\(\int \frac {1}{x^4 (a+b x^2) (c+d x^2)^{3/2}} \, dx\) [721]

Optimal result

Integrand size = 24, antiderivative size = 176 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {b^3 \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}} \]

[Out]

b^3*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(5/2)/(-a*d+b*c)^(3/2)-d/c/(-a*d+b*c)/x^3/(d*x^2+c)^(
1/2)-1/3*(-4*a*d+b*c)*(d*x^2+c)^(1/2)/a/c^2/(-a*d+b*c)/x^3+1/3*(-4*a*d+3*b*c)*(2*a*d+b*c)*(d*x^2+c)^(1/2)/a^2/
c^3/(-a*d+b*c)/x

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {483, 597, 12, 385, 211} \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {b^3 \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x^2} (3 b c-4 a d) (2 a d+b c)}{3 a^2 c^3 x (b c-a d)}-\frac {\sqrt {c+d x^2} (b c-4 a d)}{3 a c^2 x^3 (b c-a d)}-\frac {d}{c x^3 \sqrt {c+d x^2} (b c-a d)} \]

[In]

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

[Out]

-(d/(c*(b*c - a*d)*x^3*Sqrt[c + d*x^2])) - ((b*c - 4*a*d)*Sqrt[c + d*x^2])/(3*a*c^2*(b*c - a*d)*x^3) + ((3*b*c
- 4*a*d)*(b*c + 2*a*d)*Sqrt[c + d*x^2])/(3*a^2*c^3*(b*c - a*d)*x) + (b^3*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*
Sqrt[c + d*x^2])])/(a^(5/2)*(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^3 \sqrt {c+d x^2}}+\frac {\int \frac {b c-4 a d-4 b d x^2}{x^4 \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^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}-\frac {\int \frac {(3 b c-4 a d) (b c+2 a d)+2 b d (b c-4 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a c^2 (b c-a d)} \\ & = -\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {\int \frac {3 b^3 c^3}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a^2 c^3 (b c-a d)} \\ & = -\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {b^3 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{a^2 (b c-a d)} \\ & = -\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {b^3 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{a^2 (b c-a d)} \\ & = -\frac {d}{c (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {(b c-4 a d) \sqrt {c+d x^2}}{3 a c^2 (b c-a d) x^3}+\frac {(3 b c-4 a d) (b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^3 (b c-a d) x}+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {3 b^2 c^2 x^2 \left (c+d x^2\right )+a^2 d \left (c^2-4 c d x^2-8 d^2 x^4\right )+a b c \left (-c^2+c d x^2+2 d^2 x^4\right )}{3 a^2 c^3 (b c-a d) x^3 \sqrt {c+d x^2}}-\frac {b^3 \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{5/2} (b c-a d)^{3/2}} \]

[In]

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

[Out]

(3*b^2*c^2*x^2*(c + d*x^2) + a^2*d*(c^2 - 4*c*d*x^2 - 8*d^2*x^4) + a*b*c*(-c^2 + c*d*x^2 + 2*d^2*x^4))/(3*a^2*
c^3*(b*c - a*d)*x^3*Sqrt[c + d*x^2]) - (b^3*ArcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sq
rt[b*c - a*d])])/(a^(5/2)*(b*c - a*d)^(3/2))

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d \,x^{2}+c}\, \left (-5 a d \,x^{2}-3 c b \,x^{2}+a c \right )}{3 x^{3} a^{2}}+\frac {d^{3} x}{\left (a d -b c \right ) \sqrt {d \,x^{2}+c}}-\frac {b^{3} c^{3} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\left (a d -b c \right ) a^{2} \sqrt {\left (a d -b c \right ) a}}}{c^{3}}\)	\(125\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}\, \left (-5 a d \,x^{2}-3 c b \,x^{2}+a c \right )}{3 c^{3} a^{2} x^{3}}-\frac {b \,d^{3} \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{2 c^{3} \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {b^{4} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-a b}\, \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {b \,d^{3} \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{2 c^{3} \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {b^{4} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-a b}\, \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-\frac {a d -b c}{b}}}\)	\(641\)
default	\(\frac {-\frac {1}{3 c \,x^{3} \sqrt {d \,x^{2}+c}}-\frac {4 d \left (-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}\right )}{3 c}}{a}-\frac {b \left (-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}\right )}{a^{2}}+\frac {b^{2} \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^{2} \sqrt {-a b}}-\frac {b^{2} \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^{2} \sqrt {-a b}}\)	\(847\)

[In]

int(1/x^4/(b*x^2+a)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

(-1/3*(d*x^2+c)^(1/2)*(-5*a*d*x^2-3*b*c*x^2+a*c)/x^3/a^2+d^3/(a*d-b*c)/(d*x^2+c)^(1/2)*x-1/(a*d-b*c)/a^2*b^3*c
^3/((a*d-b*c)*a)^(1/2)*arctanh((d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2)))/c^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (156) = 312\).

Time = 0.42 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.01 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (b^{3} c^{3} d x^{5} + b^{3} c^{4} x^{3}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - {\left (3 \, a b^{3} c^{3} d - a^{2} b^{2} c^{2} d^{2} - 10 \, a^{3} b c d^{3} + 8 \, a^{4} d^{4}\right )} x^{4} - {\left (3 \, a b^{3} c^{4} - 2 \, a^{2} b^{2} c^{3} d - 5 \, a^{3} b c^{2} d^{2} + 4 \, a^{4} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{3} b^{2} c^{5} d - 2 \, a^{4} b c^{4} d^{2} + a^{5} c^{3} d^{3}\right )} x^{5} + {\left (a^{3} b^{2} c^{6} - 2 \, a^{4} b c^{5} d + a^{5} c^{4} d^{2}\right )} x^{3}\right )}}, \frac {3 \, {\left (b^{3} c^{3} d x^{5} + b^{3} c^{4} x^{3}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - {\left (3 \, a b^{3} c^{3} d - a^{2} b^{2} c^{2} d^{2} - 10 \, a^{3} b c d^{3} + 8 \, a^{4} d^{4}\right )} x^{4} - {\left (3 \, a b^{3} c^{4} - 2 \, a^{2} b^{2} c^{3} d - 5 \, a^{3} b c^{2} d^{2} + 4 \, a^{4} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left ({\left (a^{3} b^{2} c^{5} d - 2 \, a^{4} b c^{4} d^{2} + a^{5} c^{3} d^{3}\right )} x^{5} + {\left (a^{3} b^{2} c^{6} - 2 \, a^{4} b c^{5} d + a^{5} c^{4} d^{2}\right )} x^{3}\right )}}\right ] \]

[In]

integrate(1/x^4/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

[1/12*(3*(b^3*c^3*d*x^5 + b^3*c^4*x^3)*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^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 - (3*a*b^3*c^3*d - a^2*b^2*c^2*d^2 - 10
*a^3*b*c*d^3 + 8*a^4*d^4)*x^4 - (3*a*b^3*c^4 - 2*a^2*b^2*c^3*d - 5*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*x^2)*sqrt(d*x^
2 + c))/((a^3*b^2*c^5*d - 2*a^4*b*c^4*d^2 + a^5*c^3*d^3)*x^5 + (a^3*b^2*c^6 - 2*a^4*b*c^5*d + a^5*c^4*d^2)*x^3
), 1/6*(3*(b^3*c^3*d*x^5 + b^3*c^4*x^3)*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^2*b^2*c^4 - 2*a^3*b*c^3*d + a
^4*c^2*d^2 - (3*a*b^3*c^3*d - a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 + 8*a^4*d^4)*x^4 - (3*a*b^3*c^4 - 2*a^2*b^2*c^3
*d - 5*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*x^2)*sqrt(d*x^2 + c))/((a^3*b^2*c^5*d - 2*a^4*b*c^4*d^2 + a^5*c^3*d^3)*x^5
+ (a^3*b^2*c^6 - 2*a^4*b*c^5*d + a^5*c^4*d^2)*x^3)]

Sympy [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{4} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/x**4/(b*x**2+a)/(d*x**2+c)**(3/2),x)

[Out]

Integral(1/(x**4*(a + b*x**2)*(c + d*x**2)**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(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^4), x)

Giac [A] (verification not implemented)

none

Time = 0.86 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.56 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {b^{3} \sqrt {d} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{2} b c - a^{3} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {d^{3} x}{{\left (b c^{4} - a c^{3} d\right )} \sqrt {d x^{2} + c}} - \frac {2 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 3 \, b c^{3} \sqrt {d} + 5 \, a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} c^{2}} \]

[In]

integrate(1/x^4/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

b^3*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))/((a^2*b*c -
a^3*d)*sqrt(a*b*c*d - a^2*d^2)) - d^3*x/((b*c^4 - a*c^3*d)*sqrt(d*x^2 + c)) - 2/3*(3*(sqrt(d)*x - sqrt(d*x^2
+ c))^4*b*c*sqrt(d) + 3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*d^(3/2) - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2*sq
rt(d) - 12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*c*d^(3/2) + 3*b*c^3*sqrt(d) + 5*a*c^2*d^(3/2))/(((sqrt(d)*x - sqr
t(d*x^2 + c))^2 - c)^3*a^2*c^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{x^4\,\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int(1/(x^4*(a + b*x^2)*(c + d*x^2)^(3/2)),x)

[Out]

int(1/(x^4*(a + b*x^2)*(c + d*x^2)^(3/2)), x)