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

Optimal. Leaf size=122 \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}} \]

[Out]

(b*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*(b*c - a*d)*Sqrt[b*e - a*f]) - (d*ArcTan[(S
qrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*(b*c - a*d)*Sqrt[d*e - c*f])

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Rubi [A]  time = 0.110828, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {532, 377, 205} \[ \frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*(b*c - a*d)*Sqrt[b*e - a*f]) - (d*ArcTan[(S
qrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*(b*c - a*d)*Sqrt[d*e - c*f])

Rule 532

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt{e+f x^2}} \, dx &=\frac{b \int \frac{1}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx}{b c-a d}-\frac{d \int \frac{1}{\left (c+d x^2\right ) \sqrt{e+f x^2}} \, dx}{b c-a d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{a-(-b e+a f) x^2} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{b c-a d}-\frac{d \operatorname{Subst}\left (\int \frac{1}{c-(-d e+c f) x^2} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{b c-a d}\\ &=\frac{b \tan ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} (b c-a d) \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} (b c-a d) \sqrt{d e-c f}}\\ \end{align*}

Mathematica [A]  time = 0.176689, size = 113, normalized size = 0.93 \[ \frac{\frac{b \tan ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{a} \sqrt{e+f x^2}}\right )}{\sqrt{a} \sqrt{b e-a f}}-\frac{d \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} \sqrt{d e-c f}}}{b c-a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.037, size = 782, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} \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)/(f*x^2+e)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError