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

Optimal. Leaf size=191 \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b^2 c}{a^2 d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c+d x^2} \sqrt{a^2 f+b^2 e}}{\sqrt{e+f x^2} \sqrt{a^2 d+b^2 c}}\right )}{\sqrt{a^2 d+b^2 c} \sqrt{a^2 f+b^2 e}} \]

[Out]

-((b*ArcTanh[(Sqrt[b^2*e + a^2*f]*Sqrt[c + d*x^2])/(Sqrt[b^2*c + a^2*d]*Sqrt[e + f*x^2])])/(Sqrt[b^2*c + a^2*d
]*Sqrt[b^2*e + a^2*f])) + (Sqrt[-c]*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b^2*c)/(a^2*d)), Arc
Sin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.512984, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2113, 538, 537, 571, 93, 208} \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b^2 c}{a^2 d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c+d x^2} \sqrt{a^2 f+b^2 e}}{\sqrt{e+f x^2} \sqrt{a^2 d+b^2 c}}\right )}{\sqrt{a^2 d+b^2 c} \sqrt{a^2 f+b^2 e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*ArcTanh[(Sqrt[b^2*e + a^2*f]*Sqrt[c + d*x^2])/(Sqrt[b^2*c + a^2*d]*Sqrt[e + f*x^2])])/(Sqrt[b^2*c + a^2*d
]*Sqrt[b^2*e + a^2*f])) + (Sqrt[-c]*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b^2*c)/(a^2*d)), Arc
Sin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

Rule 2113

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

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)
])

Rule 571

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rubi steps

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

Mathematica [C]  time = 1.58234, size = 772, normalized size = 4.04 \[ \frac{2 \sqrt{d} \left (\sqrt{c}+i \sqrt{d} x\right ) \left (\sqrt{e}+i \sqrt{f} x\right ) \sqrt{\frac{\left (\sqrt{d} x+i \sqrt{c}\right ) \left (\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right )}{\left (\sqrt{d} x-i \sqrt{c}\right ) \left (\sqrt{c} \sqrt{f}+\sqrt{d} \sqrt{e}\right )}} \sqrt{\frac{\sqrt{c} \sqrt{d} \left (\sqrt{f} x+i \sqrt{e}\right )}{\left (\sqrt{d} x-i \sqrt{c}\right ) \left (\sqrt{c} \sqrt{f}-\sqrt{d} \sqrt{e}\right )}} \left (\left (b \sqrt{c}+i a \sqrt{d}\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right ) \left (\sqrt{d} x+i \sqrt{c}\right )}{\left (\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right ) \left (\sqrt{d} x-i \sqrt{c}\right )}}\right )|\frac{\left (\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right )^2}{\left (\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right )^2}\right )-2 b \sqrt{c} \Pi \left (\frac{\left (b \sqrt{c}-i a \sqrt{d}\right ) \left (\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right )}{\left (i \sqrt{d} a+b \sqrt{c}\right ) \left (\sqrt{c} \sqrt{f}-\sqrt{d} \sqrt{e}\right )};\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right ) \left (\sqrt{d} x+i \sqrt{c}\right )}{\left (\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right ) \left (\sqrt{d} x-i \sqrt{c}\right )}}\right )|\frac{\left (\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right )^2}{\left (\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right )^2}\right )\right )}{\sqrt{c+d x^2} \sqrt{e+f x^2} \left (b \sqrt{c}-i a \sqrt{d}\right ) \left (b \sqrt{c}+i a \sqrt{d}\right ) \left (\sqrt{c} \sqrt{f}-\sqrt{d} \sqrt{e}\right ) \sqrt{\frac{\sqrt{c} \sqrt{d} \left (\sqrt{e}+i \sqrt{f} x\right )}{\left (\sqrt{c}+i \sqrt{d} x\right ) \left (\sqrt{c} \sqrt{f}+\sqrt{d} \sqrt{e}\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d]*(Sqrt[c] + I*Sqrt[d]*x)*Sqrt[((Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d
]*Sqrt[e] + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]*(Sqrt[e] + I*Sqrt[f]*x)*Sqrt[(Sqrt[c]*Sqrt[d]*(I*Sqr
t[e] + Sqrt[f]*x))/((-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]*((b*Sqrt[c] + I*a*Sqrt
[d])*EllipticF[ArcSin[Sqrt[((Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + S
qrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]], (Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])^2/(Sqrt[d]*Sqrt[e] - Sqrt[c
]*Sqrt[f])^2] - 2*b*Sqrt[c]*EllipticPi[((b*Sqrt[c] - I*a*Sqrt[d])*(Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f]))/((b*Sqr
t[c] + I*a*Sqrt[d])*(-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])), ArcSin[Sqrt[((Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])*
(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]], (Sqrt[d]*Sqrt[e]
+ Sqrt[c]*Sqrt[f])^2/(Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])^2]))/((b*Sqrt[c] - I*a*Sqrt[d])*(b*Sqrt[c] + I*a*Sqrt
[d])*(-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])*Sqrt[(Sqrt[c]*Sqrt[d]*(Sqrt[e] + I*Sqrt[f]*x))/((Sqrt[d]*Sqrt[e] +
Sqrt[c]*Sqrt[f])*(Sqrt[c] + I*Sqrt[d]*x))]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.064, size = 353, normalized size = 1.9 \begin{align*}{\frac{1}{2\,ab \left ( df{x}^{4}+cf{x}^{2}+e{x}^{2}d+ce \right ) } \left ( 2\,b\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},-{\frac{{b}^{2}c}{{a}^{2}d}},{\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) \sqrt{{\frac{{a}^{4}df+{a}^{2}{b}^{2}cf+{a}^{2}{b}^{2}de+ce{b}^{4}}{{b}^{4}}}}-{\it Artanh} \left ({\frac{2\,{a}^{2}df{x}^{2}+{b}^{2}cf{x}^{2}+{b}^{2}de{x}^{2}+{a}^{2}cf+{a}^{2}de+2\,{b}^{2}ce}{2\,{b}^{2}}{\frac{1}{\sqrt{{\frac{{a}^{4}df+{a}^{2}{b}^{2}cf+{a}^{2}{b}^{2}de+ce{b}^{4}}{{b}^{4}}}}}}{\frac{1}{\sqrt{df{x}^{4}+cf{x}^{2}+e{x}^{2}d+ce}}}} \right ) \sqrt{-{\frac{d}{c}}}a\sqrt{df{x}^{4}+cf{x}^{2}+e{x}^{2}d+ce} \right ) \sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{{a}^{4}df+{a}^{2}{b}^{2}cf+{a}^{2}{b}^{2}de+ce{b}^{4}}{{b}^{4}}}}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-1/c*d)^(1/2),-b^2*c/a^2/d,(-f/e)^(1/2)/(-1/c*d
)^(1/2))*((a^4*d*f+a^2*b^2*c*f+a^2*b^2*d*e+b^4*c*e)/b^4)^(1/2)-arctanh(1/2*(2*a^2*d*f*x^2+b^2*c*f*x^2+b^2*d*e*
x^2+a^2*c*f+a^2*d*e+2*b^2*c*e)/b^2/((a^4*d*f+a^2*b^2*c*f+a^2*b^2*d*e+b^4*c*e)/b^4)^(1/2)/(d*f*x^4+c*f*x^2+d*e*
x^2+c*e)^(1/2))*(-1/c*d)^(1/2)*a*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2))*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/b/a/((a^
4*d*f+a^2*b^2*c*f+a^2*b^2*d*e+b^4*c*e)/b^4)^(1/2)/(-1/c*d)^(1/2)/(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}{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}{\left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)*(b*x + a)), 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+a)/(d*x^2+c)^(1/2)/(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\right ) \sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*x)*sqrt(c + d*x**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}{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}{\left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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