\(\int \frac {\sqrt {c+d x^2}}{x^2 (a+b x^2)^2} \, dx\) [737]

Optimal result

Integrand size = 24, antiderivative size = 113 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {3 \sqrt {c+d x^2}}{2 a^2 x}+\frac {\sqrt {c+d x^2}}{2 a x \left (a+b x^2\right )}-\frac {(3 b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} \sqrt {b c-a d}} \]

[Out]

-1/2*(-2*a*d+3*b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(5/2)/(-a*d+b*c)^(1/2)-3/2*(d*x^2+c)^
(1/2)/a^2/x+1/2*(d*x^2+c)^(1/2)/a/x/(b*x^2+a)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {480, 597, 12, 385, 211} \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=-\frac {(3 b c-2 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} \sqrt {b c-a d}}-\frac {3 \sqrt {c+d x^2}}{2 a^2 x}+\frac {\sqrt {c+d x^2}}{2 a x \left (a+b x^2\right )} \]

[In]

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

[Out]

(-3*Sqrt[c + d*x^2])/(2*a^2*x) + Sqrt[c + d*x^2]/(2*a*x*(a + b*x^2)) - ((3*b*c - 2*a*d)*ArcTan[(Sqrt[b*c - a*d
]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*Sqrt[b*c - a*d])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 480

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*n*(p + 1))), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^
n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + 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] && LtQ[0, q, 1] && IntBinomialQ[a, b,
c, d, e, m, 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+d x^2}}{2 a x \left (a+b x^2\right )}-\frac {\int \frac {-3 c-2 d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a} \\ & = -\frac {3 \sqrt {c+d x^2}}{2 a^2 x}+\frac {\sqrt {c+d x^2}}{2 a x \left (a+b x^2\right )}-\frac {\int \frac {c (3 b c-2 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2 c} \\ & = -\frac {3 \sqrt {c+d x^2}}{2 a^2 x}+\frac {\sqrt {c+d x^2}}{2 a x \left (a+b x^2\right )}-\frac {(3 b c-2 a d) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2} \\ & = -\frac {3 \sqrt {c+d x^2}}{2 a^2 x}+\frac {\sqrt {c+d x^2}}{2 a x \left (a+b x^2\right )}-\frac {(3 b c-2 a d) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a^2} \\ & = -\frac {3 \sqrt {c+d x^2}}{2 a^2 x}+\frac {\sqrt {c+d x^2}}{2 a x \left (a+b x^2\right )}-\frac {(3 b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} \sqrt {b c-a d}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1650\) vs. \(2(113)=226\).

Time = 7.99 (sec) , antiderivative size = 1650, normalized size of antiderivative = 14.60 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {\sqrt {a} \left (2 a+3 b x^2\right ) \left (4 c^2+5 c d x^2+d^2 x^4-4 c^{3/2} \sqrt {c+d x^2}-3 \sqrt {c} d x^2 \sqrt {c+d x^2}\right )}{x \left (a+b x^2\right ) \left (-4 c^{3/2}-3 \sqrt {c} d x^2+4 c \sqrt {c+d x^2}+d x^2 \sqrt {c+d x^2}\right )}+\frac {5 a b^{3/2} c^{3/2} d \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{(b c-a d)^{3/2} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {3 b^2 c^2 \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{(b c-a d) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 a^2 d^2 \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{(b c-a d) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {3 b^{5/2} c^{5/2} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{(b c-a d)^{3/2} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 a^2 \sqrt {b} \sqrt {c} d^2 \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{(b c-a d)^{3/2} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {3 b^2 c^2 \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{(b c-a d) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 a^2 d^2 \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{(b c-a d) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {3 b^{5/2} c^{5/2} \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{(b c-a d)^{3/2} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 a^2 \sqrt {b} \sqrt {c} d^2 \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{(b c-a d)^{3/2} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {5 a b c d \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{(b c-a d) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {5 a b^{3/2} c^{3/2} d \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{(b c-a d)^{3/2} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {5 a b c d \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{(b c-a d) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}}{2 a^{5/2}} \]

[In]

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

[Out]

(-((Sqrt[a]*(2*a + 3*b*x^2)*(4*c^2 + 5*c*d*x^2 + d^2*x^4 - 4*c^(3/2)*Sqrt[c + d*x^2] - 3*Sqrt[c]*d*x^2*Sqrt[c
+ d*x^2]))/(x*(a + b*x^2)*(-4*c^(3/2) - 3*Sqrt[c]*d*x^2 + 4*c*Sqrt[c + d*x^2] + d*x^2*Sqrt[c + d*x^2]))) + (5*
a*b^(3/2)*c^(3/2)*d*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[
c + d*x^2]))])/((b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (3*b^2*c^2*ArcTan[(
Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/((b*c - a*d)*
Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (2*a^2*d^2*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[
c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/((b*c - a*d)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[
c]*Sqrt[b*c - a*d]]) + (3*b^(5/2)*c^(5/2)*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sq
rt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/((b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]])
+ (2*a^2*Sqrt[b]*Sqrt[c]*d^2*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[
c] - Sqrt[c + d*x^2]))])/((b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (3*b^2*c^
2*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/((b
*c - a*d)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (2*a^2*d^2*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqr
t[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/((b*c - a*d)*Sqrt[2*b*c - a*d + 2*Sqr
t[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (3*b^(5/2)*c^(5/2)*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*
d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/((b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b
*c - a*d]]) + (2*a^2*Sqrt[b]*Sqrt[c]*d^2*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqr
t[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/((b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]])
+ (5*a*b*c*d*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d
*x^2]))])/((b*c - a*d)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (5*a*b^(3/2)*c^(3/2)*d*ArcTan[
(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/((b*c - a*d
)^(3/2)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (5*a*b*c*d*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[
b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/((b*c - a*d)*Sqrt[2*b*c - a*d + 2*Sqrt
[b]*Sqrt[c]*Sqrt[b*c - a*d]]))/(2*a^(5/2))

Maple [A] (verified)

Time = 3.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d \,x^{2}+c}}{x}-\frac {b \sqrt {d \,x^{2}+c}\, x}{2 \left (b \,x^{2}+a \right )}+\frac {\left (2 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{2 \sqrt {\left (a d -b c \right ) a}}}{a^{2}}\)	\(92\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}}{a^{2} x}-\frac {\frac {\left (a d -b c \right ) \left (\frac {b \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b}+\frac {\left (a d -b c \right ) \left (\frac {b \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b}+\frac {\left (a d -3 b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a d -3 b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}}{a^{2}}\)	\(865\)
default	\(\text {Expression too large to display}\)	\(2024\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (93) = 186\).

Time = 0.33 (sec) , antiderivative size = 458, normalized size of antiderivative = 4.05 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=\left [\frac {{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{3} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (2 \, a^{2} b c - 2 \, a^{3} d + 3 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left ({\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3} + {\left (a^{4} b c - a^{5} d\right )} x\right )}}, -\frac {{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{3} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} b c - 2 \, a^{3} d + 3 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3} + {\left (a^{4} b c - a^{5} d\right )} x\right )}}\right ] \]

[In]

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

[Out]

[1/8*(((3*b^2*c - 2*a*b*d)*x^3 + (3*a*b*c - 2*a^2*d)*x)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2
*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(
d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(2*a^2*b*c - 2*a^3*d + 3*(a*b^2*c - a^2*b*d)*x^2)*sqrt(d*x^2 + c)
)/((a^3*b^2*c - a^4*b*d)*x^3 + (a^4*b*c - a^5*d)*x), -1/4*(((3*b^2*c - 2*a*b*d)*x^3 + (3*a*b*c - 2*a^2*d)*x)*s
qrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^
2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + 2*(2*a^2*b*c - 2*a^3*d + 3*(a*b^2*c - a^2*b*d)*x^2)*sqrt(d*x^2 + c))/((a^3*
b^2*c - a^4*b*d)*x^3 + (a^4*b*c - a^5*d)*x)]

Sympy [F]

\[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c + d x^{2}}}{x^{2} \left (a + b x^{2}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (93) = 186\).

Time = 0.87 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.91 \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=\frac {{\left (3 \, b c \sqrt {d} - 2 \, a d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{2}} + \frac {3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} + 10 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 3 \, b c^{3} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d - b c^{3}\right )} a^{2}} \]

[In]

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

[Out]

1/2*(3*b*c*sqrt(d) - 2*a*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d -
a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a^2) + (3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c*sqrt(d) - 2*(sqrt(d)*x - sqrt
(d*x^2 + c))^4*a*d^(3/2) - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2*sqrt(d) + 10*(sqrt(d)*x - sqrt(d*x^2 + c))^
2*a*c*d^(3/2) + 3*b*c^3*sqrt(d))/(((sqrt(d)*x - sqrt(d*x^2 + c))^6*b - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c +
4*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d + 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2 - 4*(sqrt(d)*x - sqrt(d*x^2 +
c))^2*a*c*d - b*c^3)*a^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2}}{x^2 \left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {d\,x^2+c}}{x^2\,{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

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

[Out]

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