3.1.62 \(\int \frac {1}{(a+b x^2) (c+d x^2) \sqrt {e+f x^2}} \, dx\) [62]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, 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 = {546, 385, 211} \begin {gather*} \frac {b \text {ArcTan}\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 \text {ArcTan}\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 211

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

Rule 385

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 546

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 \text {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 \text {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 = 0.38, size = 153, normalized size = 1.25 \begin {gather*} \frac {-\frac {b \tan ^{-1}\left (\frac {a \sqrt {f}+b x \left (\sqrt {f} x-\sqrt {e+f x^2}\right )}{\sqrt {a} \sqrt {b e-a f}}\right )}{\sqrt {a} \sqrt {b e-a f}}+\frac {d \tan ^{-1}\left (\frac {c \sqrt {f}+d x \left (\sqrt {f} x-\sqrt {e+f x^2}\right )}{\sqrt {c} \sqrt {d e-c f}}\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[(a*Sqrt[f] + b*x*(Sqrt[f]*x - Sqrt[e + f*x^2]))/(Sqrt[a]*Sqrt[b*e - a*f])])/(Sqrt[a]*Sqrt[b*e - a
*f])) + (d*ArcTan[(c*Sqrt[f] + d*x*(Sqrt[f]*x - Sqrt[e + f*x^2]))/(Sqrt[c]*Sqrt[d*e - c*f])])/(Sqrt[c]*Sqrt[d*
e - c*f]))/(b*c - a*d)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(781\) vs. \(2(102)=204\).
time = 0.12, size = 782, normalized size = 6.41

method	result	size
default	\(\frac {b \,d^{2} \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}+\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}-\frac {b^{2} d \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}+\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} f +\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-\frac {a f -b e}{b}}}+\frac {b^{2} d \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}-\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} f -\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-\frac {a f -b e}{b}}}-\frac {b \,d^{2} \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}-\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}\)	\(782\)

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,method=_RETURNVERBOSE)

[Out]

1/2*b*d^2/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)/((-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^2*d/(-a*b)^(1/2)/(b*(-c*d)^(1/2)+(
-a*b)^(1/2)*d)/((-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)/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)/((-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/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)/((-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))

________________________________________________________________________________________

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^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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (108) = 216\).
time = 58.27, size = 1377, normalized size = 11.29 \begin {gather*} \left [-\frac {{\left (b c^{2} f - b c d e\right )} \sqrt {a^{2} f - a b e} \log \left (\frac {8 \, a^{2} f^{2} x^{4} - 4 \, {\left (2 \, a f x^{3} - {\left (b x^{3} - a x\right )} e\right )} \sqrt {a^{2} f - a b e} \sqrt {f x^{2} + e} + {\left (b^{2} x^{4} - 6 \, a b x^{2} + a^{2}\right )} e^{2} - 8 \, {\left (a b f x^{4} - a^{2} f x^{2}\right )} e}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + {\left (a^{2} d f - a b d e\right )} \sqrt {c^{2} f - c d e} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, \frac {2 \, {\left (a^{2} d f - a b d e\right )} \sqrt {-c^{2} f + c d e} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right ) - {\left (b c^{2} f - b c d e\right )} \sqrt {a^{2} f - a b e} \log \left (\frac {8 \, a^{2} f^{2} x^{4} - 4 \, {\left (2 \, a f x^{3} - {\left (b x^{3} - a x\right )} e\right )} \sqrt {a^{2} f - a b e} \sqrt {f x^{2} + e} + {\left (b^{2} x^{4} - 6 \, a b x^{2} + a^{2}\right )} e^{2} - 8 \, {\left (a b f x^{4} - a^{2} f x^{2}\right )} e}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, -\frac {2 \, {\left (b c^{2} f - b c d e\right )} \sqrt {-a^{2} f + a b e} \arctan \left (\frac {{\left (2 \, a f x^{2} - {\left (b x^{2} - a\right )} e\right )} \sqrt {-a^{2} f + a b e} \sqrt {f x^{2} + e}}{2 \, {\left (a^{2} f^{2} x^{3} - a b x e^{2} - {\left (a b f x^{3} - a^{2} f x\right )} e\right )}}\right ) + {\left (a^{2} d f - a b d e\right )} \sqrt {c^{2} f - c d e} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\right )}}, -\frac {{\left (b c^{2} f - b c d e\right )} \sqrt {-a^{2} f + a b e} \arctan \left (\frac {{\left (2 \, a f x^{2} - {\left (b x^{2} - a\right )} e\right )} \sqrt {-a^{2} f + a b e} \sqrt {f x^{2} + e}}{2 \, {\left (a^{2} f^{2} x^{3} - a b x e^{2} - {\left (a b f x^{3} - a^{2} f x\right )} e\right )}}\right ) - {\left (a^{2} d f - a b d e\right )} \sqrt {-c^{2} f + c d e} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right )}{2 \, {\left ({\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} f^{2} - {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} f e + {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e^{2}\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="fricas")

[Out]

[-1/4*((b*c^2*f - b*c*d*e)*sqrt(a^2*f - a*b*e)*log((8*a^2*f^2*x^4 - 4*(2*a*f*x^3 - (b*x^3 - a*x)*e)*sqrt(a^2*f
- a*b*e)*sqrt(f*x^2 + e) + (b^2*x^4 - 6*a*b*x^2 + a^2)*e^2 - 8*(a*b*f*x^4 - a^2*f*x^2)*e)/(b^2*x^4 + 2*a*b*x^
2 + a^2)) + (a^2*d*f - a*b*d*e)*sqrt(c^2*f - c*d*e)*log((8*c^2*f^2*x^4 + 4*(2*c*f*x^3 - (d*x^3 - c*x)*e)*sqrt(
c^2*f - c*d*e)*sqrt(f*x^2 + e) + (d^2*x^4 - 6*c*d*x^2 + c^2)*e^2 - 8*(c*d*f*x^4 - c^2*f*x^2)*e)/(d^2*x^4 + 2*c
*d*x^2 + c^2)))/((a^2*b*c^3 - a^3*c^2*d)*f^2 - (a*b^2*c^3 - a^3*c*d^2)*f*e + (a*b^2*c^2*d - a^2*b*c*d^2)*e^2),
1/4*(2*(a^2*d*f - a*b*d*e)*sqrt(-c^2*f + c*d*e)*arctan(1/2*(2*c*f*x^2 - (d*x^2 - c)*e)*sqrt(-c^2*f + c*d*e)*s
qrt(f*x^2 + e)/(c^2*f^2*x^3 - c*d*x*e^2 - (c*d*f*x^3 - c^2*f*x)*e)) - (b*c^2*f - b*c*d*e)*sqrt(a^2*f - a*b*e)*
log((8*a^2*f^2*x^4 - 4*(2*a*f*x^3 - (b*x^3 - a*x)*e)*sqrt(a^2*f - a*b*e)*sqrt(f*x^2 + e) + (b^2*x^4 - 6*a*b*x^
2 + a^2)*e^2 - 8*(a*b*f*x^4 - a^2*f*x^2)*e)/(b^2*x^4 + 2*a*b*x^2 + a^2)))/((a^2*b*c^3 - a^3*c^2*d)*f^2 - (a*b^
2*c^3 - a^3*c*d^2)*f*e + (a*b^2*c^2*d - a^2*b*c*d^2)*e^2), -1/4*(2*(b*c^2*f - b*c*d*e)*sqrt(-a^2*f + a*b*e)*ar
ctan(1/2*(2*a*f*x^2 - (b*x^2 - a)*e)*sqrt(-a^2*f + a*b*e)*sqrt(f*x^2 + e)/(a^2*f^2*x^3 - a*b*x*e^2 - (a*b*f*x^
3 - a^2*f*x)*e)) + (a^2*d*f - a*b*d*e)*sqrt(c^2*f - c*d*e)*log((8*c^2*f^2*x^4 + 4*(2*c*f*x^3 - (d*x^3 - c*x)*e
)*sqrt(c^2*f - c*d*e)*sqrt(f*x^2 + e) + (d^2*x^4 - 6*c*d*x^2 + c^2)*e^2 - 8*(c*d*f*x^4 - c^2*f*x^2)*e)/(d^2*x^
4 + 2*c*d*x^2 + c^2)))/((a^2*b*c^3 - a^3*c^2*d)*f^2 - (a*b^2*c^3 - a^3*c*d^2)*f*e + (a*b^2*c^2*d - a^2*b*c*d^2
)*e^2), -1/2*((b*c^2*f - b*c*d*e)*sqrt(-a^2*f + a*b*e)*arctan(1/2*(2*a*f*x^2 - (b*x^2 - a)*e)*sqrt(-a^2*f + a*
b*e)*sqrt(f*x^2 + e)/(a^2*f^2*x^3 - a*b*x*e^2 - (a*b*f*x^3 - a^2*f*x)*e)) - (a^2*d*f - a*b*d*e)*sqrt(-c^2*f +
c*d*e)*arctan(1/2*(2*c*f*x^2 - (d*x^2 - c)*e)*sqrt(-c^2*f + c*d*e)*sqrt(f*x^2 + e)/(c^2*f^2*x^3 - c*d*x*e^2 -
(c*d*f*x^3 - c^2*f*x)*e)))/((a^2*b*c^3 - a^3*c^2*d)*f^2 - (a*b^2*c^3 - a^3*c*d^2)*f*e + (a*b^2*c^2*d - a^2*b*c
*d^2)*e^2)]

________________________________________________________________________________________

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)

________________________________________________________________________________________

Giac [A]
time = 1.27, 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)))

________________________________________________________________________________________

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)

________________________________________________________________________________________