Integrand size = 21, antiderivative size = 265 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, dx=-\frac {\left (11 b^2+20 a b c+8 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{16 c^3 (b+a c)^2 x^2}+\frac {(3 b+4 a c) d \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{8 c^3 (b+a c) x^4}-\frac {\left (c+d x^2\right )^3 \left (\frac {b+a c+a d x^2}{c+d x^2}\right )^{3/2}}{6 c^2 (b+a c) x^6}+\frac {b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{16 c^{7/2} (b+a c)^{5/2}} \]
-1/6*(d*x^2+c)^3*((a*d*x^2+a*c+b)/(d*x^2+c))^(3/2)/c^2/(a*c+b)/x^6+1/16*b* (8*a^2*c^2+12*a*b*c+5*b^2)*d^3*arctanh(c^(1/2)*((a*d*x^2+a*c+b)/(d*x^2+c)) ^(1/2)/(a*c+b)^(1/2))/c^(7/2)/(a*c+b)^(5/2)-1/16*(8*a^2*c^2+20*a*b*c+11*b^ 2)*d^2*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^3/(a*c+b)^2/x^2+1/8*( 4*a*c+3*b)*d*(d*x^2+c)^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^3/(a*c+b)/x^4
Time = 0.40 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, dx=-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (8 a^2 c^2 \left (c^2-c d x^2+d^2 x^4\right )+2 a b c \left (8 c^2-9 c d x^2+13 d^2 x^4\right )+b^2 \left (8 c^2-10 c d x^2+15 d^2 x^4\right )\right )}{48 c^3 (b+a c)^2 x^6}-\frac {b \left (5 b^2+12 a b c+8 a^2 c^2\right ) d^3 \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{16 c^{7/2} (-b-a c)^{5/2}} \]
-1/48*((c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(8*a^2*c^2*(c^2 - c*d*x^2 + d^2*x^4) + 2*a*b*c*(8*c^2 - 9*c*d*x^2 + 13*d^2*x^4) + b^2*(8*c^ 2 - 10*c*d*x^2 + 15*d^2*x^4)))/(c^3*(b + a*c)^2*x^6) - (b*(5*b^2 + 12*a*b* c + 8*a^2*c^2)*d^3*ArcTan[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/ Sqrt[-b - a*c]])/(16*c^(7/2)*(-b - a*c)^(5/2))
Time = 0.48 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2057, 2053, 2052, 27, 366, 27, 360, 27, 298, 221}
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.
| \(\displaystyle \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{x^7}dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{x^8}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle -b d \int \frac {d^2 x^4 \left (a-x^4\right )^2}{\left (-c x^4+b+a c\right )^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b d^3 \int \frac {x^4 \left (a-x^4\right )^2}{\left (-c x^4+b+a c\right )^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 366 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^6}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\int \frac {3 x^4 \left (2 c (b+a c) x^4+b^2-2 a^2 c^2\right )}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{6 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^6}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\int \frac {x^4 \left (2 c (b+a c) x^4+b^2-2 a^2 c^2\right )}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{2 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^6}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (4 a c+3 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c \left (a c+b-c x^4\right )^2}-\frac {\int \frac {c \left (8 c (b+a c) x^4+b (3 b+4 a c)\right )}{\left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c^2}}{2 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^6}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (4 a c+3 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c \left (a c+b-c x^4\right )^2}-\frac {\int \frac {8 c (b+a c) x^4+b (3 b+4 a c)}{\left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c}}{2 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^6}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (4 a c+3 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c \left (a c+b-c x^4\right )^2}-\frac {\frac {\left (8 a^2 c^2+20 a b c+11 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 (a c+b) \left (a c+b-c x^4\right )}-\frac {\left (8 a^2 c^2+12 a b c+5 b^2\right ) \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{2 (a c+b)}}{4 c}}{2 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^6}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (4 a c+3 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c \left (a c+b-c x^4\right )^2}-\frac {\frac {\left (8 a^2 c^2+20 a b c+11 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 (a c+b) \left (a c+b-c x^4\right )}-\frac {\left (8 a^2 c^2+12 a b c+5 b^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{2 \sqrt {c} (a c+b)^{3/2}}}{4 c}}{2 c^2 (a c+b)}\right )\) |
-(b*d^3*((b^2*x^6)/(6*c^2*(b + a*c)*(b + a*c - c*x^4)^3) - ((b*(3*b + 4*a* c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(4*c*(b + a*c - c*x^4)^2) - (((1 1*b^2 + 20*a*b*c + 8*a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(2*(b + a*c)*(b + a*c - c*x^4)) - ((5*b^2 + 12*a*b*c + 8*a^2*c^2)*ArcTanh[(Sqrt [c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]])/(2*Sqrt[c]*(b + a*c)^(3/2)))/(4*c))/(2*c^2*(b + a*c))))
3.4.24.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p , -1]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Time = 0.18 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (8 a^{2} c^{2} d^{2} x^{4}+26 a c \,d^{2} b \,x^{4}-8 a^{2} c^{3} d \,x^{2}+15 b^{2} d^{2} x^{4}-18 a b \,c^{2} d \,x^{2}+8 a^{2} c^{4}-10 b^{2} c d \,x^{2}+16 a b \,c^{3}+8 b^{2} c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{48 c^{3} x^{6} \left (a c +b \right )^{2}}+\frac {d^{3} b \left (8 a^{2} c^{2}+12 a b c +5 b^{2}\right ) \ln \left (\frac {2 a \,c^{2}+2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{32 \left (a c +b \right )^{2} c^{3} \sqrt {a \,c^{2}+b c}\, \left (a d \,x^{2}+a c +b \right )}\) | \(317\) |
| default | \(\text {Expression too large to display}\) | \(1518\) |
-1/48*(d*x^2+c)*(8*a^2*c^2*d^2*x^4+26*a*b*c*d^2*x^4-8*a^2*c^3*d*x^2+15*b^2 *d^2*x^4-18*a*b*c^2*d*x^2+8*a^2*c^4-10*b^2*c*d*x^2+16*a*b*c^3+8*b^2*c^2)/c ^3/x^6/(a*c+b)^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/32*d^3*b*(8*a^2*c^2+1 2*a*b*c+5*b^2)/(a*c+b)^2/c^3/(a*c^2+b*c)^(1/2)*ln((2*a*c^2+2*b*c+(2*a*c*d+ b*d)*x^2+2*(a*c^2+b*c)^(1/2)*(a*c^2+b*c+(2*a*c*d+b*d)*x^2+a*d^2*x^4)^(1/2) )/x^2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2) /(a*d*x^2+a*c+b)
Time = 0.64 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.85 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, dx=\left [\frac {3 \, {\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \sqrt {a c^{2} + b c} d^{3} x^{6} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (8 \, a^{3} c^{7} + {\left (8 \, a^{3} c^{4} + 34 \, a^{2} b c^{3} + 41 \, a b^{2} c^{2} + 15 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} + {\left (8 \, a^{2} b c^{4} + 13 \, a b^{2} c^{3} + 5 \, b^{3} c^{2}\right )} d^{2} x^{4} - 2 \, {\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, {\left (a^{3} c^{7} + 3 \, a^{2} b c^{6} + 3 \, a b^{2} c^{5} + b^{3} c^{4}\right )} x^{6}}, -\frac {3 \, {\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} \sqrt {-a c^{2} - b c} d^{3} x^{6} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{7} + {\left (8 \, a^{3} c^{4} + 34 \, a^{2} b c^{3} + 41 \, a b^{2} c^{2} + 15 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} + {\left (8 \, a^{2} b c^{4} + 13 \, a b^{2} c^{3} + 5 \, b^{3} c^{2}\right )} d^{2} x^{4} - 2 \, {\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, {\left (a^{3} c^{7} + 3 \, a^{2} b c^{6} + 3 \, a b^{2} c^{5} + b^{3} c^{4}\right )} x^{6}}\right ] \]
[1/192*(3*(8*a^2*b*c^2 + 12*a*b^2*c + 5*b^3)*sqrt(a*c^2 + b*c)*d^3*x^6*log (((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 + 4*((2*a*c + b)*d^2*x^4 + 2*a* c^3 + (4*a*c^2 + 3*b*c)*d*x^2 + 2*b*c^2)*sqrt(a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/x^4) - 4*(8*a^3*c^7 + (8*a^3*c^4 + 34*a^2*b*c^3 + 41*a*b^2*c^2 + 15*b^3*c)*d^3*x^6 + 24*a^2*b*c^6 + 24*a*b^2*c^5 + 8*b^3*c^4 + (8*a^2*b*c^4 + 13*a*b^2*c^3 + 5*b^3*c^2)*d^2*x^4 - 2*(a^2*b*c^5 + 2*a*b ^2*c^4 + b^3*c^3)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^7 + 3*a^2*b*c^6 + 3*a*b^2*c^5 + b^3*c^4)*x^6), -1/96*(3*(8*a^2*b*c^2 + 12*a* b^2*c + 5*b^3)*sqrt(-a*c^2 - b*c)*d^3*x^6*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt(-a*c^2 - b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/ (a^2*c^3 + 2*a*b*c^2 + (a^2*c^2 + a*b*c)*d*x^2 + b^2*c)) + 2*(8*a^3*c^7 + (8*a^3*c^4 + 34*a^2*b*c^3 + 41*a*b^2*c^2 + 15*b^3*c)*d^3*x^6 + 24*a^2*b*c^ 6 + 24*a*b^2*c^5 + 8*b^3*c^4 + (8*a^2*b*c^4 + 13*a*b^2*c^3 + 5*b^3*c^2)*d^ 2*x^4 - 2*(a^2*b*c^5 + 2*a*b^2*c^4 + b^3*c^3)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^7 + 3*a^2*b*c^6 + 3*a*b^2*c^5 + b^3*c^4)*x^6)]
\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, dx=\int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{7}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (241) = 482\).
Time = 0.31 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.10 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, dx=-\frac {{\left (8 \, a^{2} b c^{2} + 12 \, a b^{2} c + 5 \, b^{3}\right )} d^{3} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{32 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {3 \, {\left (8 \, a^{2} b c^{4} + 20 \, a b^{2} c^{3} + 11 \, b^{3} c^{2}\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{4} + 18 \, a^{2} b^{2} c^{3} + 17 \, a b^{3} c^{2} + 5 \, b^{4} c\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{4} + 28 \, a^{3} b^{2} c^{3} + 37 \, a^{2} b^{3} c^{2} + 22 \, a b^{4} c + 5 \, b^{5}\right )} d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{5} c^{8} + 5 \, a^{4} b c^{7} + 10 \, a^{3} b^{2} c^{6} + 10 \, a^{2} b^{3} c^{5} + 5 \, a b^{4} c^{4} + b^{5} c^{3} - \frac {{\left (a^{2} c^{8} + 2 \, a b c^{7} + b^{2} c^{6}\right )} {\left (a d x^{2} + a c + b\right )}^{3}}{{\left (d x^{2} + c\right )}^{3}} + \frac {3 \, {\left (a^{3} c^{8} + 3 \, a^{2} b c^{7} + 3 \, a b^{2} c^{6} + b^{3} c^{5}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {3 \, {\left (a^{4} c^{8} + 4 \, a^{3} b c^{7} + 6 \, a^{2} b^{2} c^{6} + 4 \, a b^{3} c^{5} + b^{4} c^{4}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \]
-1/32*(8*a^2*b*c^2 + 12*a*b^2*c + 5*b^3)*d^3*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - sqrt((a*c + b)*c))/(c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) + sqrt((a*c + b)*c)))/((a^2*c^5 + 2*a*b*c^4 + b^2*c^3)*sqrt((a*c + b)* c)) - 1/48*(3*(8*a^2*b*c^4 + 20*a*b^2*c^3 + 11*b^3*c^2)*d^3*((a*d*x^2 + a* c + b)/(d*x^2 + c))^(5/2) - 8*(6*a^3*b*c^4 + 18*a^2*b^2*c^3 + 17*a*b^3*c^2 + 5*b^4*c)*d^3*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) + 3*(8*a^4*b*c^4 + 28*a^3*b^2*c^3 + 37*a^2*b^3*c^2 + 22*a*b^4*c + 5*b^5)*d^3*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^5*c^8 + 5*a^4*b*c^7 + 10*a^3*b^2*c^6 + 10*a^2*b ^3*c^5 + 5*a*b^4*c^4 + b^5*c^3 - (a^2*c^8 + 2*a*b*c^7 + b^2*c^6)*(a*d*x^2 + a*c + b)^3/(d*x^2 + c)^3 + 3*(a^3*c^8 + 3*a^2*b*c^7 + 3*a*b^2*c^6 + b^3* c^5)*(a*d*x^2 + a*c + b)^2/(d*x^2 + c)^2 - 3*(a^4*c^8 + 4*a^3*b*c^7 + 6*a^ 2*b^2*c^6 + 4*a*b^3*c^5 + b^4*c^4)*(a*d*x^2 + a*c + b)/(d*x^2 + c))
Leaf count of result is larger than twice the leaf count of optimal. 1414 vs. \(2 (241) = 482\).
Time = 0.42 (sec) , antiderivative size = 1414, normalized size of antiderivative = 5.34 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, dx=\text {Too large to display} \]
-1/48*(3*(8*a^2*b*c^2*d^3 + 12*a*b^2*c*d^3 + 5*b^3*d^3)*arctan(-(sqrt(a*d^ 2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))/sqrt(-a*c^ 2 - b*c))/((a^2*c^5 + 2*a*b*c^4 + b^2*c^3)*sqrt(-a*c^2 - b*c)) + (64*a^(11 /2)*c^8*d^2*abs(d) + 192*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^5*c^7*d^3 + 192*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x ^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^(9/2)*c^6*d^2*abs(d) + 304* a^(9/2)*b*c^7*d^2*abs(d) + 64*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d* x^2 + b*d*x^2 + a*c^2 + b*c))^3*a^4*c^5*d^3 + 744*(sqrt(a*d^2)*x^2 - sqrt( a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^4*b*c^6*d^3 + 528*(sqr t(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^ (7/2)*b*c^5*d^2*abs(d) + 576*a^(7/2)*b^2*c^6*d^2*abs(d) + 64*(sqrt(a*d^2)* x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*a^3*b*c^4*d ^3 + 1116*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^ 2 + b*c))*a^3*b^2*c^5*d^3 + 480*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c* d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^(5/2)*b^2*c^4*d^2*abs(d) + 544*a^(5/2) *b^3*c^5*d^2*abs(d) + 24*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^5*a^2*b*c^2*d^3 - 96*(sqrt(a*d^2)*x^2 - sqrt(a*d^ 2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*a^2*b^2*c^3*d^3 + 801*(sqr t(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^2* b^3*c^4*d^3 + 144*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d...
Timed out. \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^7} \, dx=\int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^7} \,d x \]