\(\int \frac {\sqrt {a+b x^2}}{x^2 \sqrt {c+d x^2}} \, dx\) [942]

Optimal result

Integrand size = 26, antiderivative size = 232 \[ \int \frac {\sqrt {a+b x^2}}{x^2 \sqrt {c+d x^2}} \, dx=\frac {d x \sqrt {a+b x^2}}{c \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{c x}-\frac {\sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b \sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {486, 21, 433, 429, 506, 422} \[ \int \frac {\sqrt {a+b x^2}}{x^2 \sqrt {c+d x^2}} \, dx=\frac {b \sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d x \sqrt {a+b x^2}}{c \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{c x} \]

[In]

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

[Out]

(d*x*Sqrt[a + b*x^2])/(c*Sqrt[c + d*x^2]) - (Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x) - (Sqrt[d]*Sqrt[a + b*x^2]
*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[
c + d*x^2]) + (b*Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*S
qrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 433

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

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]

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

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {a+b x^2}}{x^2 \sqrt {c+d x^2}} \, dx=\frac {-\left (\left (a+b x^2\right ) \left (c+d x^2\right )\right )+\frac {b c x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}}}}{c x \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

[In]

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

[Out]

(-((a + b*x^2)*(c + d*x^2)) + (b*c*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(b/a)]*x],
(a*d)/(b*c)])/Sqrt[-(b/a)])/(c*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 4.89 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.72

method	result	size
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-\sqrt {-\frac {b}{a}}\, b d \,x^{4}+b c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, x E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )-\sqrt {-\frac {b}{a}}\, a d \,x^{2}-\sqrt {-\frac {b}{a}}\, b c \,x^{2}-\sqrt {-\frac {b}{a}}\, a c \right )}{\left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) c x \sqrt {-\frac {b}{a}}}\)	\(168\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{c x}+\frac {b \left (\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{c \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(276\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{c x}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(277\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {a+b x^2}}{x^2 \sqrt {c+d x^2}} \, dx=\frac {\sqrt {a c} b x \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {a c} {\left (a + b\right )} x \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a}{a c x} \]

[In]

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

[Out]

(sqrt(a*c)*b*x*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - sqrt(a*c)*(a + b)*x*sqrt(-b/a)*ellipti
c_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a)/(a*c*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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