3.349 \(\int \frac {1}{x^5 \sqrt {a+\frac {b}{c+d x^2}}} \, dx\)

Optimal. Leaf size=177 \[ \frac {b d^2 (4 a c+b) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{8 c^{3/2} (a c+b)^{5/2}}+\frac {d (4 a c+b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{8 c x^2 (a c+b)^2}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c x^4 (a c+b)} \]

[Out]

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

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Rubi [A]  time = 0.47, antiderivative size = 218, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6722, 1975, 446, 96, 94, 93, 208} \[ \frac {b d^2 (4 a c+b) \sqrt {a \left (c+d x^2\right )+b} \tanh ^{-1}\left (\frac {\sqrt {a c+b} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a \left (c+d x^2\right )+b}}\right )}{8 c^{3/2} (a c+b)^{5/2} \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {d (4 a c+b) \left (a \left (c+d x^2\right )+b\right )}{8 c x^2 (a c+b)^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}{4 c x^4 (a c+b) \sqrt {a+\frac {b}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[a + b/(c + d*x^2)]),x]

[Out]

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

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

Rule 96

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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 446

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]]

Rule 1975

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 6722

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[(a + b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b + a/
v^n)^FracPart[p]), Int[u*v^(n*p)*(b + a/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[p] && ILtQ[n, 0] &
& BinomialQ[v, x] &&  !LinearQ[v, x]

Rubi steps

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

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Mathematica [A]  time = 0.48, size = 269, normalized size = 1.52 \[ \frac {c \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-2 \sqrt {c (a c+b)} \left (c+d x^2\right ) \left (2 a^2 c \left (c^2-d^2 x^4\right )+a b \left (4 c^2+c d x^2+d^2 x^4\right )+b^2 \left (2 c+d x^2\right )\right )-2 b d^2 x^4 \log (x) (4 a c+b) \sqrt {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}+b d^2 x^4 (4 a c+b) \sqrt {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )} \log \left (2 \sqrt {c (a c+b)} \sqrt {\left (c+d x^2\right ) \left (a c+a d x^2+b\right )}+2 a c \left (c+d x^2\right )+b \left (2 c+d x^2\right )\right )\right )}{16 x^4 (c (a c+b))^{5/2} \left (a \left (c+d x^2\right )+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[a + b/(c + d*x^2)]),x]

[Out]

(c*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-2*Sqrt[c*(b + a*c)]*(c + d*x^2)*(b^2*(2*c + d*x^2) + 2*a^2*c*(c^2 -
d^2*x^4) + a*b*(4*c^2 + c*d*x^2 + d^2*x^4)) - 2*b*(b + 4*a*c)*d^2*x^4*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*L
og[x] + b*(b + 4*a*c)*d^2*x^4*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) +
2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(16*(c*(b + a*c))^(5/2)*x^4*(b + a*(c + d*x^2)))

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fricas [A]  time = 0.94, size = 593, normalized size = 3.35 \[ \left [\frac {{\left (4 \, a b c + b^{2}\right )} \sqrt {a c^{2} + b c} d^{2} x^{4} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + a b c^{2} - b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} + 3 \, {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2}\right )} x^{4}}, -\frac {{\left (4 \, a b c + b^{2}\right )} \sqrt {-a c^{2} - b c} d^{2} x^{4} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) + 2 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + a b c^{2} - b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} + 3 \, {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2}\right )} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 10.39, size = 815, normalized size = 4.60 \[ -\frac {{\left (4 \, a b c d^{2} + 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 )}{8 \, {\left (a^{2} c^{3} \mathrm {sgn}\left (d x^{2} + c\right ) + 2 \, a b c^{2} \mathrm {sgn}\left (d x^{2} + c\right ) + b^{2} c \mathrm {sgn}\left (d x^{2} + c\right )\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 |} + 16 \, a^{\frac {5}{2}} b c^{4} d {\left | d \right |} + 28 \, {\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} + 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 )}^{2} a^{\frac {3}{2}} b c^{2} d {\left | d \right |} + 8 \, 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} + 13 \, {\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 \, {\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} \sqrt {a} b^{2} c d {\left | d \right |} + {\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} + {\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}}{8 \, {\left (a^{2} c^{3} \mathrm {sgn}\left (d x^{2} + c\right ) + 2 \, a b c^{2} \mathrm {sgn}\left (d x^{2} + c\right ) + b^{2} c \mathrm {sgn}\left (d x^{2} + c\right )\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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*a*b*c*d^2 + 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^2*c^3*sgn(d*x^2 + c) + 2*a*b*c^2*sgn(d*x^2 + c) + b^2*c*sgn(d*x^2 + c))*sqrt(-a*c^2 -
b*c)) + 1/8*(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) + 16*a^(5/2)*b*c^4*d*abs(d) + 28*(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 + 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))^2*a^(3/2)*b
*c^2*d*abs(d) + 8*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 + 13*(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*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*sqrt(a)*b^2*c*d*abs(d) +
(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*b^2*d^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))*b^3*c*d^2)/((a^2*c^3*sgn(d*x^2 + c) + 2*a*b*c^2*sgn(d*x^2 +
c) + b^2*c*sgn(d*x^2 + c))*(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)

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maple [B]  time = 0.05, size = 922, normalized size = 5.21 \[ \frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (4 a^{3} b \,c^{5} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )+9 a^{2} b^{2} c^{4} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )+6 a \,b^{3} c^{3} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )+b^{4} c^{2} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )-12 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a^{2} c \,d^{3} x^{6}-2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a b \,d^{3} x^{6}-20 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a^{2} c^{2} d^{2} x^{4}-12 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a b c \,d^{2} x^{4}-2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} b^{2} d^{2} x^{4}+12 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a c d \,x^{2}+2 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} b d \,x^{2}-4 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+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}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+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 )}\, \left (a c +b \right )^{3} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)*(-12*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b
*c)^(3/2)*a^2*c*d^3*x^6+4*a^3*b*c^5*d^2*x^4*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+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))/x^2)-2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(3
/2)*a*b*d^3*x^6+9*a^2*b^2*c^4*d^2*x^4*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+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))/x^2)-20*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(3/2)*a
^2*c^2*d^2*x^4+6*a*b^3*c^3*d^2*x^4*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+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))/x^2)-12*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(3/2)*a*b*
c*d^2*x^4+b^4*c^2*d^2*x^4*ln((2*a*c*d*x^2+b*d*x^2+2*a*c^2+2*b*c+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))/x^2)-2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(3/2)*b^2*d^2*x^4+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^2+b*c)^(3/2)*a*c*d*x^2+2*(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*d*x^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)/(a*c+b)^3/c^2/x^4/(a*c^2+b*c)^(3/2)

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maxima [B]  time = 2.37, size = 359, normalized size = 2.03 \[ -\frac {{\left (4 \, a b c + b^{2}\right )} d^{2} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{16 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + b^{2} c\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {{\left (4 \, a b c^{2} + b^{2} c\right )} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (4 \, a^{2} b c^{2} + 3 \, a b^{2} c - b^{3}\right )} d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{4} c^{5} + 4 \, a^{3} b c^{4} + 6 \, a^{2} b^{2} c^{3} + 4 \, a b^{3} c^{2} + b^{4} c + \frac {{\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {2 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*a*b*c + 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^2*c^3 + 2*a*b*c^2 + b^2*c)*sqrt((a*c + b)*c)) - 1/8*((4*a*b*c
^2 + b^2*c)*d^2*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) - (4*a^2*b*c^2 + 3*a*b^2*c - b^3)*d^2*sqrt((a*d*x^2 +
a*c + b)/(d*x^2 + c)))/(a^4*c^5 + 4*a^3*b*c^4 + 6*a^2*b^2*c^3 + 4*a*b^3*c^2 + b^4*c + (a^2*c^5 + 2*a*b*c^4 + b
^2*c^3)*(a*d*x^2 + a*c + b)^2/(d*x^2 + c)^2 - 2*(a^3*c^5 + 3*a^2*b*c^4 + 3*a*b^2*c^3 + b^3*c^2)*(a*d*x^2 + a*c
+ b)/(d*x^2 + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{5} \sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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