\(\int \frac {1}{x^2 (a+b x^2) (c+d x^2)^{3/2}} \, dx\) [719]

Optimal result

Integrand size = 24, antiderivative size = 124 \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d}{c (b c-a d) x \sqrt {c+d x^2}}-\frac {(b c-2 a d) \sqrt {c+d x^2}}{a c^2 (b c-a d) x}-\frac {b^2 \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {483, 597, 12, 385, 211} \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {b^2 \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^2} (b c-2 a d)}{a c^2 x (b c-a d)}-\frac {d}{c x \sqrt {c+d x^2} (b c-a d)} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 483

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

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {-b c \left (c+d x^2\right )+a d \left (c+2 d x^2\right )}{a c^2 (b c-a d) x \sqrt {c+d x^2}}+\frac {b^2 \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{3/2} (b c-a d)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.01 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d \,x^{2}+c}}{a x}-\frac {d^{2} x}{\left (a d -b c \right ) \sqrt {d \,x^{2}+c}}+\frac {b^{2} c^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\left (a d -b c \right ) a \sqrt {\left (a d -b c \right ) a}}}{c^{2}}\)	\(107\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}}{c^{2} a x}-\frac {b \,d^{2} \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{2 c^{2} \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (b \sqrt {-c d}-d \sqrt {-a b}\right ) \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {b \,d^{2} \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{2 c^{2} \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (b \sqrt {-c d}-d \sqrt {-a b}\right ) \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {b^{3} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a \sqrt {-a b}\, \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (b \sqrt {-c d}-d \sqrt {-a b}\right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {b^{3} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a \sqrt {-a b}\, \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (b \sqrt {-c d}-d \sqrt {-a b}\right ) \sqrt {-\frac {a d -b c}{b}}}\)	\(623\)
default	\(\frac {-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}}{a}-\frac {b \left (-\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}}}\right )}{2 a \sqrt {-a b}}+\frac {b \left (-\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}}}\right )}{2 a \sqrt {-a b}}\)	\(779\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (110) = 220\).

Time = 0.34 (sec) , antiderivative size = 560, normalized size of antiderivative = 4.52 \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{2} b^{2} c^{4} d - 2 \, a^{3} b c^{3} d^{2} + a^{4} c^{2} d^{3}\right )} x^{3} + {\left (a^{2} b^{2} c^{5} - 2 \, a^{3} b c^{4} d + a^{4} c^{3} d^{2}\right )} x\right )}}, -\frac {{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a^{2} b^{2} c^{4} d - 2 \, a^{3} b c^{3} d^{2} + a^{4} c^{2} d^{3}\right )} x^{3} + {\left (a^{2} b^{2} c^{5} - 2 \, a^{3} b c^{4} d + a^{4} c^{3} d^{2}\right )} x\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.76 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {b^{2} \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} {\left (a b c - a^{2} d\right )}} + \frac {d^{2} x}{{\left (b c^{3} - a c^{2} d\right )} \sqrt {d x^{2} + c}} + \frac {2 \, \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )} a c} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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