\(\int \frac {(c+d x^2)^{3/2}}{x^3 (a+b x^2)^2} \, dx\) [747]

Optimal result

Integrand size = 24, antiderivative size = 170 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {(2 b c-a d) \sqrt {c+d x^2}}{2 a^2 \left (a+b x^2\right )}-\frac {c \sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )}+\frac {\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 \sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 100, 156, 162, 65, 214} \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 \sqrt {b}}-\frac {\sqrt {c+d x^2} (2 b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac {c \sqrt {c+d x^2}}{2 a x^2 \left (a+b x^2\right )} \]

[In]

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

[Out]

-1/2*((2*b*c - a*d)*Sqrt[c + d*x^2])/(a^2*(a + b*x^2)) - (c*Sqrt[c + d*x^2])/(2*a*x^2*(a + b*x^2)) + (Sqrt[c]*
(4*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3) - (Sqrt[b*c - a*d]*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*Sq
rt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^3*Sqrt[b])

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 100

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

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(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*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\frac {a \sqrt {c+d x^2} \left (-a c-2 b c x^2+a d x^2\right )}{x^2 \left (a+b x^2\right )}+\frac {\left (4 b^2 c^2-5 a b c d+a^2 d^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} \sqrt {-b c+a d}}+\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3} \]

[In]

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

[Out]

((a*Sqrt[c + d*x^2]*(-(a*c) - 2*b*c*x^2 + a*d*x^2))/(x^2*(a + b*x^2)) + ((4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Arc
Tan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*Sqrt[-(b*c) + a*d]) + Sqrt[c]*(4*b*c - 3*a*d)*ArcT
anh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3)

Maple [A] (verified)

Time = 3.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(\frac {\frac {x^{2} \left (b \,x^{2}+a \right ) \left (a d -b c \right ) \left (a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2}+2 \sqrt {\left (a d -b c \right ) b}\, \left (\left (b \,x^{2}+a \right ) \left (c^{\frac {3}{2}} b -\frac {3 a d \sqrt {c}}{4}\right ) x^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )-\frac {\left (2 c b \,x^{2}+a \left (-d \,x^{2}+c \right )\right ) a \sqrt {d \,x^{2}+c}}{4}\right )}{x^{2} \sqrt {\left (a d -b c \right ) b}\, \left (b \,x^{2}+a \right ) a^{3}}\)	\(165\)
risch	\(-\frac {c \sqrt {d \,x^{2}+c}}{2 a^{2} x^{2}}-\frac {\frac {\sqrt {c}\, \left (3 a d -4 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a}-\frac {2 c \left (a d -b c \right ) \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 )}{a \sqrt {-\frac {a d -b c}{b}}}-\frac {2 c \left (a d -b c \right ) \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 )}{a \sqrt {-\frac {a d -b c}{b}}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \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 )}{2 b \sqrt {-a b}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \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 )}{2 b \sqrt {-a b}}}{2 a^{2}}\)	\(939\)
default	\(\text {Expression too large to display}\)	\(3521\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 1034, normalized size of antiderivative = 6.08 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\left [-\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{b}} \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 (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{4} + {\left (4 \, a b c - 3 \, a^{2} 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^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {4 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{4} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {\frac {b c - a d}{b}} \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 (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{4} + {\left (4 \, a b c - 3 \, a^{2} 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 ) + 2 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{4} + {\left (4 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{4} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 2 \, {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.27 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b c d - 2 \, \sqrt {d x^{2} + c} b c^{2} d - {\left (d x^{2} + c\right )}^{\frac {3}{2}} a d^{2} + 2 \, \sqrt {d x^{2} + c} a c d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )} a^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.57 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.59 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {b^2\,c^2\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {b^2\,c-a\,b\,d}}{\frac {a^2\,b\,c\,d^7}{4}-\frac {5\,a\,b^2\,c^2\,d^6}{4}+b^3\,c^3\,d^5}-\frac {b\,c\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {b^2\,c-a\,b\,d}}{4\,\left (\frac {a\,b\,c\,d^7}{4}-\frac {5\,b^2\,c^2\,d^6}{4}+\frac {b^3\,c^3\,d^5}{a}\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-4\,b\,c\right )}{2\,a^3\,b}-\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {3\,b\,\sqrt {c}\,d^7\,\sqrt {d\,x^2+c}}{4\,\left (\frac {3\,b\,c\,d^7}{4}-\frac {7\,b^2\,c^2\,d^6}{4\,a}+\frac {b^3\,c^3\,d^5}{a^2}\right )}-\frac {7\,b^2\,c^{3/2}\,d^6\,\sqrt {d\,x^2+c}}{4\,\left (\frac {3\,a\,b\,c\,d^7}{4}-\frac {7\,b^2\,c^2\,d^6}{4}+\frac {b^3\,c^3\,d^5}{a}\right )}+\frac {b^3\,c^{5/2}\,d^5\,\sqrt {d\,x^2+c}}{\frac {3\,a^2\,b\,c\,d^7}{4}-\frac {7\,a\,b^2\,c^2\,d^6}{4}+b^3\,c^3\,d^5}\right )\,\left (3\,a\,d-4\,b\,c\right )}{2\,a^3}-\frac {\frac {\left (a\,c\,d^2-b\,c^2\,d\right )\,\sqrt {d\,x^2+c}}{a^2}-\frac {d\,{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d-2\,b\,c\right )}{2\,a^2}}{\left (d\,x^2+c\right )\,\left (a\,d-2\,b\,c\right )+b\,{\left (d\,x^2+c\right )}^2+b\,c^2-a\,c\,d} \]

[In]

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

[Out]

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