3.2.8 \(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{(e+f x^2)^{3/2}} \, dx\) [108]

Optimal. Leaf size=484 \[ -\frac {(d e-c f) x \sqrt {a+b x^2}}{e f \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {c} \sqrt {d e-c f} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{e f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} (b e-a f) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a e f \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac {d e}{d e-c f};\sin ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

-c^(3/2)*(-a*f+b*e)*(1/(1+x^2*(-c*f+d*e)/c/(f*x^2+e)))^(1/2)*(1+x^2*(-c*f+d*e)/c/(f*x^2+e))^(1/2)*EllipticF(x*
(-c*f+d*e)^(1/2)/c^(1/2)/(f*x^2+e)^(1/2)/(1+x^2*(-c*f+d*e)/c/(f*x^2+e))^(1/2),(-(-a*d+b*c)*e/a/(-c*f+d*e))^(1/
2))*(b*x^2+a)^(1/2)/a/e/f/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+(1/(1+x^2*(-c*f+d*e
)/c/(f*x^2+e)))^(1/2)*(1+x^2*(-c*f+d*e)/c/(f*x^2+e))^(1/2)*EllipticE(x*(-c*f+d*e)^(1/2)/c^(1/2)/(f*x^2+e)^(1/2
)/(1+x^2*(-c*f+d*e)/c/(f*x^2+e))^(1/2),(-(-a*d+b*c)*e/a/(-c*f+d*e))^(1/2))*c^(1/2)*(-c*f+d*e)^(1/2)*(b*x^2+a)^
(1/2)/e/f/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-(-c*f+d*e)*x*(b*x^2+a)^(1/2)/e/f/(d*x^2+c)^(1/2)/(f*
x^2+e)^(1/2)+b*c*EllipticPi(x*(-c*f+d*e)^(1/2)/e^(1/2)/(d*x^2+c)^(1/2),d*e/(-c*f+d*e),(-(-a*d+b*c)*e/a/(-c*f+d
*e))^(1/2))*e^(1/2)*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)/a/f/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/(d*x^2
+c))^(1/2)/(f*x^2+e)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.51, antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {571, 567, 551, 568, 433, 429, 506, 422} \begin {gather*} \frac {b c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac {d e}{d e-c f};\text {ArcSin}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {d x^2+c}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt {e+f x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) F\left (\text {ArcTan}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {f x^2+e}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a e f \sqrt {c+d x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {d e-c f} E\left (\text {ArcTan}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {f x^2+e}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \sqrt {a+b x^2} (d e-c f)}{e f \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) + (Sqrt[c]*Sqrt[d*e - c*f]*Sqrt[a + b
*x^2]*EllipticE[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(e
*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b*e - a*f)*Sqrt[a + b*x^2]*EllipticF[Arc
Tan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*e*f*Sqrt[d*e - c*f
]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*c*Sqrt[e]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(
e*(c + d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c -
a*d)*e)/(a*(d*e - c*f)))])/(a*f*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*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 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 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 551

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

Rule 567

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

Rule 568

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[
c + d*x^2]*(Sqrt[a*((e + f*x^2)/(e*(a + b*x^2)))]/(a*Sqrt[e + f*x^2]*Sqrt[a*((c + d*x^2)/(c*(a + b*x^2)))])),
Subst[Int[Sqrt[1 - (b*c - a*d)*(x^2/c)]/Sqrt[1 - (b*e - a*f)*(x^2/e)], x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[
{a, b, c, d, e, f}, x]

Rule 571

Int[(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2])/((e_) + (f_.)*(x_)^2)^(3/2), x_Symbol] :> Dist[b/f,
Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]), x], x] - Dist[(b*e - a*f)/f, Int[Sqrt[c + d*x^2]/(Sqrt[
a + b*x^2]*(e + f*x^2)^(3/2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]
time = 24.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]

[Out]

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2), x]

________________________________________________________________________________________

Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{\left (f \,x^{2}+e \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(f*x^2 + e)^(3/2), x)

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}{\left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)/(e + f*x**2)**(3/2), x)

________________________________________________________________________________________

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

[Out]

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(f*x^2 + e)^(3/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________