3.99 \(\int \frac{\sqrt{c-d x^2} \sqrt{e+f x^2}}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=359 \[ -\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),-\frac{c f}{d e}\right )}{2 a b^2 \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (a^2 d f+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 b^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1}} \]

[Out]

(x*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) + (Sqrt[c]*Sqrt[d]*Sqrt[1 - (d*x^2)/c]*Sqrt[e + f*x^2]*E
llipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*b*Sqrt[c - d*x^2]*Sqrt[1 + (f*x^2)/e]) - (Sqrt[c]*
Sqrt[d]*(b*e + a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*
e))])/(2*a*b^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]) + (Sqrt[c]*(b^2*c*e + a^2*d*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f
*x^2)/e]*EllipticPi[-((b*c)/(a*d)), ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a^2*b^2*Sqrt[d]*Sqrt[c -
d*x^2]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.323645, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {548, 524, 427, 426, 424, 421, 419, 538, 537} \[ \frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (a^2 d f+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 b^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b^2 \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) + (Sqrt[c]*Sqrt[d]*Sqrt[1 - (d*x^2)/c]*Sqrt[e + f*x^2]*E
llipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*b*Sqrt[c - d*x^2]*Sqrt[1 + (f*x^2)/e]) - (Sqrt[c]*
Sqrt[d]*(b*e + a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*
e))])/(2*a*b^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]) + (Sqrt[c]*(b^2*c*e + a^2*d*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f
*x^2)/e]*EllipticPi[-((b*c)/(a*d)), ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a^2*b^2*Sqrt[d]*Sqrt[c -
d*x^2]*Sqrt[e + f*x^2])

Rule 548

Int[(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*Sqrt[c
+ d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)), x] + (Dist[(d*f)/(2*a*b^2), Int[(a - b*x^2)/(Sqrt[c + d*x^2]*Sqr
t[e + f*x^2]), x], x] + Dist[(b^2*c*e - a^2*d*f)/(2*a*b^2), Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]
), x], x]) /; FreeQ[{a, b, c, d, e, f}, x]

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rule 427

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{\sqrt{c-d x^2} \sqrt{e+f x^2}}{\left (a+b x^2\right )^2} \, dx &=\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}-\frac{(d f) \int \frac{a-b x^2}{\sqrt{c-d x^2} \sqrt{e+f x^2}} \, dx}{2 a b^2}+\frac{1}{2} \left (\frac{c e}{a}+\frac{a d f}{b^2}\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c-d x^2} \sqrt{e+f x^2}} \, dx\\ &=\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{d \int \frac{\sqrt{e+f x^2}}{\sqrt{c-d x^2}} \, dx}{2 a b}-\frac{(d (b e+a f)) \int \frac{1}{\sqrt{c-d x^2} \sqrt{e+f x^2}} \, dx}{2 a b^2}+\frac{\left (\left (\frac{c e}{a}+\frac{a d f}{b^2}\right ) \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2}} \, dx}{2 \sqrt{c-d x^2}}\\ &=\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{\left (d \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{e+f x^2}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{2 a b \sqrt{c-d x^2}}-\frac{\left (d (b e+a f) \sqrt{1+\frac{f x^2}{e}}\right ) \int \frac{1}{\sqrt{c-d x^2} \sqrt{1+\frac{f x^2}{e}}} \, dx}{2 a b^2 \sqrt{e+f x^2}}+\frac{\left (\left (\frac{c e}{a}+\frac{a d f}{b^2}\right ) \sqrt{1-\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}}\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{1-\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}}} \, dx}{2 \sqrt{c-d x^2} \sqrt{e+f x^2}}\\ &=\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{\sqrt{c} \left (\frac{c e}{a}+\frac{a d f}{b^2}\right ) \sqrt{1-\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}} \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{\left (d \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2}\right ) \int \frac{\sqrt{1+\frac{f x^2}{e}}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{2 a b \sqrt{c-d x^2} \sqrt{1+\frac{f x^2}{e}}}-\frac{\left (d (b e+a f) \sqrt{1-\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}}\right ) \int \frac{1}{\sqrt{1-\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}}} \, dx}{2 a b^2 \sqrt{c-d x^2} \sqrt{e+f x^2}}\\ &=\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b \sqrt{c-d x^2} \sqrt{1+\frac{f x^2}{e}}}-\frac{\sqrt{c} \sqrt{d} (b e+a f) \sqrt{1-\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b^2 \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \left (\frac{c e}{a}+\frac{a d f}{b^2}\right ) \sqrt{1-\frac{d x^2}{c}} \sqrt{1+\frac{f x^2}{e}} \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2}}\\ \end{align*}

Mathematica [C]  time = 2.41177, size = 422, normalized size = 1.18 \[ \frac{-\frac{i c \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{d}{c}}\right ),-\frac{c f}{d e}\right )}{b^2}+\frac{i a c f \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{b^2}+\frac{i d e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{a \left (-\frac{d}{c}\right )^{3/2}}+\frac{c e x}{a+b x^2}+\frac{c f x^3}{a+b x^2}-\frac{d e x^3}{a+b x^2}-\frac{d f x^5}{a+b x^2}+\frac{i c e \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{b}}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*e*x)/(a + b*x^2) - (d*e*x^3)/(a + b*x^2) + (c*f*x^3)/(a + b*x^2) - (d*f*x^5)/(a + b*x^2) + (I*c*Sqrt[-(d/c
)]*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/b - (I*c*Sq
rt[-(d/c)]*(b*e + a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d
*e))])/b^2 + (I*d*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*
x], -((c*f)/(d*e))])/(a*(-(d/c))^(3/2)) + (I*a*c*Sqrt[-(d/c)]*f*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ellipt
icPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/b^2)/(2*a*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.043, size = 793, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*((d/c)^(1/2)*x^5*a*b^2*d*f+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellip
ticF(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*x^2*a^2*b*d*f+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(d/c)^
(1/2),(-c*f/d/e)^(1/2))*x^2*a*b^2*d*e-(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(d/c)^(1/2),(-c*f/d
/e)^(1/2))*x^2*a*b^2*d*e-(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/
2)/(d/c)^(1/2))*x^2*a^2*b*d*f-(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e
)^(1/2)/(d/c)^(1/2))*x^2*b^3*c*e-(d/c)^(1/2)*x^3*a*b^2*c*f+(d/c)^(1/2)*x^3*a*b^2*d*e+(-(d*x^2-c)/c)^(1/2)*((f*
x^2+e)/e)^(1/2)*EllipticF(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*a^3*d*f+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ell
ipticF(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*a^2*b*d*e-(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(d/c)^(1
/2),(-c*f/d/e)^(1/2))*a^2*b*d*e-(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f
/e)^(1/2)/(d/c)^(1/2))*a^3*d*f-(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/
e)^(1/2)/(d/c)^(1/2))*a*b^2*c*e-(d/c)^(1/2)*x*a*b^2*c*e)/(d*f*x^4-c*f*x^2+d*e*x^2-c*e)/a^2/(b*x^2+a)/b^2/(d/c)
^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-d x^{2} + c} \sqrt{f x^{2} + e}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - d x^{2}} \sqrt{e + f x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-d x^{2} + c} \sqrt{f x^{2} + e}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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