\(\int \frac {1}{x^4 (a+b x^2)^2 \sqrt {c+d x^2}} \, dx\) [766]

Optimal result

Integrand size = 24, antiderivative size = 206 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}} \]

[Out]

1/2*b^2*(-6*a*d+5*b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(7/2)/(-a*d+b*c)^(3/2)-1/6*(-2*a*d
+5*b*c)*(d*x^2+c)^(1/2)/a^2/c/(-a*d+b*c)/x^3+1/6*(-4*a^2*d^2-8*a*b*c*d+15*b^2*c^2)*(d*x^2+c)^(1/2)/a^3/c^2/(-a
*d+b*c)/x+1/2*b*(d*x^2+c)^(1/2)/a/(-a*d+b*c)/x^3/(b*x^2+a)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 206, 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} \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^2} (5 b c-2 a d)}{6 a^2 c x^3 (b c-a d)}+\frac {\sqrt {c+d x^2} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{6 a^3 c^2 x (b c-a d)}+\frac {b \sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]

[In]

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

[Out]

-1/6*((5*b*c - 2*a*d)*Sqrt[c + d*x^2])/(a^2*c*(b*c - a*d)*x^3) + ((15*b^2*c^2 - 8*a*b*c*d - 4*a^2*d^2)*Sqrt[c
+ d*x^2])/(6*a^3*c^2*(b*c - a*d)*x) + (b*Sqrt[c + d*x^2])/(2*a*(b*c - a*d)*x^3*(a + b*x^2)) + (b^2*(5*b*c - 6*
a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*(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 {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac {\int \frac {-5 b c+2 a d-4 b d x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {\int \frac {-15 b^2 c^2+8 a b c d+4 a^2 d^2-2 b d (5 b c-2 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^2 c (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac {\int -\frac {3 b^2 c^2 (5 b c-6 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^3 c^2 (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {\left (b^2 (5 b c-6 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^3 (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {\left (b^2 (5 b c-6 a d)\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a^3 (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2} \left (15 b^3 c^2 x^4+2 a b^2 c x^2 \left (5 c-4 d x^2\right )+2 a^3 d \left (c-2 d x^2\right )-2 a^2 b \left (c^2+3 c d x^2+2 d^2 x^4\right )\right )}{6 a^3 c^2 (-b c+a d) x^3 \left (a+b x^2\right )}-\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{7/2} (b c-a d)^{3/2}} \]

[In]

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

[Out]

-1/6*(Sqrt[c + d*x^2]*(15*b^3*c^2*x^4 + 2*a*b^2*c*x^2*(5*c - 4*d*x^2) + 2*a^3*d*(c - 2*d*x^2) - 2*a^2*b*(c^2 +
3*c*d*x^2 + 2*d^2*x^4)))/(a^3*c^2*(-(b*c) + a*d)*x^3*(a + b*x^2)) - (b^2*(5*b*c - 6*a*d)*ArcTan[(a*Sqrt[d] +
b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(2*a^(7/2)*(b*c - a*d)^(3/2))

Maple [A] (verified)

Time = 3.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}-6 c b \,x^{2}+a c \right )}{3 x^{3}}-\frac {b^{2} c^{2} \left (\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (6 a d -5 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 \left (a d -b c \right )}}{a^{3} c^{2}}\)	\(131\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}-6 c b \,x^{2}+a c \right )}{3 c^{2} a^{3} x^{3}}+\frac {b^{2} \left (-\frac {\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a 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}}}}{4 b}-\frac {\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a 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}}}}{4 b}-\frac {5 \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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {5 \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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a^{3}}\)	\(856\)
default	\(\frac {-\frac {\sqrt {d \,x^{2}+c}}{3 c \,x^{3}}+\frac {2 d \sqrt {d \,x^{2}+c}}{3 c^{2} x}}{a^{2}}+\frac {2 b \sqrt {d \,x^{2}+c}}{a^{3} c x}-\frac {b \left (\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a 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 )}{4 a^{3}}-\frac {b \left (\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a 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 )}{4 a^{3}}-\frac {5 b^{2} \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 )}{4 a^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {5 b^{2} \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 )}{4 a^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\)	\(885\)

[In]

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

[Out]

1/a^3*(-1/3*(d*x^2+c)^(1/2)*(-2*a*d*x^2-6*b*c*x^2+a*c)/x^3-1/2*b^2*c^2/(a*d-b*c)*(b*(d*x^2+c)^(1/2)*x/(b*x^2+a
)-(6*a*d-5*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^2

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 758, normalized size of antiderivative = 3.68 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{5} + {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} 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 (2 \, a^{3} b^{2} c^{3} - 4 \, a^{4} b c^{2} d + 2 \, a^{5} c d^{2} - {\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{3} - 8 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2} + 2 \, a^{5} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left ({\left (a^{4} b^{3} c^{4} - 2 \, a^{5} b^{2} c^{3} d + a^{6} b c^{2} d^{2}\right )} x^{5} + {\left (a^{5} b^{2} c^{4} - 2 \, a^{6} b c^{3} d + a^{7} c^{2} d^{2}\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{5} + {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} 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 (2 \, a^{3} b^{2} c^{3} - 4 \, a^{4} b c^{2} d + 2 \, a^{5} c d^{2} - {\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{3} - 8 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2} + 2 \, a^{5} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{4} b^{3} c^{4} - 2 \, a^{5} b^{2} c^{3} d + a^{6} b c^{2} d^{2}\right )} x^{5} + {\left (a^{5} b^{2} c^{4} - 2 \, a^{6} b c^{3} d + a^{7} c^{2} d^{2}\right )} x^{3}\right )}}\right ] \]

[In]

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

[Out]

[-1/24*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^5 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*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*(2*a^3*b^2*c^3 - 4*a^4*b*c^2*d + 2*a^5*
c*d^2 - (15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^4 - 2*(5*a^2*b^3*c^3 - 8*a^3*b^2*c
^2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^2)*sqrt(d*x^2 + c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^5 + (
a^5*b^2*c^4 - 2*a^6*b*c^3*d + a^7*c^2*d^2)*x^3), 1/12*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^5 + (5*a*b^3*c^3 - 6*a
^2*b^2*c^2*d)*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*(2*a^3*b^2*c^3 - 4*a^4*b*c^2*d + 2*a^5*c*d^2 - (15*a*
b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^4 - 2*(5*a^2*b^3*c^3 - 8*a^3*b^2*c^2*d + a^4*b*c
*d^2 + 2*a^5*d^3)*x^2)*sqrt(d*x^2 + c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^5 + (a^5*b^2*c^4 -
2*a^6*b*c^3*d + a^7*c^2*d^2)*x^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (182) = 364\).

Time = 0.99 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.82 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {1}{6} \, d^{\frac {7}{2}} {\left (\frac {3 \, {\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {6 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} d - b^{3} c^{2}\right )}}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} - \frac {8 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + 3 \, b c^{2} + a c d\right )}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3} d^{3}}\right )} \]

[In]

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

[Out]

1/6*d^(7/2)*(3*(5*b^3*c - 6*a*b^2*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^3*b*c*d^3 - a^4*d^4)*sqrt(a*b*c*d - a^2*d^2)) - 6*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b^3*c - 2
*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b^2*d - b^3*c^2)/((a^3*b*c*d^3 - a^4*d^4)*((sqrt(d)*x - sqrt(d*x^2 + c))^4*
b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)) - 8*(3*(sqrt(d)*x
- sqrt(d*x^2 + c))^4*b - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + 3*b*c
^2 + a*c*d)/(((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3*a^3*d^3))

Mupad [F(-1)]

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

[In]

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

[Out]

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