3.4.60 \(\int \frac {1}{x^5 (a+\frac {b}{c+d x^2})^{3/2}} \, dx\) [360]

Optimal. Leaf size=212 \[ -\frac {a b d^2}{(b+a c)^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(3 b-4 a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{8 (b+a c)^3 x^2}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{4 (b+a c)^2 x^4}-\frac {3 b (b-4 a c) d^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{8 \sqrt {c} (b+a c)^{7/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.31, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1985, 1981, 1980, 467, 464, 214} \begin {gather*} -\frac {a b d^2}{(a c+b)^3 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {3 b d^2 (b-4 a c) \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 \sqrt {c} (a c+b)^{7/2}}-\frac {d (3 b-4 a c) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{8 x^2 (a c+b)^3}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 x^4 (a c+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b/(c + d*x^2))^(3/2)),x]

[Out]

-((a*b*d^2)/((b + a*c)^3*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])) - ((3*b - 4*a*c)*d*(c + d*x^2)*Sqrt[(b + a*c
+ a*d*x^2)/(c + d*x^2)])/(8*(b + a*c)^3*x^2) - ((c + d*x^2)^2*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(4*(b + a
*c)^2*x^4) - (3*b*(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*S
qrt[c]*(b + a*c)^(7/2))

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 464

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

Rule 467

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[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[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 {1}{x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {\left (c+d x^2\right )^{3/2}}{x^5 \left (b+a \left (c+d x^2\right )\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {\left (c+d x^2\right )^{3/2}}{x^5 \left (b+a c+a d x^2\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{x^3 (b+a c+a d x)^{3/2}} \, 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 )^2}{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 ) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{x^2 (b+a c+a d x)^{3/2}} \, 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 (c+d x^2\right )}{8 c (b+a c)^2 x^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (c+d x^2\right )^2}{4 c (b+a c) x^4 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b (b-4 a c) d^2 \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x (b+a c+a d x)^{3/2}} \, 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 {3 b (b-4 a c) d^2}{8 c (b+a c)^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(b-4 a c) d \left (c+d x^2\right )}{8 c (b+a c)^2 x^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (c+d x^2\right )^2}{4 c (b+a c) x^4 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b (b-4 a c) d^2 \sqrt {b+a \left (c+d x^2\right )}\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 (b+a c)^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {3 b (b-4 a c) d^2}{8 c (b+a c)^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(b-4 a c) d \left (c+d x^2\right )}{8 c (b+a c)^2 x^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (c+d x^2\right )^2}{4 c (b+a c) x^4 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b (b-4 a c) d^2 \sqrt {b+a \left (c+d x^2\right )}\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 (b+a c)^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {3 b (b-4 a c) d^2}{8 c (b+a c)^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(b-4 a c) d \left (c+d x^2\right )}{8 c (b+a c)^2 x^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (c+d x^2\right )^2}{4 c (b+a c) x^4 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 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 \sqrt {c} (b+a c)^{7/2} \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.35, size = 192, normalized size = 0.91 \begin {gather*} -\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (b^2 \left (2 c+5 d x^2\right )+2 a^2 c \left (c^2-d^2 x^4\right )+a b \left (4 c^2+5 c d x^2+13 d^2 x^4\right )\right )}{8 (b+a c)^3 x^4 \left (b+a \left (c+d x^2\right )\right )}-\frac {3 b (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 \sqrt {c} (-b-a c)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b/(c + d*x^2))^(3/2)),x]

[Out]

-1/8*((c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b^2*(2*c + 5*d*x^2) + 2*a^2*c*(c^2 - d^2*x^4) + a*b*(
4*c^2 + 5*c*d*x^2 + 13*d^2*x^4)))/((b + a*c)^3*x^4*(b + a*(c + d*x^2))) - (3*b*(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*Sqrt[c]*(-b - a*c)^(7/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1946\) vs. \(2(190)=380\).
time = 0.14, size = 1947, normalized size = 9.18

method	result	size
risch	\(-\frac {\left (a d \,x^{2}+a c +b \right ) \left (-2 a c d \,x^{2}+5 b d \,x^{2}+2 c^{2} a +2 b c \right )}{8 \left (a c +b \right )^{3} x^{4} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {\left (\frac {3 d^{2} b \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) a c}{4 \left (a c +b \right )^{3} \sqrt {c^{2} a +b c}}-\frac {3 d^{2} b^{2} \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right )}{16 \left (a c +b \right )^{3} \sqrt {c^{2} a +b c}}-\frac {d^{3} b a \,x^{2}}{\left (a c +b \right )^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {d^{2} b a c}{\left (a c +b \right )^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(427\)
default	\(\text {Expression too large to display}\)	\(1947\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(a+b/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/16*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)*(-8*(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^2*c^2*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)*a*b*c*d*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^3*c*d^4*x^8+6*(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^2*d*x^2+8*(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*b*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)*a^2*c^3+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^2*c+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^2*b*c*d^3*x^6+16*(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*b*c^2*d^2*x^
4+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^5*c^2*d^2*x^4-6*(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^3*d^2*x^4+16*(a*c^
2+b*c)^(3/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a^2*b*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)*(a*c^2+b*c)^(3/2)*a*b^2*c*d^2*x^4+16*(a*c^2+b*c)^(3/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b^2*c*d^2*x^4
-33*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^2*c^5*d^2*x^4+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)*a*b^4*c^2*d^3*x^6+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^3*c^3*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)^(1/2)*(a*c^2+b*c)^(3/2)*a*b^2*d^3*x^6-27*l
n((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^3*c^4*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^2*c*d^2*x^4-3*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)*a*
b^4*c^3*d^2*x^4+6*(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*b*d^2*x^4-12*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^4*b*c^5
*d^3*x^6-6*(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*b*d^4*x^8-21*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^2*c^4*d^
3*x^6+32*(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^3*c^2*d^3*x^6-12*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^4*b*c^6*d^2*
x^4-6*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^3*c^3*d^3*x^6)/c/((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)/(a*c+b)^4/x^4/(a*c^2+b*c)^(3/2)/(a*d*x^2+a*c+
b)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (192) = 384\).
time = 0.55, size = 450, normalized size = 2.12 \begin {gather*} -\frac {3 \, {\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^{3} c^{3} + 3 \, a^{2} b c^{2} + 3 \, a b^{2} c + b^{3}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {8 \, {\left (a^{3} b c^{2} + 2 \, a^{2} b^{2} c + a b^{3}\right )} d^{2} + \frac {3 \, {\left (4 \, a b c^{2} - b^{2} c\right )} {\left (a d x^{2} + a c + b\right )}^{2} d^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {5 \, {\left (4 \, a^{2} b c^{2} + 3 \, a b^{2} c - b^{3}\right )} {\left (a d x^{2} + a c + b\right )} d^{2}}{d x^{2} + c}}{8 \, {\left ({\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 2 \, {\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\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + {\left (a^{5} c^{5} + 5 \, a^{4} b c^{4} + 10 \, a^{3} b^{2} c^{3} + 10 \, a^{2} b^{3} c^{2} + 5 \, a b^{4} c + b^{5}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a+b/(d*x^2+c))^(3/2),x, algorithm="maxima")

[Out]

-3/16*(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^3*c^3 + 3*a^2*b*c^2 + 3*a*b^2*c + b^3)*sqrt((a*c + b)*c)) - 1
/8*(8*(a^3*b*c^2 + 2*a^2*b^2*c + a*b^3)*d^2 + 3*(4*a*b*c^2 - b^2*c)*(a*d*x^2 + a*c + b)^2*d^2/(d*x^2 + c)^2 -
5*(4*a^2*b*c^2 + 3*a*b^2*c - b^3)*(a*d*x^2 + a*c + b)*d^2/(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*d*x^2 + a*c + b)/(d*x^2 + c))^(5/2) - 2*(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*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) + (a^5*c^5 + 5*a^4*b*c^4 + 10*a^3*b^2*c^3 + 10*a^2*b^3*c^2 + 5*a
*b^4*c + b^5)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (192) = 384\).
time = 0.77, size = 961, normalized size = 4.53 \begin {gather*} \left [\frac {3 \, {\left ({\left (4 \, a^{2} b c - a b^{2}\right )} d^{3} x^{6} + {\left (4 \, a^{2} b c^{2} + 3 \, a b^{2} c - b^{3}\right )} d^{2} x^{4}\right )} \sqrt {a c^{2} + b c} \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 ({\left (2 \, a^{3} c^{3} - 11 \, a^{2} b c^{2} - 13 \, a b^{2} c\right )} d^{3} x^{6} - 2 \, a^{3} c^{6} - 6 \, a^{2} b c^{5} - 6 \, a b^{2} c^{4} + {\left (2 \, a^{3} c^{4} - 16 \, a^{2} b c^{3} - 23 \, a b^{2} c^{2} - 5 \, b^{3} c\right )} d^{2} x^{4} - 2 \, b^{3} c^{3} - {\left (2 \, a^{3} c^{5} + 11 \, a^{2} b c^{4} + 16 \, a b^{2} c^{3} + 7 \, b^{3} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left ({\left (a^{5} c^{5} + 4 \, a^{4} b c^{4} + 6 \, a^{3} b^{2} c^{3} + 4 \, a^{2} b^{3} c^{2} + a b^{4} c\right )} d x^{6} + {\left (a^{5} c^{6} + 5 \, a^{4} b c^{5} + 10 \, a^{3} b^{2} c^{4} + 10 \, a^{2} b^{3} c^{3} + 5 \, a b^{4} c^{2} + b^{5} c\right )} x^{4}\right )}}, -\frac {3 \, {\left ({\left (4 \, a^{2} b c - a b^{2}\right )} d^{3} x^{6} + {\left (4 \, a^{2} b c^{2} + 3 \, a b^{2} c - b^{3}\right )} d^{2} x^{4}\right )} \sqrt {-a c^{2} - b c} \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 ({\left (2 \, a^{3} c^{3} - 11 \, a^{2} b c^{2} - 13 \, a b^{2} c\right )} d^{3} x^{6} - 2 \, a^{3} c^{6} - 6 \, a^{2} b c^{5} - 6 \, a b^{2} c^{4} + {\left (2 \, a^{3} c^{4} - 16 \, a^{2} b c^{3} - 23 \, a b^{2} c^{2} - 5 \, b^{3} c\right )} d^{2} x^{4} - 2 \, b^{3} c^{3} - {\left (2 \, a^{3} c^{5} + 11 \, a^{2} b c^{4} + 16 \, a b^{2} c^{3} + 7 \, b^{3} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left ({\left (a^{5} c^{5} + 4 \, a^{4} b c^{4} + 6 \, a^{3} b^{2} c^{3} + 4 \, a^{2} b^{3} c^{2} + a b^{4} c\right )} d x^{6} + {\left (a^{5} c^{6} + 5 \, a^{4} b c^{5} + 10 \, a^{3} b^{2} c^{4} + 10 \, a^{2} b^{3} c^{3} + 5 \, a b^{4} c^{2} + b^{5} c\right )} x^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a+b/(d*x^2+c))^(3/2),x, algorithm="fricas")

[Out]

[1/32*(3*((4*a^2*b*c - a*b^2)*d^3*x^6 + (4*a^2*b*c^2 + 3*a*b^2*c - b^3)*d^2*x^4)*sqrt(a*c^2 + b*c)*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^3*c^3 - 11*a^2*b*c^2 - 13*a*b^2*c)*d^3*x^6 - 2*a^3*c^6 - 6*a^2*b*c^5 - 6*a*b^
2*c^4 + (2*a^3*c^4 - 16*a^2*b*c^3 - 23*a*b^2*c^2 - 5*b^3*c)*d^2*x^4 - 2*b^3*c^3 - (2*a^3*c^5 + 11*a^2*b*c^4 +
16*a*b^2*c^3 + 7*b^3*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^5*c^5 + 4*a^4*b*c^4 + 6*a^3*b^2*c^
3 + 4*a^2*b^3*c^2 + a*b^4*c)*d*x^6 + (a^5*c^6 + 5*a^4*b*c^5 + 10*a^3*b^2*c^4 + 10*a^2*b^3*c^3 + 5*a*b^4*c^2 +
b^5*c)*x^4), -1/16*(3*((4*a^2*b*c - a*b^2)*d^3*x^6 + (4*a^2*b*c^2 + 3*a*b^2*c - b^3)*d^2*x^4)*sqrt(-a*c^2 - b*
c)*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^3*c^3 - 11*a^2*b*c^2 - 13*a*b^2*c)*d^3*x^6 -
2*a^3*c^6 - 6*a^2*b*c^5 - 6*a*b^2*c^4 + (2*a^3*c^4 - 16*a^2*b*c^3 - 23*a*b^2*c^2 - 5*b^3*c)*d^2*x^4 - 2*b^3*c
^3 - (2*a^3*c^5 + 11*a^2*b*c^4 + 16*a*b^2*c^3 + 7*b^3*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^5
*c^5 + 4*a^4*b*c^4 + 6*a^3*b^2*c^3 + 4*a^2*b^3*c^2 + a*b^4*c)*d*x^6 + (a^5*c^6 + 5*a^4*b*c^5 + 10*a^3*b^2*c^4
+ 10*a^2*b^3*c^3 + 5*a*b^4*c^2 + b^5*c)*x^4)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{5} \left (\frac {a c + a d x^{2} + b}{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**5/(a+b/(d*x**2+c))**(3/2),x)

[Out]

Integral(1/(x**5*((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2)), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a+b/(d*x^2+c))^(3/2),x, algorithm="giac")

[Out]

undef

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^5\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b/(c + d*x^2))^(3/2)),x)

[Out]

int(1/(x^5*(a + b/(c + d*x^2))^(3/2)), x)

________________________________________________________________________________________