3.4.64 \(\int \frac {1}{x^2 (a+\frac {b}{c+d x^2})^{3/2}} \, dx\) [364]

Optimal. Leaf size=410 \[ -\frac {b}{a (b+a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(b-a c) \left (b+a c+a d x^2\right )}{a (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d x \left (b+a c+a d x^2\right )}{a (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\sqrt {c} (b-a c) \sqrt {d} \left (b+a c+a d x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {c^{3/2} \sqrt {d} \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{(b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

[Out]

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

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Rubi [A]
time = 0.30, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {c^{3/2} \sqrt {d} \left (a c+a d x^2+b\right ) F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{(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 {\sqrt {c} \sqrt {d} (b-a c) \left (a c+a d x^2+b\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a (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 {(b-a c) \left (a c+a d x^2+b\right )}{a x (a c+b)^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {d x (b-a c) \left (a c+a d x^2+b\right )}{a (a c+b)^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {b}{a x (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b/(c + d*x^2))^(3/2)),x]

[Out]

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 10.45, size = 268, normalized size = 0.65 \begin {gather*} -\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} \left (c+d x^2\right ) \left (b \left (c-d x^2\right )+a c \left (c+d x^2\right )\right )+i c (-b+a c) d x \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 )+2 i b c d x \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 )}{(b+a c)^2 \sqrt {\frac {a d}{b+a c}} x \left (b+a \left (c+d x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b/(c + d*x^2))^(3/2)),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*(c - d*x^2) + a*c*(c + d*x^2))
+ I*c*(-b + a*c)*d*x*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)*b*c*d*x*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)]))/((b + a*c)^2*Sqrt[(a*d)/(b + a*c)]*x*(b + a*(c + d*x^2))))

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Maple [A]
time = 0.07, size = 686, normalized size = 1.67

method	result	size
default	\(\frac {\left (-\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a c \,d^{2} x^{4}+\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {-\frac {a d}{a c +b}}\, b \,d^{2} x^{4}+a \,c^{2} d \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) x \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}-2 \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,c^{2} d \,x^{2}+2 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b c d x -\sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b c d x +\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {-\frac {a d}{a c +b}}\, b c d \,x^{2}-\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b c d \,x^{2}-\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,c^{3}-\sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b \,c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {-\frac {a d}{a c +b}}\, x \left (a c +b \right )^{2} \left (a d \,x^{2}+a c +b \right )}\)	\(686\)
risch	\(-\frac {c \left (a d \,x^{2}+a c +b \right )}{\left (a c +b \right )^{2} x \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {d \left (\frac {-\frac {2 a^{2} c 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 )}{\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^{2} 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 {2 a b 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}}+\frac {b^{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}}}{a}-\frac {b^{2} \left (a c +b \right ) \left (-\frac {\left (a \,d^{2} x^{2}+a c d \right ) x}{\left (a c +b \right ) b d \sqrt {\left (x^{2}+\frac {a c +b}{a d}\right ) \left (a \,d^{2} x^{2}+a c d \right )}}+\frac {\left (\frac {1}{a c +b}+\frac {a c}{\left (a c +b \right ) b}\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 \left (a c +b \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 )}\right )}{a}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{\left (a c +b \right )^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(1061\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

(-(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*c*d^2*x^4+(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)*b*d^2*x^4+a*c^2*d*((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*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)-2*(-a*d/(a*c+b))^(1/2)*((d*x^2+c
)*(a*d*x^2+a*c+b))^(1/2)*a*c^2*d*x^2+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))*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*b*c*d*x-((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)*b*c
*d*x+(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)*b*c*d*x^2-(-a*d/(a*c+b))^(1/2)*((d*x
^2+c)*(a*d*x^2+a*c+b))^(1/2)*b*c*d*x^2-(-a*d/(a*c+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*c^3-(-a*d/(a*c
+b))^(1/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*b*c^2)*((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)/x/(a*c+b)^2/(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(1/x^2/(a+b/(d*x^2+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a + b/(d*x^2 + c))^(3/2)*x^2), 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(1/x^2/(a+b/(d*x^2+c))^(3/2),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 {1}{x^{2} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b/(d*x**2+c))**(3/2),x)

[Out]

Integral(1/(x**2*((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2)), 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(1/x^2/(a+b/(d*x^2+c))^(3/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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