\(\int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx\) [354]

Optimal result

Integrand size = 21, antiderivative size = 435 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\frac {-b-a c-a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(b-a c) d^2 x \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d^{3/2} \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {a \sqrt {c} d^{3/2} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

[Out]

1/3*(-a*d*x^2-a*c-b)/(a*c+b)/x^3/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/3*(-a*c+b)*d*(a*d*x^2+a*c+b)/c/(a*c+b)^2/
x/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/3*(-a*c+b)*d^2*x*(a*d*x^2+a*c+b)/c/(a*c+b)^2/(d*x^2+c)/((a*d*x^2+a*c+b)/
(d*x^2+c))^(1/2)-1/3*(-a*c+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))/(a*c+b)^2/(d*x^2+c)/c^(1/2)/((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*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)^2/(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)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986, 486, 597, 545, 429, 506, 422} \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {a \sqrt {c} d^{3/2} \left (a c+a d x^2+b\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 (a c+b)^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {d^{3/2} (b-a c) \left (a c+a d x^2+b\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (a c+b)^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d^2 x (b-a c) \left (a c+a d x^2+b\right )}{3 c (a c+b)^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {d (b-a c) \left (a c+a d x^2+b\right )}{3 c x (a c+b)^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {a c+a d x^2+b}{3 x^3 (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \]

[In]

Int[1/(x^4*Sqrt[a + b/(c + d*x^2)]),x]

[Out]

-1/3*(b + a*c + a*d*x^2)/((b + a*c)*x^3*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]) - ((b - a*c)*d*(b + a*c + a*d*x
^2))/(3*c*(b + a*c)^2*x*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]) + ((b - a*c)*d^2*x*(b + a*c + a*d*x^2))/(3*c*(b
+ a*c)^2*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]) - ((b - a*c)*d^(3/2)*(b + a*c + a*d*x^2)*Elliptic
E[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(3*Sqrt[c]*(b + a*c)^2*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c +
d*x^2)]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]) - (a*Sqrt[c]*d^(3/2)*(b + a*c + a*d*x^2)*Ellipt
icF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(3*(b + a*c)^2*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)
]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[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]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 486

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*
x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x
], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] &&
IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 506

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] - Dist[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]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 1985

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]

Rule 1986

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^(r_.))^(p_), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^4 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \, dx \\ & = \frac {\sqrt {b+a c+a d x^2} \int \frac {\sqrt {c+d x^2}}{x^4 \sqrt {b+a c+a d x^2}} \, dx}{\sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\sqrt {b+a c+a d x^2} \int \frac {(b-a c) d-a d^2 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 (b+a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {b+a c+a d x^2} \int \frac {a c (b+a c) d^2-a (b-a c) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c (b+a c)^2 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\left (a d^2 \sqrt {b+a c+a d x^2}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 (b+a c) \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (a (b-a c) d^3 \sqrt {b+a c+a d x^2}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 c (b+a c)^2 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(b-a c) d^2 x \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {a \sqrt {c} d^{3/2} \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^2 \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 {\left ((b-a c) d^2 \sqrt {b+a c+a d x^2}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 (b+a c)^2 \sqrt {c+d x^2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}} \\ & = -\frac {b+a c+a d x^2}{3 (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(b-a c) d^2 x \left (b+a c+a d x^2\right )}{3 c (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d^{3/2} \left (b+a c+a d x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 \sqrt {c} (b+a c)^2 \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 {a \sqrt {c} d^{3/2} \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^2 \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 )}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.85 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {\sqrt {\frac {d}{c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (b^2 \left (c+d x^2\right )+a^2 c \left (c^2-d^2 x^4\right )+a b \left (2 c^2+c d x^2+d^2 x^4\right )\right )+i \left (b^2-a^2 c^2\right ) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-i b (b+a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 (b+a c)^2 d x^3 \left (b+a \left (c+d x^2\right )\right )} \]

[In]

Integrate[1/(x^4*Sqrt[a + b/(c + d*x^2)]),x]

[Out]

-1/3*(Sqrt[d/c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[d/c]*(c + d*x^2)*(b^2*(c + d*x^2) + a^2*c*(c^2 - d
^2*x^4) + a*b*(2*c^2 + c*d*x^2 + d^2*x^4)) + I*(b^2 - a^2*c^2)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqr
t[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)] - 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)^2*d*x^3*
(b + a*(c + d*x^2)))

Maple [A] (verified)

Time = 7.00 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.31

method	result	size
risch	\(-\frac {\left (a d \,x^{2}+a c +b \right ) \left (-a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )}{3 \left (a c +b \right )^{2} x^{3} c \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {d^{2} a \left (\frac {a \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {b c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (a c d -b d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{3 \left (a c +b \right )^{2} c \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(568\)
default	\(-\frac {\left (-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c \,d^{3} x^{6}+\sqrt {-\frac {a d}{a c +b}}\, a b \,d^{3} x^{6}+\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} c^{2} d^{2} x^{3}-\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{2} d^{2} x^{4}+2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}-\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a b c \,d^{2} x^{3}+2 \sqrt {-\frac {a d}{a c +b}}\, a b c \,d^{2} x^{4}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{3} d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} d^{2} x^{4}+3 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{2} d \,x^{2}+\sqrt {-\frac {a d}{a c +b}}\, a^{2} c^{4}+2 \sqrt {-\frac {a d}{a c +b}}\, b^{2} c d \,x^{2}+2 \sqrt {-\frac {a d}{a c +b}}\, a b \,c^{3}+\sqrt {-\frac {a d}{a c +b}}\, b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, c \,x^{3} \left (a c +b \right )^{2} \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\)	\(593\)

[In]

int(1/x^4/(a+b/(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(a*d*x^2+a*c+b)*(-a*c*d*x^2+b*d*x^2+a*c^2+b*c)/(a*c+b)^2/x^3/c/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/3*d^2*
a/(a*c+b)^2/c*(a*c^2/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2
+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))+b*c/(-a*d/(a*c+b))^
(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(
x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-2*(a*c*d-b*d)*(a*c^2+b*c)/(-a*d/(a*c+b))^(1/2)*(1+a*d/(
a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(2*a*c*d+2*b*d)*(Ellipti
cF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-EllipticE(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d
/c/a)^(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)/(d*x^2+c)

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=-\frac {{\left (a^{2} c - a b\right )} \sqrt {-\frac {a d}{a c + b}} d^{3} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c - a b\right )} d^{3} + {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} d^{2}\right )} \sqrt {-\frac {a d}{a c + b}} x^{3} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c^{2} - b^{2}\right )} d^{2} x^{4} - a^{2} c^{4} - 2 \, a b c^{3} - b^{2} c^{2} - 2 \, {\left (a b c^{2} + b^{2} c\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, {\left (a^{3} c^{4} + 3 \, a^{2} b c^{3} + 3 \, a b^{2} c^{2} + b^{3} c\right )} x^{3}} \]

[In]

integrate(1/x^4/(a+b/(d*x^2+c))^(1/2),x, algorithm="fricas")

[Out]

-1/3*((a^2*c - a*b)*sqrt(-a*d/(a*c + b))*d^3*x^3*sqrt((a*c^2 + b*c)/d^2)*elliptic_e(arcsin(sqrt(-a*d/(a*c + b)
)*x), (a*c + b)/(a*c)) - ((a^2*c - a*b)*d^3 + (a^2*c^2 + 2*a*b*c + b^2)*d^2)*sqrt(-a*d/(a*c + b))*x^3*sqrt((a*
c^2 + b*c)/d^2)*elliptic_f(arcsin(sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a*c)) - ((a^2*c^2 - b^2)*d^2*x^4 - a^2*c
^4 - 2*a*b*c^3 - b^2*c^2 - 2*(a*b*c^2 + b^2*c)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^3*c^4 + 3*a^2
*b*c^3 + 3*a*b^2*c^2 + b^3*c)*x^3)

Sympy [F]

\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^{4} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \]

[In]

integrate(1/x**4/(a+b/(d*x**2+c))**(1/2),x)

[Out]

Integral(1/(x**4*sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))), x)

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{d x^{2} + c}} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(a+b/(d*x^2+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a + b/(d*x^2 + c))*x^4), x)

Giac [F]

\[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int { \frac {1}{\sqrt {a + \frac {b}{d x^{2} + c}} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(a+b/(d*x^2+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/(d*x^2 + c))*x^4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+\frac {b}{c+d x^2}}} \, dx=\int \frac {1}{x^4\,\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \]

[In]

int(1/(x^4*(a + b/(c + d*x^2))^(1/2)),x)

[Out]

int(1/(x^4*(a + b/(c + d*x^2))^(1/2)), x)