\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2)^2} \, dx\) [79]

Optimal result

Integrand size = 21, antiderivative size = 101 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx=-\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {390, 385, 214} \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx=\frac {(2 b c-a d) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}-\frac {d x \sqrt {a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)} \]

[In]

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

[Out]

-1/2*(d*x*Sqrt[a + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)) + ((2*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*S
qrt[a + b*x^2])])/(2*c^(3/2)*(b*c - a*d)^(3/2))

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 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 390

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

Rubi steps \begin{align*} \text {integral}& = -\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)} \\ & = -\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 c (b c-a d)} \\ & = -\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx=-\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{2 c^{3/2} (-b c+a d)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.39 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {\frac {d \sqrt {b \,x^{2}+a}\, x}{d \,x^{2}+c}-\frac {\left (a d -2 b c \right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{\sqrt {\left (a d -b c \right ) c}}}{2 \left (a d -b c \right ) c}\)	\(87\)
default	\(-\frac {-\frac {d \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {b \sqrt {-c d}\, \ln \left (\frac {\frac {2 a d -2 b c}{d}+\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{\left (a d -b c \right ) \sqrt {\frac {a d -b c}{d}}}}{4 d c}-\frac {-\frac {d \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {b \sqrt {-c d}\, \ln \left (\frac {\frac {2 a d -2 b c}{d}-\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{\left (a d -b c \right ) \sqrt {\frac {a d -b c}{d}}}}{4 d c}+\frac {\ln \left (\frac {\frac {2 a d -2 b c}{d}-\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{4 c \sqrt {-c d}\, \sqrt {\frac {a d -b c}{d}}}-\frac {\ln \left (\frac {\frac {2 a d -2 b c}{d}+\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{4 c \sqrt {-c d}\, \sqrt {\frac {a d -b c}{d}}}\)	\(810\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (85) = 170\).

Time = 0.36 (sec) , antiderivative size = 463, normalized size of antiderivative = 4.58 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx=\left [-\frac {4 \, {\left (b c^{2} d - a c d^{2}\right )} \sqrt {b x^{2} + a} x - {\left (2 \, b c^{2} - a c d + {\left (2 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{8 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}, -\frac {2 \, {\left (b c^{2} d - a c d^{2}\right )} \sqrt {b x^{2} + a} x + {\left (2 \, b c^{2} - a c d + {\left (2 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{4 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/8*(4*(b*c^2*d - a*c*d^2)*sqrt(b*x^2 + a)*x - (2*b*c^2 - a*c*d + (2*b*c*d - a*d^2)*x^2)*sqrt(b*c^2 - a*c*d)
*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*((2*b*c - a*d)*x^3 +
a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^
2 + (b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3)*x^2), -1/4*(2*(b*c^2*d - a*c*d^2)*sqrt(b*x^2 + a)*x + (2*b*c^2 -
a*c*d + (2*b*c*d - a*d^2)*x^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)
*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2 - a^2*c*d)*x)))/(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2 + (b
^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3)*x^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (85) = 170\).

Time = 0.29 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.40 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \, b^{\frac {3}{2}} {\left (\frac {{\left (2 \, b c - a d\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{2} c^{2} - a b c d\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {2 \, {\left (2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )} {\left (b^{2} c^{2} - a b c d\right )}}\right )} \]

[In]

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

[Out]

1/2*b^(3/2)*((2*b*c - a*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))/((b^2*c^2 - a*b*c*d)*sqrt(-b^2*c^2 + a*b*c*d)) - 2*(2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - (sqrt(b)*x -
sqrt(b*x^2 + a))^2*a*d + a^2*d)/(((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c -
2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)*(b^2*c^2 - a*b*c*d)))

Mupad [F(-1)]

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

[In]

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

[Out]

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