3.8.20 \(\int \frac {1}{x^3 (a+b x^2) (c+d x^2)^{3/2}} \, dx\) [720]

Optimal. Leaf size=156 \[ -\frac {d (b c-3 a d)}{2 a c^2 (b c-a d) \sqrt {c+d x^2}}-\frac {1}{2 a c x^2 \sqrt {c+d x^2}}+\frac {(2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{5/2}}-\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 (b c-a d)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 105, 157, 162, 65, 214} \begin {gather*} -\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 (b c-a d)^{3/2}}+\frac {(3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{5/2}}-\frac {d (b c-3 a d)}{2 a c^2 \sqrt {c+d x^2} (b c-a d)}-\frac {1}{2 a c x^2 \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(d*(b*c - 3*a*d))/(a*c^2*(b*c - a*d)*Sqrt[c + d*x^2]) - 1/(2*a*c*x^2*Sqrt[c + d*x^2]) + ((2*b*c + 3*a*d)*
ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*c^(5/2)) - (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]
])/(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 157

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] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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

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Mathematica [A]
time = 0.39, size = 142, normalized size = 0.91 \begin {gather*} \frac {\frac {a \left (-b c \left (c+d x^2\right )+a d \left (c+3 d x^2\right )\right )}{c^2 (b c-a d) x^2 \sqrt {c+d x^2}}-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{5/2}}}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(-(b*c*(c + d*x^2)) + a*d*(c + 3*d*x^2)))/(c^2*(b*c - a*d)*x^2*Sqrt[c + d*x^2]) - (2*b^(5/2)*ArcTan[(Sqrt[
b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2) + ((2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[
c]])/c^(5/2))/(2*a^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(846\) vs. \(2(130)=260\).
time = 0.14, size = 847, normalized size = 5.43

method	result	size
risch	\(-\frac {\sqrt {d \,x^{2}+c}}{2 c^{2} a \,x^{2}}+\frac {b \,d^{3} \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{2 c^{2} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {3 \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) d}{2 c^{\frac {5}{2}} a}+\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) b}{c^{\frac {3}{2}} a^{2}}-\frac {b^{3} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-\frac {a d -b c}{b}}}-\frac {b \,d^{3} \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{2 c^{2} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {b^{3} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right ) \sqrt {-\frac {a d -b c}{b}}}\)	\(686\)
default	\(\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^{2}}+\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^{2}}+\frac {-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}}{a}-\frac {b \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{a^{2}}\)	\(847\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(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^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (130) = 260\).
time = 2.27, size = 1291, normalized size = 8.28 \begin {gather*} \left [-\frac {{\left (b^{2} c^{3} d x^{4} + b^{2} c^{4} 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 ) - {\left ({\left (2 \, b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\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 b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{4} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{2}\right )}}, -\frac {2 \, {\left ({\left (2 \, b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (b^{2} c^{3} d x^{4} + b^{2} c^{4} 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 ) + 2 \, {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{4} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{2}\right )}}, \frac {2 \, {\left (b^{2} c^{3} d x^{4} + b^{2} c^{4} 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 ) + {\left ({\left (2 \, b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\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 b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{4} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{2}\right )}}, \frac {{\left (b^{2} c^{3} d x^{4} + b^{2} c^{4} 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 ) - {\left ({\left (2 \, b^{2} c^{2} d + a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (a b c^{3} - a^{2} c^{2} d + {\left (a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{4} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.28, size = 172, normalized size = 1.10 \begin {gather*} \frac {b^{3} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{2} b c - a^{3} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {{\left (d x^{2} + c\right )} b c d - 3 \, {\left (d x^{2} + c\right )} a d^{2} + 2 \, a c d^{2}}{2 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} {\left ({\left (d x^{2} + c\right )}^{\frac {3}{2}} - \sqrt {d x^{2} + c} c\right )}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.21, size = 2500, normalized size = 16.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^2/(b*c^2 - a*c*d) + (d*(c + d*x^2)*(3*a*d - b*c))/(2*a*c^2*(a*d - b*c)))/(c*(c + d*x^2)^(1/2) - (c + d*x^2)
^(3/2)) + (atan((((-b^5*(a*d - b*c)^3)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^10*c^13*d^2 - 320*a^4*b^9*c^12*d^3
+ 16*a^5*b^8*c^11*d^4 + 496*a^6*b^7*c^10*d^5 - 160*a^7*b^6*c^9*d^6 - 544*a^8*b^5*c^8*d^7 + 528*a^9*b^4*c^7*d^
8 - 144*a^10*b^3*c^6*d^9))/2 - ((-b^5*(a*d - b*c)^3)^(1/2)*(416*a^8*b^6*c^12*d^5 - 32*a^6*b^8*c^14*d^3 - 1024*
a^9*b^5*c^11*d^6 + 1056*a^10*b^4*c^10*d^7 - 512*a^11*b^3*c^9*d^8 + 96*a^12*b^2*c^8*d^9 + ((-b^5*(a*d - b*c)^3)
^(1/2)*(c + d*x^2)^(1/2)*(512*a^7*b^8*c^16*d^2 - 2816*a^8*b^7*c^15*d^3 + 6400*a^9*b^6*c^14*d^4 - 7680*a^10*b^5
*c^13*d^5 + 5120*a^11*b^4*c^12*d^6 - 1792*a^12*b^3*c^11*d^7 + 256*a^13*b^2*c^10*d^8))/(4*a^2*(a*d - b*c)^3)))/
(2*a^2*(a*d - b*c)^3))*1i)/(a^2*(a*d - b*c)^3) + ((-b^5*(a*d - b*c)^3)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^10
*c^13*d^2 - 320*a^4*b^9*c^12*d^3 + 16*a^5*b^8*c^11*d^4 + 496*a^6*b^7*c^10*d^5 - 160*a^7*b^6*c^9*d^6 - 544*a^8*
b^5*c^8*d^7 + 528*a^9*b^4*c^7*d^8 - 144*a^10*b^3*c^6*d^9))/2 - ((-b^5*(a*d - b*c)^3)^(1/2)*(32*a^6*b^8*c^14*d^
3 - 416*a^8*b^6*c^12*d^5 + 1024*a^9*b^5*c^11*d^6 - 1056*a^10*b^4*c^10*d^7 + 512*a^11*b^3*c^9*d^8 - 96*a^12*b^2
*c^8*d^9 + ((-b^5*(a*d - b*c)^3)^(1/2)*(c + d*x^2)^(1/2)*(512*a^7*b^8*c^16*d^2 - 2816*a^8*b^7*c^15*d^3 + 6400*
a^9*b^6*c^14*d^4 - 7680*a^10*b^5*c^13*d^5 + 5120*a^11*b^4*c^12*d^6 - 1792*a^12*b^3*c^11*d^7 + 256*a^13*b^2*c^1
0*d^8))/(4*a^2*(a*d - b*c)^3)))/(2*a^2*(a*d - b*c)^3))*1i)/(a^2*(a*d - b*c)^3))/(32*a^2*b^10*c^11*d^3 - 144*a^
3*b^9*c^10*d^4 + 96*a^4*b^8*c^9*d^5 + 256*a^5*b^7*c^8*d^6 - 384*a^6*b^6*c^7*d^7 + 144*a^7*b^5*c^6*d^8 - ((-b^5
*(a*d - b*c)^3)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^10*c^13*d^2 - 320*a^4*b^9*c^12*d^3 + 16*a^5*b^8*c^11*d^4
+ 496*a^6*b^7*c^10*d^5 - 160*a^7*b^6*c^9*d^6 - 544*a^8*b^5*c^8*d^7 + 528*a^9*b^4*c^7*d^8 - 144*a^10*b^3*c^6*d^
9))/2 - ((-b^5*(a*d - b*c)^3)^(1/2)*(416*a^8*b^6*c^12*d^5 - 32*a^6*b^8*c^14*d^3 - 1024*a^9*b^5*c^11*d^6 + 1056
*a^10*b^4*c^10*d^7 - 512*a^11*b^3*c^9*d^8 + 96*a^12*b^2*c^8*d^9 + ((-b^5*(a*d - b*c)^3)^(1/2)*(c + d*x^2)^(1/2
)*(512*a^7*b^8*c^16*d^2 - 2816*a^8*b^7*c^15*d^3 + 6400*a^9*b^6*c^14*d^4 - 7680*a^10*b^5*c^13*d^5 + 5120*a^11*b
^4*c^12*d^6 - 1792*a^12*b^3*c^11*d^7 + 256*a^13*b^2*c^10*d^8))/(4*a^2*(a*d - b*c)^3)))/(2*a^2*(a*d - b*c)^3)))
/(a^2*(a*d - b*c)^3) + ((-b^5*(a*d - b*c)^3)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^10*c^13*d^2 - 320*a^4*b^9*c^
12*d^3 + 16*a^5*b^8*c^11*d^4 + 496*a^6*b^7*c^10*d^5 - 160*a^7*b^6*c^9*d^6 - 544*a^8*b^5*c^8*d^7 + 528*a^9*b^4*
c^7*d^8 - 144*a^10*b^3*c^6*d^9))/2 - ((-b^5*(a*d - b*c)^3)^(1/2)*(32*a^6*b^8*c^14*d^3 - 416*a^8*b^6*c^12*d^5 +
1024*a^9*b^5*c^11*d^6 - 1056*a^10*b^4*c^10*d^7 + 512*a^11*b^3*c^9*d^8 - 96*a^12*b^2*c^8*d^9 + ((-b^5*(a*d - b
*c)^3)^(1/2)*(c + d*x^2)^(1/2)*(512*a^7*b^8*c^16*d^2 - 2816*a^8*b^7*c^15*d^3 + 6400*a^9*b^6*c^14*d^4 - 7680*a^
10*b^5*c^13*d^5 + 5120*a^11*b^4*c^12*d^6 - 1792*a^12*b^3*c^11*d^7 + 256*a^13*b^2*c^10*d^8))/(4*a^2*(a*d - b*c)
^3)))/(2*a^2*(a*d - b*c)^3)))/(a^2*(a*d - b*c)^3)))*(-b^5*(a*d - b*c)^3)^(1/2)*1i)/(a^2*(a*d - b*c)^3) + (atan
h((440*a^4*b^8*c^11*d^5*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(440*a^4*b^8*c^9*d^5 - 240*a^3*b^9*c^10*d^4 + 480*a^5*
b^7*c^8*d^6 - 1464*a^6*b^6*c^7*d^7 + 496*a^7*b^5*c^6*d^8 + 936*a^8*b^4*c^5*d^9 - 864*a^9*b^3*c^4*d^10 + 216*a^
10*b^2*c^3*d^11)) - (240*a^3*b^9*c^12*d^4*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(440*a^4*b^8*c^9*d^5 - 240*a^3*b^9*c
^10*d^4 + 480*a^5*b^7*c^8*d^6 - 1464*a^6*b^6*c^7*d^7 + 496*a^7*b^5*c^6*d^8 + 936*a^8*b^4*c^5*d^9 - 864*a^9*b^3
*c^4*d^10 + 216*a^10*b^2*c^3*d^11)) + (480*a^5*b^7*c^10*d^6*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(440*a^4*b^8*c^9*d
^5 - 240*a^3*b^9*c^10*d^4 + 480*a^5*b^7*c^8*d^6 - 1464*a^6*b^6*c^7*d^7 + 496*a^7*b^5*c^6*d^8 + 936*a^8*b^4*c^5
*d^9 - 864*a^9*b^3*c^4*d^10 + 216*a^10*b^2*c^3*d^11)) - (1464*a^6*b^6*c^9*d^7*(c + d*x^2)^(1/2))/((c^5)^(1/2)*
(440*a^4*b^8*c^9*d^5 - 240*a^3*b^9*c^10*d^4 + 480*a^5*b^7*c^8*d^6 - 1464*a^6*b^6*c^7*d^7 + 496*a^7*b^5*c^6*d^8
+ 936*a^8*b^4*c^5*d^9 - 864*a^9*b^3*c^4*d^10 + 216*a^10*b^2*c^3*d^11)) + (496*a^7*b^5*c^8*d^8*(c + d*x^2)^(1/
2))/((c^5)^(1/2)*(440*a^4*b^8*c^9*d^5 - 240*a^3*b^9*c^10*d^4 + 480*a^5*b^7*c^8*d^6 - 1464*a^6*b^6*c^7*d^7 + 49
6*a^7*b^5*c^6*d^8 + 936*a^8*b^4*c^5*d^9 - 864*a^9*b^3*c^4*d^10 + 216*a^10*b^2*c^3*d^11)) + (936*a^8*b^4*c^7*d^
9*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(440*a^4*b^8*c^9*d^5 - 240*a^3*b^9*c^10*d^4 + 480*a^5*b^7*c^8*d^6 - 1464*a^6
*b^6*c^7*d^7 + 496*a^7*b^5*c^6*d^8 + 936*a^8*b^4*c^5*d^9 - 864*a^9*b^3*c^4*d^10 + 216*a^10*b^2*c^3*d^11)) - (8
64*a^9*b^3*c^6*d^10*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(440*a^4*b^8*c^9*d^5 - 240*a^3*b^9*c^10*d^4 + 480*a^5*b^7*
c^8*d^6 - 1464*a^6*b^6*c^7*d^7 + 496*a^7*b^5*c^6*d^8 + 936*a^8*b^4*c^5*d^9 - 864*a^9*b^3*c^4*d^10 + 216*a^10*b
^2*c^3*d^11)) + (216*a^10*b^2*c^5*d^11*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(440*a^4*b^8*c^9*d^5 - 240*a^3*b^9*c^10
*d^4 + 480*a^5*b^7*c^8*d^6 - 1464*a^6*b^6*c^7*d^7 + 496*a^7*b^5*c^6*d^8 + 936*a^8*b^4*c^5*d^9 - 864*a^9*b^3*c^
4*d^10 + 216*a^10*b^2*c^3*d^11)))*(3*a*d + 2*b*...

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