3.4.43 \(\int \frac {(a+\frac {b}{c+d x^2})^{3/2}}{x^6} \, dx\) [343]

Optimal. Leaf size=494 \[ \frac {b \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c x^5}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c^4 (b+a c)}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c^2 x^5}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c^3 x^3}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c^4 (b+a c) x}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^{5/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{7/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {a (8 b+a c) d^{5/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{5/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

[Out]

b*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c/x^5+1/5*(a^2*c^2+16*a*b*c+16*b^2)*d^3*x*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2
)/c^4/(a*c+b)-1/5*(a*c+6*b)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^2/x^5+1/5*(a*c+8*b)*d*(d*x^2+c)*((a*
d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^3/x^3-1/5*(a^2*c^2+16*a*b*c+16*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-1/5*(a^2*c^2+16*a*b*c+16*b^2)*d^(5/2)*(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*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^(7/2)/(a*c+b)/(c*(a*d
*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)+1/5*a*(a*c+8*b)*d^(5/2)*(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))*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^(5/2)/(a*c+b)/(c*(
a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)

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Rubi [A]
time = 0.56, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986, 479, 597, 545, 429, 506, 422} \begin {gather*} -\frac {d^{5/2} \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{7/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d^3 x \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c^4 (a c+b)}-\frac {d^2 \left (a^2 c^2+16 a b c+16 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c^4 x (a c+b)}+\frac {a d^{5/2} (a c+8 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{5/2} (a c+b) \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d (a c+8 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c^3 x^3}-\frac {(a c+6 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c^2 x^5}+\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/(c + d*x^2))^(3/2)/x^6,x]

[Out]

(b*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(c*x^5) + ((16*b^2 + 16*a*b*c + a^2*c^2)*d^3*x*Sqrt[(b + a*c + a*d*x
^2)/(c + d*x^2)])/(5*c^4*(b + a*c)) - ((6*b + a*c)*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(5*c^2*x
^5) + ((8*b + a*c)*d*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(5*c^3*x^3) - ((16*b^2 + 16*a*b*c + a^
2*c^2)*d^2*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(5*c^4*(b + a*c)*x) - ((16*b^2 + 16*a*b*c + 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)])/(5*c^(
7/2)*(b + a*c)*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]) + (a*(8*b + 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)])/(5*c^(5/2)*(b + a*c)*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 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 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*} \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\left (b+a \left (c+d x^2\right )\right )^{3/2}}{x^6 \left (c+d x^2\right )^{3/2}} \, 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 {\left (b+a c+a d x^2\right )^{3/2}}{x^6 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{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 {-(b+a c) (6 b+a c) d-a (5 b+a c) d^2 x^2}{x^6 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{c d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 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 {-3 (b+a c)^2 (8 b+a c) d^2-3 a (b+a c) (6 b+a c) d^3 x^2}{x^4 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 c^2 (b+a c) d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+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}}}{5 c^3 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 {-3 (b+a c)^2 \left (16 b^2+16 a b c+a^2 c^2\right ) d^3-3 a (b+a c)^2 (8 b+a c) d^4 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 c^3 (b+a c)^2 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+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}}}{5 c^3 x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+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}}}{5 c^4 (b+a c) 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 {3 a c (b+a c)^3 (8 b+a c) d^4+3 a (b+a c)^2 \left (16 b^2+16 a b c+a^2 c^2\right ) d^5 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 c^4 (b+a c)^3 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+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}}}{5 c^3 x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+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}}}{5 c^4 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (a (8 b+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}{5 c^3 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (a \left (16 b^2+16 a b c+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}{5 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+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}}}{5 c^3 x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+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}}}{5 c^4 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}+\frac {a (8 b+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 )}{5 c^{5/2} (b+a c) \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 (16 b^2+16 a b c+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}{5 c^3 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {b \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c^2 x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(8 b+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}}}{5 c^3 x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+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}}}{5 c^4 (b+a c) x \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (16 b^2+16 a b c+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 )}{5 c^{7/2} (b+a c) \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 (8 b+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 )}{5 c^{5/2} (b+a c) \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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.76, size = 430, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {\frac {a d}{b+a c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} \left (b^3 \left (c^3-2 c^2 d x^2+8 c d^2 x^4+16 d^3 x^6\right )+a^3 c^2 \left (c^4+c^3 d x^2+c d^3 x^6+d^4 x^8\right )+a^2 b c \left (3 c^4+5 c^2 d^2 x^4+24 c d^3 x^6+16 d^4 x^8\right )+a b^2 \left (3 c^4-3 c^3 d x^2+13 c^2 d^2 x^4+40 c d^3 x^6+16 d^4 x^8\right )\right )+i a c \left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x^5 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )-i a b c (8 b+7 a c) d^3 x^5 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )\right )}{5 a c^4 d x^5 \left (b+a \left (c+d x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/(c + d*x^2))^(3/2)/x^6,x]

[Out]

-1/5*(Sqrt[(a*d)/(b + a*c)]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(b^3*(c^3 - 2*c^2*d*x
^2 + 8*c*d^2*x^4 + 16*d^3*x^6) + a^3*c^2*(c^4 + c^3*d*x^2 + c*d^3*x^6 + d^4*x^8) + a^2*b*c*(3*c^4 + 5*c^2*d^2*
x^4 + 24*c*d^3*x^6 + 16*d^4*x^8) + a*b^2*(3*c^4 - 3*c^3*d*x^2 + 13*c^2*d^2*x^4 + 40*c*d^3*x^6 + 16*d^4*x^8)) +
I*a*c*(16*b^2 + 16*a*b*c + 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)] - I*a*b*c*(8*b + 7*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)]))/(a*c^4*d*x^5*(b + a*(
c + d*x^2)))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1665\) vs. \(2(526)=1052\).
time = 0.09, size = 1666, normalized size = 3.37

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (a^{2} c^{2} d^{2} x^{4}+11 a c \,d^{2} b \,x^{4}-a^{2} c^{3} d \,x^{2}+11 b^{2} d^{2} x^{4}-4 a b \,c^{2} d \,x^{2}+a^{2} c^{4}-3 b^{2} c d \,x^{2}+2 a b \,c^{3}+b^{2} c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{5 c^{4} x^{5} \left (a c +b \right )}+\frac {d^{3} \left (-\frac {2 \left (a^{3} c^{2} d +11 a^{2} b c d +11 a \,b^{2} d \right ) \left (c^{2} a +b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-\EllipticE \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}+c^{2} a +b c}\, \left (2 a c d +2 b d \right )}+\frac {a^{3} c^{3} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \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}+c^{2} a +b c}}+\frac {4 a^{2} b \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \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}+c^{2} a +b c}}+\frac {3 a \,b^{2} c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \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}+c^{2} a +b c}}-5 b^{2} c \left (a c +b \right ) \left (\frac {\left (a \,d^{2} x^{2}+a c d +b d \right ) x}{c b d \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (a \,d^{2} x^{2}+a c d +b d \right )}}+\frac {\left (\frac {1}{c}-\frac {a c d +b d}{c b d}\right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \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}+c^{2} a +b c}}+\frac {2 a d \left (c^{2} a +b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-\EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{b c \sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \left (2 a c d +2 b d \right )}\right )\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{5 c^{4} \left (a c +b \right ) \left (a d \,x^{2}+a c +b \right )}\)	\(1170\)
default	\(\text {Expression too large to display}\)	\(1666\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/(d*x^2+c))^(3/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*(5*(-a*d/(a*c+b))^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*b^3*d^3*x^6+(-a*d/(a*c+b))^(1/2)*
((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^3*c^6+(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*b^3*c^3+11*(-
a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2*b*c*d^4*x^8+19*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x
^2+a*c+b))^(1/2)*a^2*b*c^2*d^3*x^6+5*(-a*d/(a*c+b))^(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*b^
2*d^4*x^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))*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^3*c^3*d^3*x^5+(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)
*a^3*c^2*d^4*x^8+(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^3*c^3*d^3*x^6+30*(-a*d/(a*c+b))^(1/2
)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b^2*c*d^3*x^6+5*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a
^2*b*c^3*d^2*x^4+13*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b^2*c^2*d^2*x^4-3*(-a*d/(a*c+b))^
(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b^2*c^3*d*x^2-16*((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))*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2*b*c^2*d^3*x^5+8*(
(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))*((d*x
^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b^2*c*d^3*x^5-16*((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))*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b^2*c*d^3*x^5+7*((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))*((d*x^2+c)*(a*d*x^
2+a*c+b))^(1/2)*a^2*b*c^2*d^3*x^5+5*(-a*d/(a*c+b))^(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^2*b
*c*d^4*x^8+5*(-a*d/(a*c+b))^(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^2*b*c^2*d^3*x^6+10*(-a*d/(
a*c+b))^(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*b^2*c*d^3*x^6+11*(-a*d/(a*c+b))^(1/2)*((d*x^2+
c)*(a*d*x^2+a*c+b))^(1/2)*a*b^2*d^4*x^8+(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^3*c^5*d*x^2+8
*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*b^3*c*d^2*x^4-2*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x
^2+a*c+b))^(1/2)*b^3*c^2*d*x^2+11*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*b^3*d^3*x^6+3*(-a*d/(
a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2*b*c^5+3*(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(
1/2)*a*b^2*c^4)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/x^5/(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/(a*c+b))^(1/2)/c^4/(a*d*x^2+a*c+b)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(3/2)/x^6,x, algorithm="maxima")

[Out]

integrate((a + b/(d*x^2 + c))^(3/2)/x^6, x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(3/2)/x^6,x, algorithm="fricas")

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x**2+c))**(3/2)/x**6,x)

[Out]

Integral(((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2)/x**6, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(3/2)/x^6,x, algorithm="giac")

[Out]

integrate((a + b/(d*x^2 + c))^(3/2)/x^6, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/(c + d*x^2))^(3/2)/x^6,x)

[Out]

int((a + b/(c + d*x^2))^(3/2)/x^6, x)

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