\(\int \frac {1}{x (a+b x^2)^2 \sqrt {c+d x^2}} \, dx\) [763]

Optimal result

Integrand size = 24, antiderivative size = 130 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 \sqrt {c}}+\frac {\sqrt {b} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 105, 162, 65, 214} \[ \int \frac {1}{x \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 \sqrt {c}}+\frac {b \sqrt {c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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 457

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {b c-a d+\frac {b d x}{2}}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a (b c-a d)} \\ & = \frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2}-\frac {(b (2 b c-3 a d)) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2 (b c-a d)} \\ & = \frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a^2 d}-\frac {(b (2 b c-3 a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^2 d (b c-a d)} \\ & = \frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 \sqrt {c}}+\frac {\sqrt {b} (2 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {-\frac {a b \sqrt {c+d x^2}}{(-b c+a d) \left (a+b x^2\right )}+\frac {\sqrt {b} (2 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {\left (b \,x^{2}+a \right ) b \left (b c -\frac {3 a d}{2}\right ) \sqrt {c}\, \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\frac {\left (2 \left (b \,x^{2}+a \right ) \left (a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+b \sqrt {d \,x^{2}+c}\, a \sqrt {c}\right ) \sqrt {\left (a d -b c \right ) b}}{2}}{\sqrt {c}\, \sqrt {\left (a d -b c \right ) b}\, a^{2} \left (a d -b c \right ) \left (b \,x^{2}+a \right )}\)	\(145\)
default	\(-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a^{2} \sqrt {c}}+\frac {\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 {-\frac {a d -b c}{b}}}+\frac {\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 {-\frac {a d -b c}{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 a \sqrt {-a 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 a \sqrt {-a b}}\)	\(847\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.56 (sec) , antiderivative size = 1037, normalized size of antiderivative = 7.98 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\left [\frac {4 \, \sqrt {d x^{2} + c} a b c + {\left (2 \, a b c^{2} - 3 \, a^{2} c d + {\left (2 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right )}{8 \, {\left (a^{3} b c^{2} - a^{4} c d + {\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{2}\right )}}, \frac {4 \, \sqrt {d x^{2} + c} a b c + 8 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, a b c^{2} - 3 \, a^{2} c d + {\left (2 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \, {\left (a^{3} b c^{2} - a^{4} c d + {\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{2}\right )}}, \frac {2 \, \sqrt {d x^{2} + c} a b c - {\left (2 \, a b c^{2} - 3 \, a^{2} c d + {\left (2 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + 2 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right )}{4 \, {\left (a^{3} b c^{2} - a^{4} c d + {\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{2}\right )}}, \frac {2 \, \sqrt {d x^{2} + c} a b c - {\left (2 \, a b c^{2} - 3 \, a^{2} c d + {\left (2 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + 4 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right )}{4 \, {\left (a^{3} b c^{2} - a^{4} c d + {\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{2}\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(1/x/(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), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {\sqrt {d x^{2} + c} b d}{2 \, {\left (a b c - a^{2} d\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} - \frac {{\left (2 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 3023, normalized size of antiderivative = 23.25 \[ \int \frac {1}{x \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan((((((c + d*x^2)^(1/2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d^3))/(2*(a^4*d^2 + a^2*b^2*c^2 - 2*a
^3*b*c*d)) - (((4*a^6*b^2*d^5 - 6*a^5*b^3*c*d^4 + 2*a^4*b^4*c^2*d^3)/(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d) - (
(c + d*x^2)^(1/2)*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2)*(16*a^7*b^2*d^5 - 64*a^6*b^3*c*d^4 - 32*a^4*b^5*c^3
*d^2 + 80*a^5*b^4*c^2*d^3))/(8*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d
- 3*a^4*b*c*d^2)))*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2))/(4*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a
^4*b*c*d^2)))*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2)*1i)/(4*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4
*b*c*d^2)) + ((((c + d*x^2)^(1/2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d^3))/(2*(a^4*d^2 + a^2*b^2*c^2
- 2*a^3*b*c*d)) + (((4*a^6*b^2*d^5 - 6*a^5*b^3*c*d^4 + 2*a^4*b^4*c^2*d^3)/(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*
d) + ((c + d*x^2)^(1/2)*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2)*(16*a^7*b^2*d^5 - 64*a^6*b^3*c*d^4 - 32*a^4*b
^5*c^3*d^2 + 80*a^5*b^4*c^2*d^3))/(8*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*
c^2*d - 3*a^4*b*c*d^2)))*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2))/(4*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d
- 3*a^4*b*c*d^2)))*(3*a*d - 2*b*c)*(-b*(a*d - b*c)^3)^(1/2)*1i)/(4*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d -
3*a^4*b*c*d^2)))/(((3*a*b^3*d^4)/2 - b^4*c*d^3)/(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d) - ((((c + d*x^2)^(1/2)*
(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d^3))/(2*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)) - (((4*a^6*b^2*d^
5 - 6*a^5*b^3*c*d^4 + 2*a^4*b^4*c^2*d^3)/(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d) - ((c + d*x^2)^(1/2)*(3*a*d - 2
*b*c)*(-b*(a*d - b*c)^3)^(1/2)*(16*a^7*b^2*d^5 - 64*a^6*b^3*c*d^4 - 32*a^4*b^5*c^3*d^2 + 80*a^5*b^4*c^2*d^3))/
(8*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)))*(3*a*d -
2*b*c)*(-b*(a*d - b*c)^3)^(1/2))/(4*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)))*(3*a*d - 2*b*c
)*(-b*(a*d - b*c)^3)^(1/2))/(4*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)) + ((((c + d*x^2)^(1/
2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d^3))/(2*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)) + (((4*a^6*b^2
*d^5 - 6*a^5*b^3*c*d^4 + 2*a^4*b^4*c^2*d^3)/(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d) + ((c + d*x^2)^(1/2)*(3*a*d
- 2*b*c)*(-b*(a*d - b*c)^3)^(1/2)*(16*a^7*b^2*d^5 - 64*a^6*b^3*c*d^4 - 32*a^4*b^5*c^3*d^2 + 80*a^5*b^4*c^2*d^3
))/(8*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)))*(3*a*d
- 2*b*c)*(-b*(a*d - b*c)^3)^(1/2))/(4*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)))*(3*a*d - 2*
b*c)*(-b*(a*d - b*c)^3)^(1/2))/(4*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2))))*(3*a*d - 2*b*c)
*(-b*(a*d - b*c)^3)^(1/2)*1i)/(2*(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2)) - (atan((((((4*a^6
*b^2*d^5 - 6*a^5*b^3*c*d^4 + 2*a^4*b^4*c^2*d^3)/(2*(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d)) - ((c + d*x^2)^(1/2)
*(16*a^7*b^2*d^5 - 64*a^6*b^3*c*d^4 - 32*a^4*b^5*c^3*d^2 + 80*a^5*b^4*c^2*d^3))/(8*a^2*c^(1/2)*(a^4*d^2 + a^2*
b^2*c^2 - 2*a^3*b*c*d)))/(2*a^2*c^(1/2)) - ((c + d*x^2)^(1/2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d^3
))/(4*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)))*1i)/(a^2*c^(1/2)) - ((((4*a^6*b^2*d^5 - 6*a^5*b^3*c*d^4 + 2*a^4*
b^4*c^2*d^3)/(2*(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d)) + ((c + d*x^2)^(1/2)*(16*a^7*b^2*d^5 - 64*a^6*b^3*c*d^4
- 32*a^4*b^5*c^3*d^2 + 80*a^5*b^4*c^2*d^3))/(8*a^2*c^(1/2)*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)))/(2*a^2*c^(
1/2)) + ((c + d*x^2)^(1/2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d^3))/(4*(a^4*d^2 + a^2*b^2*c^2 - 2*a^
3*b*c*d)))*1i)/(a^2*c^(1/2)))/(((3*a*b^3*d^4)/2 - b^4*c*d^3)/(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d) + (((4*a^6*
b^2*d^5 - 6*a^5*b^3*c*d^4 + 2*a^4*b^4*c^2*d^3)/(2*(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d)) - ((c + d*x^2)^(1/2)*
(16*a^7*b^2*d^5 - 64*a^6*b^3*c*d^4 - 32*a^4*b^5*c^3*d^2 + 80*a^5*b^4*c^2*d^3))/(8*a^2*c^(1/2)*(a^4*d^2 + a^2*b
^2*c^2 - 2*a^3*b*c*d)))/(2*a^2*c^(1/2)) - ((c + d*x^2)^(1/2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d^3)
)/(4*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)))/(a^2*c^(1/2)) + (((4*a^6*b^2*d^5 - 6*a^5*b^3*c*d^4 + 2*a^4*b^4*c^
2*d^3)/(2*(a^5*d^2 + a^3*b^2*c^2 - 2*a^4*b*c*d)) + ((c + d*x^2)^(1/2)*(16*a^7*b^2*d^5 - 64*a^6*b^3*c*d^4 - 32*
a^4*b^5*c^3*d^2 + 80*a^5*b^4*c^2*d^3))/(8*a^2*c^(1/2)*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d)))/(2*a^2*c^(1/2))
+ ((c + d*x^2)^(1/2)*(13*a^2*b^3*d^4 + 8*b^5*c^2*d^2 - 20*a*b^4*c*d^3))/(4*(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*
d)))/(a^2*c^(1/2))))*1i)/(a^2*c^(1/2)) - (b*d*(c + d*x^2)^(1/2))/(2*(a^2*d - a*b*c)*(b*(c + d*x^2) + a*d - b*c
))