3.330 \(\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^6} \, dx\)

Optimal. Leaf size=466 \[ \frac{d^3 x \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{15 c^3 (a c+b)^2}-\frac{d^2 \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{15 c^3 x (a c+b)^2}-\frac{d^{5/2} \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 c^{5/2} (a c+b)^2 \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{a d^{5/2} (3 a c+4 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 c^{3/2} (a c+b)^2 \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{d (3 a c+4 b) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{15 c^2 x^3 (a c+b)}-\frac{\left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 c x^5} \]

[Out]

((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^3*x*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(15*c^3*(b + a*c)^2) - ((c + d*x^
2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(5*c*x^5) + ((4*b + 3*a*c)*d*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c
+ d*x^2)])/(15*c^2*(b + a*c)*x^3) - ((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^2*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/
(c + d*x^2)])/(15*c^3*(b + a*c)^2*x) - ((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^(5/2)*Sqrt[(b + a*c + a*d*x^2)/(c + d
*x^2)]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(15*c^(5/2)*(b + a*c)^2*Sqrt[(c*(b + a*c + a*d*x^2
))/((b + a*c)*(c + d*x^2))]) + (a*(4*b + 3*a*c)*d^(5/2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*EllipticF[ArcTan
[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(15*c^(3/2)*(b + a*c)^2*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^
2))])

________________________________________________________________________________________

Rubi [A]  time = 0.811981, antiderivative size = 598, normalized size of antiderivative = 1.28, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {6722, 1975, 475, 583, 531, 418, 492, 411} \[ \frac{d^3 x \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^3 (a c+b)^2 \sqrt{a \left (c+d x^2\right )+b}}-\frac{d^2 \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (c+d x^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^3 x (a c+b)^2 \sqrt{a \left (c+d x^2\right )+b}}-\frac{d^{5/2} \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 c^{5/2} (a c+b)^2 \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}}+\frac{a d^{5/2} (3 a c+4 b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 c^{3/2} (a c+b)^2 \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}}+\frac{d (3 a c+4 b) \left (c+d x^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^2 x^3 (a c+b) \sqrt{a \left (c+d x^2\right )+b}}-\frac{\left (c+d x^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{5 c x^5 \sqrt{a \left (c+d x^2\right )+b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/(c + d*x^2)]/x^6,x]

[Out]

((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^3*x*Sqrt[b + a*c + a*d*x^2]*Sqrt[a + b/(c + d*x^2)])/(15*c^3*(b + a*c)^2*Sqr
t[b + a*(c + d*x^2)]) - ((c + d*x^2)*Sqrt[b + a*c + a*d*x^2]*Sqrt[a + b/(c + d*x^2)])/(5*c*x^5*Sqrt[b + a*(c +
d*x^2)]) + ((4*b + 3*a*c)*d*(c + d*x^2)*Sqrt[b + a*c + a*d*x^2]*Sqrt[a + b/(c + d*x^2)])/(15*c^2*(b + a*c)*x^
3*Sqrt[b + a*(c + d*x^2)]) - ((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^2*(c + d*x^2)*Sqrt[b + a*c + a*d*x^2]*Sqrt[a +
b/(c + d*x^2)])/(15*c^3*(b + a*c)^2*x*Sqrt[b + a*(c + d*x^2)]) - ((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^(5/2)*Sqrt[
b + a*c + a*d*x^2]*Sqrt[a + b/(c + d*x^2)]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(15*c^(5/2)*(b
+ a*c)^2*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]*Sqrt[b + a*(c + d*x^2)]) + (a*(4*b + 3*a*c)*d^
(5/2)*Sqrt[b + a*c + a*d*x^2]*Sqrt[a + b/(c + d*x^2)]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(15
*c^(3/2)*(b + a*c)^2*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]*Sqrt[b + a*(c + d*x^2)])

Rule 6722

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[(a + b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b + a/
v^n)^FracPart[p]), Int[u*v^(n*p)*(b + a/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[p] && ILtQ[n, 0] &
& BinomialQ[v, x] &&  !LinearQ[v, x]

Rule 1975

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 475

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 583

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 531

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 418

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

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^6} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a \left (c+d x^2\right )}}{x^6 \sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a c+a d x^2}}{x^6 \sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{5 c x^5 \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{-(4 b+3 a c) d-3 a d^2 x^2}{x^4 \sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{5 c \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{5 c x^5 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(4 b+3 a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{-\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2-a (4 b+3 a c) d^3 x^2}{x^2 \sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{15 c^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{5 c x^5 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(4 b+3 a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^3 (b+a c)^2 x \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{a c (b+a c) (4 b+3 a c) d^3+a \left (8 b^2+13 a b c+3 a^2 c^2\right ) d^4 x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{15 c^3 (b+a c)^2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{5 c x^5 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(4 b+3 a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^3 (b+a c)^2 x \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left (a (4 b+3 a c) d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{15 c^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left (a \left (8 b^2+13 a b c+3 a^2 c^2\right ) d^4 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{15 c^3 (b+a c)^2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^3 (b+a c)^2 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{5 c x^5 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(4 b+3 a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^3 (b+a c)^2 x \sqrt{b+a \left (c+d x^2\right )}}+\frac{a (4 b+3 a c) d^{5/2} \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 c^{3/2} (b+a c)^2 \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c^2 (b+a c)^2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^3 (b+a c)^2 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{5 c x^5 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(4 b+3 a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{15 c^3 (b+a c)^2 x \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^{5/2} \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 c^{5/2} (b+a c)^2 \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}+\frac{a (4 b+3 a c) d^{5/2} \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{15 c^{3/2} (b+a c)^2 \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}\\ \end{align*}

Mathematica [C]  time = 1.06506, size = 402, normalized size = 0.86 \[ -\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (\left (c+d x^2\right ) \sqrt{\frac{a d}{a c+b}} \left (a^2 b c \left (-4 c^2 d x^2+9 c^3+9 c d^2 x^4+13 d^3 x^6\right )+3 a^3 c^2 \left (c^3+d^3 x^6\right )+a b^2 \left (-8 c^2 d x^2+9 c^3+17 c d^2 x^4+8 d^3 x^6\right )+b^3 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )\right )+i a c d^3 x^5 \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )-2 i a b c d^3 x^5 (3 a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )\right )}{15 c^3 x^5 (a c+b)^2 \sqrt{\frac{a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/(c + d*x^2)]/x^6,x]

[Out]

-(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(c + d*x^2)*(b^3*(3*c^2 - 4*c*d*x^2 + 8*d^2*x^4
) + 3*a^3*c^2*(c^3 + d^3*x^6) + a*b^2*(9*c^3 - 8*c^2*d*x^2 + 17*c*d^2*x^4 + 8*d^3*x^6) + a^2*b*c*(9*c^3 - 4*c^
2*d*x^2 + 9*c*d^2*x^4 + 13*d^3*x^6)) + I*a*c*(8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^3*x^5*Sqrt[(b + a*c + a*d*x^2)/(
b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - (2*I)*a*b*c*(2*b +
3*a*c)*d^3*x^5*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c
)]*x], 1 + b/(a*c)]))/(15*c^3*(b + a*c)^2*Sqrt[(a*d)/(b + a*c)]*x^5*(b + a*(c + d*x^2)))

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Maple [A]  time = 0.026, size = 955, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/(d*x^2+c))^(1/2)/x^6,x)

[Out]

-1/15*(3*(-a*d/(a*c+b))^(1/2)*x^8*a^3*c^2*d^4+13*(-a*d/(a*c+b))^(1/2)*x^8*a^2*b*c*d^4-3*((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))*x^5*a^3*c^3*d^3+8*(-a*d/(a
*c+b))^(1/2)*x^8*a*b^2*d^4+3*(-a*d/(a*c+b))^(1/2)*x^6*a^3*c^3*d^3+6*((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))*x^5*a^2*b*c^2*d^3-13*((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))*x^5*a^2*b*c^2*d^3+22*(-a*d/(a
*c+b))^(1/2)*x^6*a^2*b*c^2*d^3+4*((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))*x^5*a*b^2*c*d^3-8*((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))*x^5*a*b^2*c*d^3+25*(-a*d/(a*c+b))^(1/2)*x^6*a*b^2*c*d^3+8*(-a*d/(a*
c+b))^(1/2)*x^6*b^3*d^3+5*(-a*d/(a*c+b))^(1/2)*x^4*a^2*b*c^3*d^2+9*(-a*d/(a*c+b))^(1/2)*x^4*a*b^2*c^2*d^2+3*(-
a*d/(a*c+b))^(1/2)*x^2*a^3*c^5*d+4*(-a*d/(a*c+b))^(1/2)*x^4*b^3*c*d^2+5*(-a*d/(a*c+b))^(1/2)*x^2*a^2*b*c^4*d+(
-a*d/(a*c+b))^(1/2)*x^2*a*b^2*c^3*d+3*(-a*d/(a*c+b))^(1/2)*a^3*c^6-(-a*d/(a*c+b))^(1/2)*x^2*b^3*c^2*d+9*(-a*d/
(a*c+b))^(1/2)*a^2*b*c^5+9*(-a*d/(a*c+b))^(1/2)*a*b^2*c^4+3*(-a*d/(a*c+b))^(1/2)*b^3*c^3)*(d*x^2+c)*((a*d*x^2+
a*c+b)/(d*x^2+c))^(1/2)/(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*c+b)^2/x^5/c^3
/((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{d x^{2} + c}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^6,x, algorithm="maxima")

[Out]

integrate(sqrt(a + b/(d*x^2 + c))/x^6, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^6,x, algorithm="fricas")

[Out]

integral(sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/x^6, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a c + a d x^{2} + b}{c + d x^{2}}}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x**2+c))**(1/2)/x**6,x)

[Out]

Integral(sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))/x**6, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{d x^{2} + c}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^6,x, algorithm="giac")

[Out]

integrate(sqrt(a + b/(d*x^2 + c))/x^6, x)