3.6.23 \(\int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [523]

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

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2138, 552, 551, 585, 95, 214} \begin {gather*} \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};\text {ArcSin}\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}} \end {gather*}

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 95

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 214

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

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 552

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

Rule 585

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 2138

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]

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 \text {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 \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 3.93, size = 772, normalized size = 4.04 \begin {gather*} \frac {2 \sqrt {d} \left (\sqrt {c}+i \sqrt {d} x\right ) \sqrt {\frac {\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right ) \left (i \sqrt {c}+\sqrt {d} x\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (-i \sqrt {c}+\sqrt {d} x\right )}} \left (\sqrt {e}+i \sqrt {f} x\right ) \sqrt {\frac {\sqrt {c} \sqrt {d} \left (i \sqrt {e}+\sqrt {f} x\right )}{\left (-\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (-i \sqrt {c}+\sqrt {d} x\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 (i \sqrt {c}+\sqrt {d} x\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (-i \sqrt {c}+\sqrt {d} x\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 (b \sqrt {c}+i a \sqrt {d}\right ) \left (-\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right ) \left (i \sqrt {c}+\sqrt {d} x\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (-i \sqrt {c}+\sqrt {d} x\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 )}{\left (b \sqrt {c}-i a \sqrt {d}\right ) \left (b \sqrt {c}+i a \sqrt {d}\right ) \left (-\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \sqrt {\frac {\sqrt {c} \sqrt {d} \left (\sqrt {e}+i \sqrt {f} x\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (\sqrt {c}+i \sqrt {d} x\right )}} \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs. \(2(164)=328\).
time = 0.25, size = 339, normalized size = 1.77

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {\arctanh \left (\frac {\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+2 c e +\left (c f +d e \right ) x^{2}+\frac {2 d f \,x^{2} a^{2}}{b^{2}}}{2 \sqrt {\frac {d f \,a^{4}}{b^{4}}+\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+c e}\, \sqrt {d f \,x^{4}+c \,x^{2} f +d e \,x^{2}+c e}}\right )}{2 \sqrt {\frac {d f \,a^{4}}{b^{4}}+\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+c e}}+\frac {b \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, -\frac {b^{2} c}{d \,a^{2}}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )}{\sqrt {-\frac {d}{c}}\, a \sqrt {d f \,x^{4}+c \,x^{2} f +d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, b}\)	\(272\)
default	\(\frac {\left (2 b \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticPi \left (x \sqrt {-\frac {d}{c}}, -\frac {b^{2} c}{d \,a^{2}}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e}{b^{4}}}-\arctanh \left (\frac {2 a^{2} d f \,x^{2}+b^{2} c f \,x^{2}+b^{2} d e \,x^{2}+a^{2} c f +d e \,a^{2}+2 b^{2} c e}{2 b^{2} \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e}{b^{4}}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}\right ) \sqrt {-\frac {d}{c}}\, a \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{2 b a \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e}{b^{4}}}\, \sqrt {-\frac {d}{c}}\, \left (d f \,x^{4}+c \,x^{2} f +d e \,x^{2}+c e \right )}\)	\(339\)

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,method=_RETURNVERBOSE)

[Out]

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

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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(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)] 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right ) \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \end {gather*}

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 = 0.00, 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(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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