Integrand size = 21, antiderivative size = 292 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx=-\frac {b d^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c^4}-\frac {\left (79 b^2+108 a b c+24 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{48 c^4 (b+a c) x^2}+\frac {(11 b+12 a c) d \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{24 c^4 x^4}-\frac {\left (c+d x^2\right )^3 \left (\frac {b+a c+a d x^2}{c+d x^2}\right )^{5/2}}{6 c^2 (b+a c) x^6}+\frac {b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{16 c^{9/2} (b+a c)^{3/2}} \]
-1/6*(d*x^2+c)^3*((a*d*x^2+a*c+b)/(d*x^2+c))^(5/2)/c^2/(a*c+b)/x^6+1/16*b* (24*a^2*c^2+60*a*b*c+35*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^(9/2)/(a*c+b)^(3/2)-b*d^3*((a*d*x^2+a*c+b)/(d*x^ 2+c))^(1/2)/c^4-1/48*(24*a^2*c^2+108*a*b*c+79*b^2)*d^2*(d*x^2+c)*((a*d*x^2 +a*c+b)/(d*x^2+c))^(1/2)/c^4/(a*c+b)/x^2+1/24*(12*a*c+11*b)*d*(d*x^2+c)^2* ((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^4/x^4
Time = 0.57 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\frac {-\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (8 a^2 c^2 \left (c^3+d^3 x^6\right )+2 a b c \left (8 c^3-7 c^2 d x^2+16 c d^2 x^4+55 d^3 x^6\right )+b^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )\right )}{(b+a c) x^6}+\frac {3 b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{(-b-a c)^{3/2}}}{48 c^{9/2}} \]
(-((Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(8*a^2*c^2*(c^3 + d^3*x^ 6) + 2*a*b*c*(8*c^3 - 7*c^2*d*x^2 + 16*c*d^2*x^4 + 55*d^3*x^6) + b^2*(8*c^ 3 - 14*c^2*d*x^2 + 35*c*d^2*x^4 + 105*d^3*x^6)))/((b + a*c)*x^6)) + (3*b*( 35*b^2 + 60*a*b*c + 24*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]])/(-b - a*c)^(3/2))/(48*c^(9/2))
Time = 0.57 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {2057, 2053, 2052, 27, 366, 360, 25, 1471, 27, 299, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {\left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}{x^7}dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (\frac {a d x^2+b+a c}{d x^2+c}\right )^{3/2}}{x^8}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle -b d \int \frac {d^2 x^8 \left (a-x^4\right )^2}{\left (-c x^4+b+a c\right )^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b d^3 \int \frac {x^8 \left (a-x^4\right )^2}{\left (-c x^4+b+a c\right )^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 366 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^{10}}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\int \frac {x^8 \left (6 c (b+a c) x^4+5 b^2-6 a^2 c^2\right )}{\left (-c x^4+b+a c\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{6 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^{10}}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {\int -\frac {24 c^3 (b+a c) x^8+4 b c^2 (11 b+12 a c) x^4+b c (b+a c) (11 b+12 a c)}{\left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c^3}+\frac {b (a c+b) (12 a c+11 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}}{6 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^{10}}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (a c+b) (12 a c+11 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}-\frac {\int \frac {24 c^3 (b+a c) x^8+4 b c^2 (11 b+12 a c) x^4+b c (b+a c) (11 b+12 a c)}{\left (-c x^4+b+a c\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c^3}}{6 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^{10}}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (a c+b) (12 a c+11 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}-\frac {\frac {c \left (24 a^2 c^2+108 a b c+79 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a c+b-c x^4\right )}-\frac {\int \frac {3 c (b+a c) \left (16 c (b+a c) x^4+19 b^2+8 a^2 c^2+28 a b c\right )}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{2 (a c+b)}}{4 c^3}}{6 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^{10}}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (a c+b) (12 a c+11 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}-\frac {\frac {c \left (24 a^2 c^2+108 a b c+79 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a c+b-c x^4\right )}-\frac {3}{2} c \int \frac {16 c (b+a c) x^4+19 b^2+8 a^2 c^2+28 a b c}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c^3}}{6 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^{10}}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (a c+b) (12 a c+11 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}-\frac {\frac {c \left (24 a^2 c^2+108 a b c+79 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a c+b-c x^4\right )}-\frac {3}{2} c \left (\left (24 a^2 c^2+60 a b c+35 b^2\right ) \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}-16 (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}\right )}{4 c^3}}{6 c^2 (a c+b)}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -b d^3 \left (\frac {b^2 x^{10}}{6 c^2 (a c+b) \left (a c+b-c x^4\right )^3}-\frac {\frac {b (a c+b) (12 a c+11 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 \left (a c+b-c x^4\right )^2}-\frac {\frac {c \left (24 a^2 c^2+108 a b c+79 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a c+b-c x^4\right )}-\frac {3}{2} c \left (\frac {\left (24 a^2 c^2+60 a b c+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{\sqrt {c} \sqrt {a c+b}}-16 (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}\right )}{4 c^3}}{6 c^2 (a c+b)}\right )\) |
-(b*d^3*((b^2*x^10)/(6*c^2*(b + a*c)*(b + a*c - c*x^4)^3) - ((b*(b + a*c)* (11*b + 12*a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(4*c^2*(b + a*c - c *x^4)^2) - ((c*(79*b^2 + 108*a*b*c + 24*a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/ (c + d*x^2)])/(2*(b + a*c - c*x^4)) - (3*c*(-16*(b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)] + ((35*b^2 + 60*a*b*c + 24*a^2*c^2)*ArcTanh[(Sqrt[c] *Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]])/(Sqrt[c]*Sqrt[b + a*c])))/2)/(4*c^3))/(6*c^2*(b + a*c))))
3.4.37.3.1 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[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -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 = 0.26 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.27
| method | result | size |
| risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (8 a^{2} c^{2} d^{2} x^{4}+62 a c \,d^{2} b \,x^{4}-8 a^{2} c^{3} d \,x^{2}+57 b^{2} d^{2} x^{4}-30 a b \,c^{2} d \,x^{2}+8 a^{2} c^{4}-22 b^{2} c d \,x^{2}+16 a b \,c^{3}+8 b^{2} c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{48 c^{4} x^{6} \left (a c +b \right )}-\frac {d^{3} b \left (-\frac {\left (24 a^{2} c^{2}+60 a b c +35 b^{2}\right ) \ln \left (\frac {2 a \,c^{2}+2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right )}{2 \sqrt {a \,c^{2}+b c}}+\frac {16 \left (a c +b \right ) \left (a d \,x^{2}+a c +b \right )}{\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{16 c^{4} \left (a c +b \right ) \left (a d \,x^{2}+a c +b \right )}\) | \(371\) |
| default | \(\text {Expression too large to display}\) | \(2605\) |
-1/48*(d*x^2+c)*(8*a^2*c^2*d^2*x^4+62*a*b*c*d^2*x^4-8*a^2*c^3*d*x^2+57*b^2 *d^2*x^4-30*a*b*c^2*d*x^2+8*a^2*c^4-22*b^2*c*d*x^2+16*a*b*c^3+8*b^2*c^2)/c ^4/x^6/(a*c+b)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/16*d^3*b/c^4/(a*c+b)*(- 1/2*(24*a^2*c^2+60*a*b*c+35*b^2)/(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)+16*(a*c+b)*(a*d*x^2+a*c+b)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2 +b*c)^(1/2))*((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 = 0.88 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.51 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\left [\frac {3 \, {\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, 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} + 118 \, a^{2} b c^{3} + 215 \, a b^{2} c^{2} + 105 \, 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 (32 \, a^{2} b c^{4} + 67 \, a b^{2} c^{3} + 35 \, b^{3} c^{2}\right )} d^{2} x^{4} - 14 \, {\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^{2} c^{7} + 2 \, a b c^{6} + b^{2} c^{5}\right )} x^{6}}, -\frac {3 \, {\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, 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} + 118 \, a^{2} b c^{3} + 215 \, a b^{2} c^{2} + 105 \, 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 (32 \, a^{2} b c^{4} + 67 \, a b^{2} c^{3} + 35 \, b^{3} c^{2}\right )} d^{2} x^{4} - 14 \, {\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^{2} c^{7} + 2 \, a b c^{6} + b^{2} c^{5}\right )} x^{6}}\right ] \]
[1/192*(3*(24*a^2*b*c^2 + 60*a*b^2*c + 35*b^3)*sqrt(a*c^2 + b*c)*d^3*x^6*l og(((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 + 118*a^2*b*c^3 + 215*a*b^2*c^2 + 105*b^3*c)*d^3*x^6 + 24*a^2*b*c^6 + 24*a*b^2*c^5 + 8*b^ 3*c^4 + (32*a^2*b*c^4 + 67*a*b^2*c^3 + 35*b^3*c^2)*d^2*x^4 - 14*(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^2*c^7 + 2*a*b*c^6 + b^2*c^5)*x^6), -1/96*(3*(24*a^2*b*c^2 + 60*a*b^2*c + 35*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 + 118*a^2*b*c^3 + 215*a*b^2*c^2 + 105*b^3*c)*d^3*x^6 + 24*a^2*b*c^6 + 24*a*b^2*c^5 + 8*b^3*c^4 + (32*a^2*b*c^4 + 67*a*b^2*c^3 + 35*b^3*c^2)*d^2 *x^4 - 14*(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^2*c^7 + 2*a*b*c^6 + b^2*c^5)*x^6)]
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{7}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (266) = 532\).
Time = 0.32 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx=-\frac {{\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, b^{3}\right )} d^{3} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{32 \, {\left (a c^{5} + b c^{4}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {b d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{c^{4}} - \frac {3 \, {\left (8 \, a^{2} b c^{4} + 36 \, a b^{2} c^{3} + 29 \, b^{3} c^{2}\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{4} + 30 \, a^{2} b^{2} c^{3} + 41 \, a b^{3} c^{2} + 17 \, b^{4} c\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{4} + 44 \, a^{3} b^{2} c^{3} + 83 \, a^{2} b^{3} c^{2} + 66 \, a b^{4} c + 19 \, b^{5}\right )} d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{4} c^{8} + 4 \, a^{3} b c^{7} + 6 \, a^{2} b^{2} c^{6} + 4 \, a b^{3} c^{5} + b^{4} c^{4} - \frac {{\left (a c^{8} + b c^{7}\right )} {\left (a d x^{2} + a c + b\right )}^{3}}{{\left (d x^{2} + c\right )}^{3}} + \frac {3 \, {\left (a^{2} c^{8} + 2 \, a b c^{7} + b^{2} c^{6}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {3 \, {\left (a^{3} c^{8} + 3 \, a^{2} b c^{7} + 3 \, a b^{2} c^{6} + b^{3} c^{5}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \]
-1/32*(24*a^2*b*c^2 + 60*a*b^2*c + 35*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*c^5 + b*c^4)*sqrt((a*c + b)*c)) - b*d^3*sq rt((a*d*x^2 + a*c + b)/(d*x^2 + c))/c^4 - 1/48*(3*(8*a^2*b*c^4 + 36*a*b^2* c^3 + 29*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 + 30*a^2*b^2*c^3 + 41*a*b^3*c^2 + 17*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 + 44*a^3*b^2*c^3 + 83*a^2*b^3*c^2 + 66 *a*b^4*c + 19*b^5)*d^3*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(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*c^8 + b*c^7)*(a*d* x^2 + a*c + b)^3/(d*x^2 + c)^3 + 3*(a^2*c^8 + 2*a*b*c^7 + b^2*c^6)*(a*d*x^ 2 + a*c + b)^2/(d*x^2 + c)^2 - 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)/(d*x^2 + c))
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}}{x^7} \,d x \]