Integrand size = 21, antiderivative size = 490 \[ \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {b}{a (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(3 b-a c) \left (b+a c+a d x^2\right )}{3 a (b+a c)^2 x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(7 b-a c) d \left (b+a c+a d x^2\right )}{3 (b+a c)^3 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(7 b-a c) d^2 x \left (b+a c+a d x^2\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} (7 b-a c) d^{3/2} \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {\sqrt {c} (3 b-a c) d^{3/2} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
-b/a/(a*c+b)/x^3/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/3*(-a*c+3*b)*(a*d*x^2 +a*c+b)/a/(a*c+b)^2/x^3/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/3*(-a*c+7*b)*d *(a*d*x^2+a*c+b)/(a*c+b)^3/x/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/3*(-a*c+7 *b)*d^2*x*(a*d*x^2+a*c+b)/(a*c+b)^3/(d*x^2+c)/((a*d*x^2+a*c+b)/(d*x^2+c))^ (1/2)-1/3*(-a*c+7*b)*d^(3/2)*(a*d*x^2+a*c+b)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^ 2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(b/(a*c+b))^(1/2) )*c^(1/2)/(a*c+b)^3/(d*x^2+c)/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(c*(a*d*x^ 2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)+1/3*(-a*c+3*b)*d^(3/2)*(a*d*x^2+a*c+b)*( 1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1+d*x^ 2/c)^(1/2),(b/(a*c+b))^(1/2))*c^(1/2)/(a*c+b)^3/(d*x^2+c)/((a*d*x^2+a*c+b) /(d*x^2+c))^(1/2)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 10.79 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (b^2 \left (c+4 d x^2\right )+a^2 c \left (c^2-d^2 x^4\right )+a b \left (2 c^2+4 c d x^2+7 d^2 x^4\right )\right )+i \left (7 b^2+6 a b c-a^2 c^2\right ) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-4 i b (b+a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 (b+a c)^3 \sqrt {\frac {d}{c}} x^3 \left (b+a \left (c+d x^2\right )\right )} \]
-1/3*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[d/c]*(c + d*x^2)*(b^2*(c + 4*d*x^2) + a^2*c*(c^2 - d^2*x^4) + a*b*(2*c^2 + 4*c*d*x^2 + 7*d^2*x^4)) + I*(7*b^2 + 6*a*b*c - a^2*c^2)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c )]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)] - (4*I)*b*(b + a*c)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + ( d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)]))/((b + a*c)^ 3*Sqrt[d/c]*x^3*(b + a*(c + d*x^2)))
Time = 0.77 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {2057, 2058, 370, 27, 445, 27, 445, 25, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {1}{x^4 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \int \frac {\left (d x^2+c\right )^{3/2}}{x^4 \left (a d x^2+b+a c\right )^{3/2}}dx}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 370 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {\int \frac {d \left ((2 b-a c) d x^2+c (3 b-a c)\right )}{x^4 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{a d (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {\int \frac {(2 b-a c) d x^2+c (3 b-a c)}{x^4 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {\int \frac {a c d \left ((3 b-a c) d x^2+c (7 b-a c)\right )}{x^2 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 c (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {a d \int \frac {(3 b-a c) d x^2+c (7 b-a c)}{x^2 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {a d \left (-\frac {\int -\frac {c d \left (a (7 b-a c) d x^2+(3 b-a c) (b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c (a c+b)}-\frac {(7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{x (a c+b)}\right )}{3 (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {a d \left (\frac {\int \frac {c d \left (a (7 b-a c) d x^2+(3 b-a c) (b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c (a c+b)}-\frac {(7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{x (a c+b)}\right )}{3 (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {a d \left (\frac {d \int \frac {a (7 b-a c) d x^2+(3 b-a c) (b+a c)}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{a c+b}-\frac {(7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{x (a c+b)}\right )}{3 (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {a d \left (\frac {d \left ((3 b-a c) (a c+b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+a d (7 b-a c) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx\right )}{a c+b}-\frac {(7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{x (a c+b)}\right )}{3 (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {a d \left (\frac {d \left (a d (7 b-a c) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {\sqrt {c} (3 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{a c+b}-\frac {(7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{x (a c+b)}\right )}{3 (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {a d \left (\frac {d \left (a d (7 b-a c) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {\sqrt {c} (3 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{a c+b}-\frac {(7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{x (a c+b)}\right )}{3 (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (-\frac {-\frac {a d \left (\frac {d \left (a d (7 b-a c) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {\sqrt {c} (3 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{a c+b}-\frac {(7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{x (a c+b)}\right )}{3 (a c+b)}-\frac {(3 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 x^3 (a c+b)}}{a (a c+b)}-\frac {b \sqrt {c+d x^2}}{a x^3 (a c+b) \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
(Sqrt[b + a*c + a*d*x^2]*(-((b*Sqrt[c + d*x^2])/(a*(b + a*c)*x^3*Sqrt[b + a*c + a*d*x^2])) - (-1/3*((3*b - a*c)*Sqrt[c + d*x^2]*Sqrt[b + a*c + a*d*x ^2])/((b + a*c)*x^3) - (a*d*(-(((7*b - a*c)*Sqrt[c + d*x^2]*Sqrt[b + a*c + a*d*x^2])/((b + a*c)*x)) + (d*(a*(7*b - a*c)*d*((x*Sqrt[b + a*c + a*d*x^2 ])/(a*d*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[b + a*c + a*d*x^2]*EllipticE[ArcT an[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(a*d^(3/2)*Sqrt[c + d*x^2]*Sqrt[(c* (b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))])) + (Sqrt[c]*(3*b - a*c)*Sqr t[b + a*c + a*d*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/ (Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^ 2))])))/(b + a*c)))/(3*(b + a*c)))/(a*(b + a*c))))/(Sqrt[c + d*x^2]*Sqrt[( b + a*c + a*d*x^2)/(c + d*x^2)])
3.4.65.3.1 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[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(e*x) ^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a *d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(1079\) vs. \(2(526)=1052\).
Time = 10.81 (sec) , antiderivative size = 1080, normalized size of antiderivative = 2.20
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1080\) |
| risch | \(\text {Expression too large to display}\) | \(1142\) |
-1/3*(-((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a^2*c*d^3*x^ 6+4*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*b*d^3*x^6+((a *d*x^2+a*c+b)*(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2+c)/ c)^(1/2)*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a^2*c^2*d^2 *x^3+3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(-a*d/(a*c+b))^(1/2 )*a*b*d^3*x^6-((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a^2*c ^2*d^2*x^4+5*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)/(a*c+b))^( 1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1 /2))*a*b*c*d^2*x^3-7*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)/(a *c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b) /a/c)^(1/2))*a*b*c*d^2*x^3+8*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+ b))^(1/2)*a*b*c*d^2*x^4-3*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*((a*d*x^2+a*c+ b)/(a*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),((a *c+b)/a/c)^(1/2))*b^2*d^2*x^3+3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^ (1/2)*(-a*d/(a*c+b))^(1/2)*a*b*c*d^2*x^4+((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2) *(-a*d/(a*c+b))^(1/2)*a^2*c^3*d*x^2+4*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(- a*d/(a*c+b))^(1/2)*b^2*d^2*x^4+6*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/( a*c+b))^(1/2)*a*b*c^2*d*x^2+((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b ))^(1/2)*a^2*c^4+5*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)* b^2*c*d*x^2+2*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*...
Time = 0.12 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {{\left ({\left (a^{4} c^{2} - 7 \, a^{3} b c\right )} d^{4} x^{5} + {\left (a^{4} c^{3} - 6 \, a^{3} b c^{2} - 7 \, a^{2} b^{2} c\right )} d^{3} x^{3}\right )} \sqrt {-\frac {a d}{a c + b}} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left ({\left (a^{4} c^{2} - 7 \, a^{3} b c\right )} d^{4} + {\left (a^{4} c^{3} - a^{3} b c^{2} - 5 \, a^{2} b^{2} c - 3 \, a b^{3}\right )} d^{3}\right )} x^{5} + {\left ({\left (a^{4} c^{3} - 6 \, a^{3} b c^{2} - 7 \, a^{2} b^{2} c\right )} d^{3} + {\left (a^{4} c^{4} - 6 \, a^{2} b^{2} c^{2} - 8 \, a b^{3} c - 3 \, b^{4}\right )} d^{2}\right )} x^{3}\right )} \sqrt {-\frac {a d}{a c + b}} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) + {\left (a^{4} c^{6} - {\left (a^{4} c^{3} - 6 \, a^{3} b c^{2} - 7 \, a^{2} b^{2} c\right )} d^{3} x^{6} + 3 \, a^{3} b c^{5} + 3 \, a^{2} b^{2} c^{4} + a b^{3} c^{3} - {\left (a^{4} c^{4} - 10 \, a^{3} b c^{3} - 15 \, a^{2} b^{2} c^{2} - 4 \, a b^{3} c\right )} d^{2} x^{4} + {\left (a^{4} c^{5} + 7 \, a^{3} b c^{4} + 11 \, a^{2} b^{2} c^{3} + 5 \, a b^{3} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, {\left ({\left (a^{6} c^{5} + 4 \, a^{5} b c^{4} + 6 \, a^{4} b^{2} c^{3} + 4 \, a^{3} b^{3} c^{2} + a^{2} b^{4} c\right )} d x^{5} + {\left (a^{6} c^{6} + 5 \, a^{5} b c^{5} + 10 \, a^{4} b^{2} c^{4} + 10 \, a^{3} b^{3} c^{3} + 5 \, a^{2} b^{4} c^{2} + a b^{5} c\right )} x^{3}\right )}} \]
-1/3*(((a^4*c^2 - 7*a^3*b*c)*d^4*x^5 + (a^4*c^3 - 6*a^3*b*c^2 - 7*a^2*b^2* c)*d^3*x^3)*sqrt(-a*d/(a*c + b))*sqrt((a*c^2 + b*c)/d^2)*elliptic_e(arcsin (sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a*c)) - (((a^4*c^2 - 7*a^3*b*c)*d^4 + (a^4*c^3 - a^3*b*c^2 - 5*a^2*b^2*c - 3*a*b^3)*d^3)*x^5 + ((a^4*c^3 - 6*a^ 3*b*c^2 - 7*a^2*b^2*c)*d^3 + (a^4*c^4 - 6*a^2*b^2*c^2 - 8*a*b^3*c - 3*b^4) *d^2)*x^3)*sqrt(-a*d/(a*c + b))*sqrt((a*c^2 + b*c)/d^2)*elliptic_f(arcsin( sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a*c)) + (a^4*c^6 - (a^4*c^3 - 6*a^3*b* c^2 - 7*a^2*b^2*c)*d^3*x^6 + 3*a^3*b*c^5 + 3*a^2*b^2*c^4 + a*b^3*c^3 - (a^ 4*c^4 - 10*a^3*b*c^3 - 15*a^2*b^2*c^2 - 4*a*b^3*c)*d^2*x^4 + (a^4*c^5 + 7* a^3*b*c^4 + 11*a^2*b^2*c^3 + 5*a*b^3*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/ (d*x^2 + c)))/((a^6*c^5 + 4*a^5*b*c^4 + 6*a^4*b^2*c^3 + 4*a^3*b^3*c^2 + a^ 2*b^4*c)*d*x^5 + (a^6*c^6 + 5*a^5*b*c^5 + 10*a^4*b^2*c^4 + 10*a^3*b^3*c^3 + 5*a^2*b^4*c^2 + a*b^5*c)*x^3)
\[ \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^{4} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
\[ \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^4\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \]