3.4.23 \(\int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx\) [323]

Optimal. Leaf size=174 \[ \frac {(5 b+4 a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{4 c^2 x^4}-\frac {b (3 b+4 a c) d^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{8 c^{5/2} (b+a c)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.23, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1985, 1981, 1980, 466, 393, 214} \begin {gather*} -\frac {b d^2 (4 a c+3 b) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{8 c^{5/2} (a c+b)^{3/2}}+\frac {d (4 a c+5 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{8 c^2 x^2 (a c+b)}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/(c + d*x^2)]/x^5,x]

[Out]

((5*b + 4*a*c)*d*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(8*c^2*(b + a*c)*x^2) - ((c + d*x^2)^2*Sqr
t[(b + a*c + a*d*x^2)/(c + d*x^2)])/(4*c^2*x^4) - (b*(3*b + 4*a*c)*d^2*ArcTanh[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^
2)/(c + d*x^2)])/Sqrt[b + a*c]])/(8*c^(5/2)*(b + a*c)^(3/2))

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 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 466

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1980

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_Symbol] :> With[{q = Denominator[p]}
, Dist[q*e*(b*c - a*d), Subst[Int[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a
+ b*x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]

Rule 1981

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

Rule 1985

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u*((b + a*c + a*d*x^n)/(c + d*x^n))^p
, x] /; FreeQ[{a, b, c, d, n, p}, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a \left (c+d x^2\right )}}{x^5 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{x^5 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a c+a d x}}{x^3 \sqrt {c+d x}} \, dx,x,x^2\right )}{2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}-\frac {\left ((3 b+4 a c) d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a c+a d x}}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right )}{8 c (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(3 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}+\frac {\left (b (3 b+4 a c) d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{16 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(3 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}+\frac {\left (b (3 b+4 a c) d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {1}{-c-(-b-a c) x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{8 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(3 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}-\frac {b (3 b+4 a c) d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {b+a c} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {b+a \left (c+d x^2\right )}}\right )}{8 c^{5/2} (b+a c)^{3/2} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 152, normalized size = 0.87 \begin {gather*} -\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (b \left (2 c-3 d x^2\right )+2 a c \left (c-d x^2\right )\right )}{8 c^2 (b+a c) x^4}-\frac {b (3 b+4 a c) d^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{8 c^{5/2} (-b-a c)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/(c + d*x^2)]/x^5,x]

[Out]

-1/8*((c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b*(2*c - 3*d*x^2) + 2*a*c*(c - d*x^2)))/(c^2*(b + a*c
)*x^4) - (b*(3*b + 4*a*c)*d^2*ArcTan[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[-b - a*c]])/(8*c^(5/
2)*(-b - a*c)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(922\) vs. \(2(154)=308\).
time = 0.09, size = 923, normalized size = 5.30

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (-2 a c d \,x^{2}-3 b d \,x^{2}+2 c^{2} a +2 b c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{8 c^{2} x^{4} \left (a c +b \right )}+\frac {\left (-\frac {d^{2} b \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) a}{4 c \left (a c +b \right ) \sqrt {c^{2} a +b c}}-\frac {3 d^{2} b^{2} \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right )}{16 c^{2} \left (a c +b \right ) \sqrt {c^{2} a +b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{a d \,x^{2}+a c +b}\)	\(337\)
default	\(-\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (12 a^{2} d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, x^{6} c \left (c^{2} a +b c \right )^{\frac {3}{2}}+4 \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 c^{2} a +2 \sqrt {c^{2} a +b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}+2 b c}{x^{2}}\right ) a^{3} b \,c^{5} d^{2} x^{4}+10 a \,d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, x^{6} b \left (c^{2} a +b c \right )^{\frac {3}{2}}+11 \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 c^{2} a +2 \sqrt {c^{2} a +b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}+2 b c}{x^{2}}\right ) a^{2} b^{2} c^{4} d^{2} x^{4}+20 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, a^{2} c^{2} d^{2} x^{4} \left (c^{2} a +b c \right )^{\frac {3}{2}}+10 \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 c^{2} a +2 \sqrt {c^{2} a +b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}+2 b c}{x^{2}}\right ) a \,b^{3} c^{3} d^{2} x^{4}+28 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, a c \,d^{2} b \,x^{4} \left (c^{2} a +b c \right )^{\frac {3}{2}}+3 \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 c^{2} a +2 \sqrt {c^{2} a +b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}+2 b c}{x^{2}}\right ) b^{4} c^{2} d^{2} x^{4}+10 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, b^{2} d^{2} x^{4} \left (c^{2} a +b c \right )^{\frac {3}{2}}-12 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c \right )^{\frac {3}{2}} a c d \,x^{2} \left (c^{2} a +b c \right )^{\frac {3}{2}}-10 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c \right )^{\frac {3}{2}} b d \,x^{2} \left (c^{2} a +b c \right )^{\frac {3}{2}}+4 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c \right )^{\frac {3}{2}} \left (c^{2} a +b c \right )^{\frac {3}{2}} a \,c^{2}+4 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c \right )^{\frac {3}{2}} \left (c^{2} a +b c \right )^{\frac {3}{2}} b c \right )}{16 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, c^{3} \left (a c +b \right )^{2} x^{4} \left (c^{2} a +b c \right )^{\frac {3}{2}}}\)	\(923\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/(d*x^2+c))^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/16*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)*(12*a^2*d^3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*
x^6*c*(a*c^2+b*c)^(3/2)+4*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a
*c^2+b*c)^(1/2)+2*b*c)/x^2)*a^3*b*c^5*d^2*x^4+10*a*d^3*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*x^6*b*(
a*c^2+b*c)^(3/2)+11*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b
*c)^(1/2)+2*b*c)/x^2)*a^2*b^2*c^4*d^2*x^4+20*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*a^2*c^2*d^2*x^4*(
a*c^2+b*c)^(3/2)+10*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b
*c)^(1/2)+2*b*c)/x^2)*a*b^3*c^3*d^2*x^4+28*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*a*c*d^2*b*x^4*(a*c^
2+b*c)^(3/2)+3*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(
1/2)+2*b*c)/x^2)*b^4*c^2*d^2*x^4+10*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*b^2*d^2*x^4*(a*c^2+b*c)^(3
/2)-12*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*a*c*d*x^2*(a*c^2+b*c)^(3/2)-10*(a*d^2*x^4+2*a*c*d*x^2+b
*d*x^2+a*c^2+b*c)^(3/2)*b*d*x^2*(a*c^2+b*c)^(3/2)+4*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c
)^(3/2)*a*c^2+4*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(3/2)*b*c)/((d*x^2+c)*(a*d*x^2+a*c
+b))^(1/2)/c^3/(a*c+b)^2/x^4/(a*c^2+b*c)^(3/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (154) = 308\).
time = 0.51, size = 322, normalized size = 1.85 \begin {gather*} \frac {{\left (4 \, a b c + 3 \, b^{2}\right )} d^{2} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{16 \, {\left (a c^{3} + b c^{2}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {{\left (4 \, a b c^{2} + 5 \, b^{2} c\right )} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (4 \, a^{2} b c^{2} + 7 \, a b^{2} c + 3 \, b^{3}\right )} d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2} + \frac {{\left (a c^{5} + b c^{4}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {2 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^5,x, algorithm="maxima")

[Out]

1/16*(4*a*b*c + 3*b^2)*d^2*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - sqrt((a*c + b)*c))/(c*sqrt((a*d*x^2
+ a*c + b)/(d*x^2 + c)) + sqrt((a*c + b)*c)))/((a*c^3 + b*c^2)*sqrt((a*c + b)*c)) - 1/8*((4*a*b*c^2 + 5*b^2*c)
*d^2*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) - (4*a^2*b*c^2 + 7*a*b^2*c + 3*b^3)*d^2*sqrt((a*d*x^2 + a*c + b)/
(d*x^2 + c)))/(a^3*c^5 + 3*a^2*b*c^4 + 3*a*b^2*c^3 + b^3*c^2 + (a*c^5 + b*c^4)*(a*d*x^2 + a*c + b)^2/(d*x^2 +
c)^2 - 2*(a^2*c^5 + 2*a*b*c^4 + b^2*c^3)*(a*d*x^2 + a*c + b)/(d*x^2 + c))

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Fricas [A]
time = 0.49, size = 577, normalized size = 3.32 \begin {gather*} \left [\frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt {a c^{2} + b c} d^{2} x^{4} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + 5 \, a b c^{2} + 3 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} - {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} x^{4}}, \frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt {-a c^{2} - b c} d^{2} x^{4} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + 5 \, a b c^{2} + 3 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} - {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/32*((4*a*b*c + 3*b^2)*sqrt(a*c^2 + b*c)*d^2*x^4*log(((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a
*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a*c + b)*d^2*x^4 + 2*a*c^3 + (4*a*c^2 + 3
*b*c)*d*x^2 + 2*b*c^2)*sqrt(a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/x^4) - 4*(2*a^2*c^5 - (2*a^2*c
^3 + 5*a*b*c^2 + 3*b^2*c)*d^2*x^4 + 4*a*b*c^4 + 2*b^2*c^3 - (a*b*c^3 + b^2*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b
)/(d*x^2 + c)))/((a^2*c^5 + 2*a*b*c^4 + b^2*c^3)*x^4), 1/16*((4*a*b*c + 3*b^2)*sqrt(-a*c^2 - b*c)*d^2*x^4*arct
an(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt(-a*c^2 - b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*c^3
+ 2*a*b*c^2 + (a^2*c^2 + a*b*c)*d*x^2 + b^2*c)) - 2*(2*a^2*c^5 - (2*a^2*c^3 + 5*a*b*c^2 + 3*b^2*c)*d^2*x^4 +
4*a*b*c^4 + 2*b^2*c^3 - (a*b*c^3 + b^2*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^2*c^5 + 2*a*b*c^
4 + b^2*c^3)*x^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x**2+c))**(1/2)/x**5,x)

[Out]

Integral(sqrt((a*c + a*d*x**2 + b)/(c + d*x**2))/x**5, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 713 vs. \(2 (154) = 308\).
time = 4.45, size = 713, normalized size = 4.10 \begin {gather*} \frac {1}{8} \, {\left (\frac {{\left (4 \, a b c d^{2} + 3 \, b^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}}{\sqrt {-a c^{2} - b c}}\right )}{{\left (a c^{3} + b c^{2}\right )} \sqrt {-a c^{2} - b c}} + \frac {8 \, a^{\frac {7}{2}} c^{5} d {\left | d \right |} + 16 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{3} c^{4} d^{2} + 8 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {5}{2}} c^{3} d {\left | d \right |} + 24 \, a^{\frac {5}{2}} b c^{4} d {\left | d \right |} + 36 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} b c^{3} d^{2} + 8 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {3}{2}} b c^{2} d {\left | d \right |} + 24 \, a^{\frac {3}{2}} b^{2} c^{3} d {\left | d \right |} - 4 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a b c d^{2} + 25 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b^{2} c^{2} d^{2} + 8 \, \sqrt {a} b^{3} c^{2} d {\left | d \right |} - 3 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} b^{2} d^{2} + 5 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} b^{3} c d^{2}}{{\left (a c^{3} + b c^{2}\right )} {\left (a c^{2} - {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} + b c\right )}^{2}}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x^2+c))^(1/2)/x^5,x, algorithm="giac")

[Out]

1/8*((4*a*b*c*d^2 + 3*b^2*d^2)*arctan(-(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c
))/sqrt(-a*c^2 - b*c))/((a*c^3 + b*c^2)*sqrt(-a*c^2 - b*c)) + (8*a^(7/2)*c^5*d*abs(d) + 16*(sqrt(a*d^2)*x^2 -
sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^3*c^4*d^2 + 8*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a
*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^(5/2)*c^3*d*abs(d) + 24*a^(5/2)*b*c^4*d*abs(d) + 36*(sqrt(a*d^2)*x^2 -
sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^2*b*c^3*d^2 + 8*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2
*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^(3/2)*b*c^2*d*abs(d) + 24*a^(3/2)*b^2*c^3*d*abs(d) - 4*(sqrt(a*d^2)*x
^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*a*b*c*d^2 + 25*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4
+ 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a*b^2*c^2*d^2 + 8*sqrt(a)*b^3*c^2*d*abs(d) - 3*(sqrt(a*d^2)*x^2 - sqr
t(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*b^2*d^2 + 5*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*
x^2 + b*d*x^2 + a*c^2 + b*c))*b^3*c*d^2)/((a*c^3 + b*c^2)*(a*c^2 - (sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d
*x^2 + b*d*x^2 + a*c^2 + b*c))^2 + b*c)^2))*sgn(d*x^2 + c)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/(c + d*x^2))^(1/2)/x^5,x)

[Out]

int((a + b/(c + d*x^2))^(1/2)/x^5, x)

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