\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2) (e+f x^2)} \, dx\) [280]

Optimal result

Integrand size = 30, antiderivative size = 120 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (d e-c f)}+\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (d e-c f)} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {\frac {\sqrt {-b c+a d} \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c}}-\frac {\sqrt {-b e+a f} \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e}}}{-d e+c f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.58, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {422, 301, 224, 219, 291, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 422

\(\displaystyle \frac {d \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d e-c f}\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {d \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{d e-c f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}\right )}{d e-c f}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 422

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[-d/(b*c - a*d)   Int[(c + d*x^2)^q*(e + f*x^2)^r, 
 x], x] + Simp[b/(b*c - a*d)   Int[(c + d*x^2)^(q + 1)*((e + f*x^2)^r/(a + 
b*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LeQ[q, -1]
 

Maple [A] (verified)

Time = 1.62 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(-\frac {-\frac {\left (a d -b c \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{\sqrt {\left (a d -b c \right ) c}}+\frac {\left (a f -b e \right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\sqrt {\left (a f -b e \right ) e}}}{c f -d e}\)	\(111\)
default	\(\text {Expression too large to display}\)	\(1460\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 5.37 (sec) , antiderivative size = 977, normalized size of antiderivative = 8.14 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/4*(sqrt((b*c - a*d)/c)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^ 
2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*(a*c^2*x + (2*b*c^2 - a*c*d)*x^3 
)*sqrt(b*x^2 + a)*sqrt((b*c - a*d)/c))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + sqrt 
((b*e - a*f)/e)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*( 
4*a*b*e^2 - 3*a^2*e*f)*x^2 - 4*(a*e^2*x + (2*b*e^2 - a*e*f)*x^3)*sqrt(b*x^ 
2 + a)*sqrt((b*e - a*f)/e))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/(d*e - c*f), -1/ 
4*(2*sqrt(-(b*e - a*f)/e)*arctan(1/2*((2*b*e - a*f)*x^2 + a*e)*sqrt(b*x^2 
+ a)*sqrt(-(b*e - a*f)/e)/((b^2*e - a*b*f)*x^3 + (a*b*e - a^2*f)*x)) + sqr 
t((b*c - a*d)/c)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2* 
(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*(a*c^2*x + (2*b*c^2 - a*c*d)*x^3)*sqrt(b*x 
^2 + a)*sqrt((b*c - a*d)/c))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/(d*e - c*f), 1/ 
4*(2*sqrt(-(b*c - a*d)/c)*arctan(1/2*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 
+ a)*sqrt(-(b*c - a*d)/c)/((b^2*c - a*b*d)*x^3 + (a*b*c - a^2*d)*x)) - sqr 
t((b*e - a*f)/e)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2* 
(4*a*b*e^2 - 3*a^2*e*f)*x^2 - 4*(a*e^2*x + (2*b*e^2 - a*e*f)*x^3)*sqrt(b*x 
^2 + a)*sqrt((b*e - a*f)/e))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/(d*e - c*f), 1/ 
2*(sqrt(-(b*c - a*d)/c)*arctan(1/2*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + 
a)*sqrt(-(b*c - a*d)/c)/((b^2*c - a*b*d)*x^3 + (a*b*c - a^2*d)*x)) - sqrt( 
-(b*e - a*f)/e)*arctan(1/2*((2*b*e - a*f)*x^2 + a*e)*sqrt(b*x^2 + a)*sqrt( 
-(b*e - a*f)/e)/((b^2*e - a*b*f)*x^3 + (a*b*e - a^2*f)*x)))/(d*e - c*f)...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {a + b x^{2}}}{\left (c + d x^{2}\right ) \left (e + f x^{2}\right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )} {\left (f x^{2} + e\right )}} \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {{\left (b^{\frac {3}{2}} c - a \sqrt {b} d\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{\sqrt {-b^{2} c^{2} + a b c d} {\left (d e - c f\right )}} - \frac {{\left (b^{\frac {3}{2}} e - a \sqrt {b} f\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} f + 2 \, b e - a f}{2 \, \sqrt {-b^{2} e^{2} + a b e f}}\right )}{\sqrt {-b^{2} e^{2} + a b e f} {\left (d e - c f\right )}} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {b\,x^2+a}}{\left (d\,x^2+c\right )\,\left (f\,x^2+e\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.87 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b \,x^{2}+a}-\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) e +\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b \,x^{2}+a}+\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) e -\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}-\sqrt {f}\, \sqrt {b \,x^{2}+a}-\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) c -\sqrt {e}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {a f -b e}+\sqrt {f}\, \sqrt {b \,x^{2}+a}+\sqrt {f}\, \sqrt {b}\, x}{\sqrt {e}\, \sqrt {b}}\right ) c}{c e \left (c f -d e \right )} \] Input:

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

Output:

(sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x**2) 
- sqrt(d)*sqrt(b)*x)/(sqrt(c)*sqrt(b)))*e + sqrt(c)*sqrt(a*d - b*c)*atan(( 
sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x**2) + sqrt(d)*sqrt(b)*x)/(sqrt(c)*s 
qrt(b)))*e - sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt( 
a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*c - sqrt(e)*sqrt(a*f - 
 b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x 
)/(sqrt(e)*sqrt(b)))*c)/(c*e*(c*f - d*e))