\(\int \frac {1}{(a+b x^2) (c+d x^2)^{3/2}} \, dx\) [115]

Optimal result

Integrand size = 21, antiderivative size = 79 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d x}{c (b c-a d) \sqrt {c+d x^2}}+\frac {b \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} (b c-a d)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {d x}{c (-b c+a d) \sqrt {c+d x^2}}-\frac {b \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a} (b c-a d)^{3/2}} \] Input:

Integrate[1/((a + b*x^2)*(c + d*x^2)^(3/2)),x]
 

Output:

(d*x)/(c*(-(b*c) + a*d)*Sqrt[c + d*x^2]) - (b*ArcTan[(a*Sqrt[d] + b*x*(Sqr 
t[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(Sqrt[a]*(b*c - a*d 
)^(3/2))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {296, 291, 218}

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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 296

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

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b c-a d}-\frac {d x}{c \sqrt {c+d x^2} (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} (b c-a d)^{3/2}}-\frac {d x}{c \sqrt {c+d x^2} (b c-a d)}\)

Input:

Int[1/((a + b*x^2)*(c + d*x^2)^(3/2)),x]
 

Output:

-((d*x)/(c*(b*c - a*d)*Sqrt[c + d*x^2])) + (b*ArcTan[(Sqrt[b*c - a*d]*x)/( 
Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*(b*c - a*d)^(3/2))
 

Defintions of rubi rules used

rule 218

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

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 296

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d))   Int[ 
(a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N 
eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1 
]) && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {\frac {d x}{\sqrt {x^{2} d +c}}-\frac {b c \,\operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{\left (a d -b c \right ) c}\)	\(71\)
default	\(\frac {-\frac {b}{\left (a d -b c \right ) \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {2 d \sqrt {-a b}\, \left (2 d \left (x -\frac {\sqrt {-a b}}{b}\right )+\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{2 \sqrt {-a b}}-\frac {-\frac {b}{\left (a d -b c \right ) \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}-\frac {2 d \sqrt {-a b}\, \left (2 d \left (x +\frac {\sqrt {-a b}}{b}\right )-\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{2 \sqrt {-a b}}\)	\(733\)

Input:

int(1/(b*x^2+a)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (67) = 134\).

Time = 0.21 (sec) , antiderivative size = 442, normalized size of antiderivative = 5.59 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\left [-\frac {4 \, {\left (a b c d - a^{2} d^{2}\right )} \sqrt {d x^{2} + c} x - {\left (b c d x^{2} + b c^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, {\left (a b^{2} c^{4} - 2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} + {\left (a b^{2} c^{3} d - 2 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )} x^{2}\right )}}, -\frac {2 \, {\left (a b c d - a^{2} d^{2}\right )} \sqrt {d x^{2} + c} x - {\left (b c d x^{2} + b c^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{2 \, {\left (a b^{2} c^{4} - 2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} + {\left (a b^{2} c^{3} d - 2 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )} x^{2}\right )}}\right ] \] Input:

integrate(1/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="fricas")
 

Output:

[-1/4*(4*(a*b*c*d - a^2*d^2)*sqrt(d*x^2 + c)*x - (b*c*d*x^2 + b*c^2)*sqrt( 
-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*( 
3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a 
^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/(a*b^2*c^4 - 2*a^2*b* 
c^3*d + a^3*c^2*d^2 + (a*b^2*c^3*d - 2*a^2*b*c^2*d^2 + a^3*c*d^3)*x^2), -1 
/2*(2*(a*b*c*d - a^2*d^2)*sqrt(d*x^2 + c)*x - (b*c*d*x^2 + b*c^2)*sqrt(a*b 
*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt( 
d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)))/(a*b^2*c^4 
- 2*a^2*b*c^3*d + a^3*c^2*d^2 + (a*b^2*c^3*d - 2*a^2*b*c^2*d^2 + a^3*c*d^3 
)*x^2)]
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(b*x**2+a)/(d*x**2+c)**(3/2),x)
 

Output:

Integral(1/((a + b*x**2)*(c + d*x**2)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(3/2)), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {b \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} {\left (b c - a d\right )}} - \frac {d x}{{\left (b c^{2} - a c d\right )} \sqrt {d x^{2} + c}} \] Input:

integrate(1/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

b*sqrt(d)*arctan(-1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sq 
rt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*(b*c - a*d)) - d*x/((b*c^2 
 - a*c*d)*sqrt(d*x^2 + c))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:

int(1/((a + b*x^2)*(c + d*x^2)^(3/2)),x)
 

Output:

int(1/((a + b*x^2)*(c + d*x^2)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 507, normalized size of antiderivative = 6.42 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b \,c^{2}-\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b c d \,x^{2}-\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b \,c^{2}-\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b c d \,x^{2}+\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) b \,c^{2}+\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) b c d \,x^{2}+2 \sqrt {d \,x^{2}+c}\, a^{2} d^{2} x -2 \sqrt {d \,x^{2}+c}\, a b c d x +2 \sqrt {d}\, a^{2} c d +2 \sqrt {d}\, a^{2} d^{2} x^{2}-2 \sqrt {d}\, a b \,c^{2}-2 \sqrt {d}\, a b c d \,x^{2}}{2 a c \left (a^{2} d^{3} x^{2}-2 a b c \,d^{2} x^{2}+b^{2} c^{2} d \,x^{2}+a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:

int(1/(b*x^2+a)/(d*x^2+c)^(3/2),x)
 

Output:

( - sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) 
- 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c**2 - sq 
rt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a* 
d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*d*x**2 - sqrt 
(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b 
*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c**2 - sqrt(a)*sqrt( 
a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqr 
t(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*d*x**2 + sqrt(a)*sqrt(a*d - 
 b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b 
*x + 2*a*d + 2*b*d*x**2)*b*c**2 + sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sq 
rt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x** 
2)*b*c*d*x**2 + 2*sqrt(c + d*x**2)*a**2*d**2*x - 2*sqrt(c + d*x**2)*a*b*c* 
d*x + 2*sqrt(d)*a**2*c*d + 2*sqrt(d)*a**2*d**2*x**2 - 2*sqrt(d)*a*b*c**2 - 
 2*sqrt(d)*a*b*c*d*x**2)/(2*a*c*(a**2*c*d**2 + a**2*d**3*x**2 - 2*a*b*c**2 
*d - 2*a*b*c*d**2*x**2 + b**2*c**3 + b**2*c**2*d*x**2))