3.81 \(\int \frac{1}{(a+b x^2) (c+d x^2)^{3/2} \sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=344 \[ -\frac{d \sqrt{e} \sqrt{c+d x^2} (a d f-2 b c f+b d e) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b^2 c^{3/2} \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{d^{3/2} \sqrt{e+f x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} \sqrt{c+d x^2} (b c-a d) (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]

[Out]

-((d^(3/2)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)*(d*e
- c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])) - (d*Sqrt[e]*(b*d*e - 2*b*c*f + a*d*f)*Sqrt[c +
d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[f]*(d*e - c*f)*Sqrt[(e*
(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcT
an[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*(b*c - a*d)^2*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*
(c + d*x^2))])

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Rubi [A]  time = 0.220375, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {546, 539, 525, 418, 411} \[ \frac{b^2 c^{3/2} \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{d^{3/2} \sqrt{e+f x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} \sqrt{c+d x^2} (b c-a d) (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{d \sqrt{e} \sqrt{c+d x^2} (a d f-2 b c f+b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]

[Out]

-((d^(3/2)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)*(d*e
- c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])) - (d*Sqrt[e]*(b*d*e - 2*b*c*f + a*d*f)*Sqrt[c +
d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[f]*(d*e - c*f)*Sqrt[(e*
(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcT
an[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*(b*c - a*d)^2*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*
(c + d*x^2))])

Rule 546

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

Rule 539

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(c*Sqrt[e +
f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 525

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

Rule 418

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

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

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

Mathematica [C]  time = 0.787321, size = 365, normalized size = 1.06 \[ \frac{\sqrt{\frac{d}{c}} \left (i a d \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )-i b c^2 f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+i b c d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+i a d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+a c d e x \left (\frac{d}{c}\right )^{3/2}+a c d f x^3 \left (\frac{d}{c}\right )^{3/2}\right )}{a d \sqrt{c+d x^2} \sqrt{e+f x^2} (a d-b c) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]

[Out]

(Sqrt[d/c]*(a*c*d*(d/c)^(3/2)*e*x + a*c*d*(d/c)^(3/2)*f*x^3 + I*a*d^2*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e
]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a*d*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e
]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*b*c*d*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticP
i[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*b*c^2*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)]))/(a*d*(-(b*c) + a*d)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[
e + f*x^2])

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Maple [A]  time = 0.03, size = 413, normalized size = 1.2 \begin{align*}{\frac{1}{ac \left ( ad-bc \right ) \left ( cf-de \right ) \left ( df{x}^{4}+cf{x}^{2}+de{x}^{2}+ce \right ) } \left ( -{x}^{3}a{d}^{2}f\sqrt{-{\frac{d}{c}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) acdf\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) a{d}^{2}e\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) a{d}^{2}e\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) b{c}^{2}f\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) bcde\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-xa{d}^{2}e\sqrt{-{\frac{d}{c}}} \right ) \sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-x^3*a*d^2*f*(-d/c)^(1/2)+EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c*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))*a*d^2*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*d^2*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))*b*c^2*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))*b*c*d*e*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-x*a*d^2*e*(-d/c)^(1/2))*(f*x^2+e)^(
1/2)*(d*x^2+c)^(1/2)/a/c/(-d/c)^(1/2)/(a*d-b*c)/(c*f-d*e)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}} \sqrt{f x^{2} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}} \sqrt{e + f x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}} \sqrt{f x^{2} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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