{84 x^{14} \left (b \,x^{2}+a \right )^{5}}\) | \(80\) |
-1/84*(21*b^5*x^10+70*a*b^4*x^8+105*a^2*b^3*x^6+84*a^3*b^2*x^4+35*a^4*b*x^ 2+6*a^5)*csgn(b*x^2+a)/x^14
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \]
-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35* a^4*b*x^2 + 6*a^5)/x^14
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{15}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {b^{5}}{4 \, x^{4}} - \frac {5 \, a b^{4}}{6 \, x^{6}} - \frac {5 \, a^{2} b^{3}}{4 \, x^{8}} - \frac {a^{3} b^{2}}{x^{10}} - \frac {5 \, a^{4} b}{12 \, x^{12}} - \frac {a^{5}}{14 \, x^{14}} \]
-1/4*b^5/x^4 - 5/6*a*b^4/x^6 - 5/4*a^2*b^3/x^8 - a^3*b^2/x^10 - 5/12*a^4*b /x^12 - 1/14*a^5/x^14
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {21 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 70 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 84 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 35 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{84 \, x^{14}} \]
-1/84*(21*b^5*x^10*sgn(b*x^2 + a) + 70*a*b^4*x^8*sgn(b*x^2 + a) + 105*a^2* b^3*x^6*sgn(b*x^2 + a) + 84*a^3*b^2*x^4*sgn(b*x^2 + a) + 35*a^4*b*x^2*sgn( b*x^2 + a) + 6*a^5*sgn(b*x^2 + a))/x^14
Time = 13.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.21 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^4\,\left (b\,x^2+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{6\,x^6\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{12\,x^{12}\,\left (b\,x^2+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^8\,\left (b\,x^2+a\right )}-\frac {a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{x^{10}\,\left (b\,x^2+a\right )} \]
- (a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(14*x^14*(a + b*x^2)) - (b^5*(a^ 2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(4*x^4*(a + b*x^2)) - (5*a*b^4*(a^2 + b^2* x^4 + 2*a*b*x^2)^(1/2))/(6*x^6*(a + b*x^2)) - (5*a^4*b*(a^2 + b^2*x^4 + 2* a*b*x^2)^(1/2))/(12*x^12*(a + b*x^2)) - (5*a^2*b^3*(a^2 + b^2*x^4 + 2*a*b* x^2)^(1/2))/(4*x^8*(a + b*x^2)) - (a^3*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/ 2))/(x^10*(a + b*x^2))