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

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

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Rubi [A]  time = 0.11, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {532, 377, 205} \begin {gather*} \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}} \end {gather*}

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 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]

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 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]

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*}

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Mathematica [A]  time = 0.21, size = 113, normalized size = 0.93 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.43, size = 239, normalized size = 1.96 \begin {gather*} -\frac {b \tan ^{-1}\left (\frac {b \sqrt {f} x^2}{\sqrt {a} \sqrt {b e-a f}}-\frac {b x \sqrt {e+f x^2}}{\sqrt {a} \sqrt {b e-a f}}+\frac {\sqrt {a} \sqrt {f}}{\sqrt {b e-a f}}\right )}{\sqrt {a} (b c-a d) \sqrt {b e-a f}}-\frac {d \sqrt {d e-c f} \tan ^{-1}\left (\frac {d \sqrt {f} x^2}{\sqrt {c} \sqrt {d e-c f}}-\frac {d x \sqrt {e+f x^2}}{\sqrt {c} \sqrt {d e-c f}}+\frac {\sqrt {c} \sqrt {f}}{\sqrt {d e-c f}}\right )}{\sqrt {c} (b c-a d) (c f-d e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  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^2+a)/(d*x^2+c)/(f*x^2+e)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.52, size = 173, normalized size = 1.42 \begin {gather*} -f^{\frac {3}{2}} {\left (\frac {b \arctan \left (\frac {{\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} b + 2 \, a f - b e}{2 \, \sqrt {-a^{2} f^{2} + a b f e}}\right )}{\sqrt {-a^{2} f^{2} + a b f e} {\left (b c f - a d f\right )}} - \frac {d \arctan \left (\frac {{\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt {-c^{2} f^{2} + c d f e}}\right )}{\sqrt {-c^{2} f^{2} + c d f e} {\left (b c f - a d f\right )}}\right )} \end {gather*}

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]

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

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maple [B]  time = 0.03, size = 782, normalized size = 6.41 \begin {gather*} -\frac {b^{2} d \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) f}{b}-\frac {2 \left (a f -b e \right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} f +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) f}{b}-\frac {a f -b e}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (\sqrt {-c d}\, b +\sqrt {-a b}\, d \right ) \left (-\sqrt {-c d}\, b +\sqrt {-a b}\, d \right ) \sqrt {-\frac {a f -b e}{b}}}+\frac {b^{2} d \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) f}{b}-\frac {2 \left (a f -b e \right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} f -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) f}{b}-\frac {a f -b e}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (\sqrt {-c d}\, b +\sqrt {-a b}\, d \right ) \left (-\sqrt {-c d}\, b +\sqrt {-a b}\, d \right ) \sqrt {-\frac {a f -b e}{b}}}+\frac {b \,d^{2} \ln \left (\frac {\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \left (\sqrt {-c d}\, b +\sqrt {-a b}\, d \right ) \left (-\sqrt {-c d}\, b +\sqrt {-a b}\, d \right ) \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}-\frac {b \,d^{2} \ln \left (\frac {-\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \left (\sqrt {-c d}\, b +\sqrt {-a b}\, d \right ) \left (-\sqrt {-c d}\, b +\sqrt {-a b}\, d \right ) \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}} \end {gather*}

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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