\(\int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [977]

Optimal result

Integrand size = 26, antiderivative size = 116 \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {506, 422} \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[In]

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

[Out]

(x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 -
(b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[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 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 {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{b} \\ & = \frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {i c \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} d \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {\left (-F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )+E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )\right ) \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, c \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{d \sqrt {-\frac {b}{a}}\, \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right )}\)	\(129\)
elliptic	\(-\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\)	\(159\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {b d} c x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} c x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} d}{b d^{2} x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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