3.4.34 \(\int \frac {1}{x^{3/2} (1+x^2)^3} \, dx\)

Optimal. Leaf size=138 \[ \frac {9}{16 \sqrt {x} \left (x^2+1\right )}+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}-\frac {45}{16 \sqrt {x}}-\frac {45 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}+\frac {45 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}+\frac {45 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {45 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {290, 325, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {9}{16 \sqrt {x} \left (x^2+1\right )}+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}-\frac {45}{16 \sqrt {x}}-\frac {45 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}+\frac {45 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}+\frac {45 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {45 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(1 + x^2)^3),x]

[Out]

-45/(16*Sqrt[x]) + 1/(4*Sqrt[x]*(1 + x^2)^2) + 9/(16*Sqrt[x]*(1 + x^2)) + (45*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(32
*Sqrt[2]) - (45*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (45*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) + (
45*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 290

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

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx &=\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{8} \int \frac {1}{x^{3/2} \left (1+x^2\right )^2} \, dx\\ &=\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{16 \sqrt {x} \left (1+x^2\right )}+\frac {45}{32} \int \frac {1}{x^{3/2} \left (1+x^2\right )} \, dx\\ &=-\frac {45}{16 \sqrt {x}}+\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{16 \sqrt {x} \left (1+x^2\right )}-\frac {45}{32} \int \frac {\sqrt {x}}{1+x^2} \, dx\\ &=-\frac {45}{16 \sqrt {x}}+\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{16 \sqrt {x} \left (1+x^2\right )}-\frac {45}{16} \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {45}{16 \sqrt {x}}+\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{16 \sqrt {x} \left (1+x^2\right )}+\frac {45}{32} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )-\frac {45}{32} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {45}{16 \sqrt {x}}+\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{16 \sqrt {x} \left (1+x^2\right )}-\frac {45}{64} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\frac {45}{64} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\frac {45 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}-\frac {45 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}\\ &=-\frac {45}{16 \sqrt {x}}+\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{16 \sqrt {x} \left (1+x^2\right )}-\frac {45 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {45 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}-\frac {45 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {45 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}\\ &=-\frac {45}{16 \sqrt {x}}+\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{16 \sqrt {x} \left (1+x^2\right )}+\frac {45 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {45 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {45 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {45 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 20, normalized size = 0.14 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{4},3;\frac {3}{4};-x^2\right )}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(1 + x^2)^3),x]

[Out]

(-2*Hypergeometric2F1[-1/4, 3, 3/4, -x^2])/Sqrt[x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.27, size = 86, normalized size = 0.62 \begin {gather*} \frac {-45 x^4-81 x^2-32}{16 \sqrt {x} \left (x^2+1\right )^2}-\frac {45 \tan ^{-1}\left (\frac {\frac {x}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{\sqrt {x}}\right )}{32 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{x+1}\right )}{32 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^(3/2)*(1 + x^2)^3),x]

[Out]

(-32 - 81*x^2 - 45*x^4)/(16*Sqrt[x]*(1 + x^2)^2) - (45*ArcTan[(-(1/Sqrt[2]) + x/Sqrt[2])/Sqrt[x]])/(32*Sqrt[2]
) + (45*ArcTanh[(Sqrt[2]*Sqrt[x])/(1 + x)])/(32*Sqrt[2])

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 178, normalized size = 1.29 \begin {gather*} \frac {180 \, \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 180 \, \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) + 45 \, \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 45 \, \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (45 \, x^{4} + 81 \, x^{2} + 32\right )} \sqrt {x}}{128 \, {\left (x^{5} + 2 \, x^{3} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(x^2+1)^3,x, algorithm="fricas")

[Out]

1/128*(180*sqrt(2)*(x^5 + 2*x^3 + x)*arctan(sqrt(2)*sqrt(sqrt(2)*sqrt(x) + x + 1) - sqrt(2)*sqrt(x) - 1) + 180
*sqrt(2)*(x^5 + 2*x^3 + x)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*sqrt(x) + 4*x + 4) - sqrt(2)*sqrt(x) + 1) + 45*s
qrt(2)*(x^5 + 2*x^3 + x)*log(4*sqrt(2)*sqrt(x) + 4*x + 4) - 45*sqrt(2)*(x^5 + 2*x^3 + x)*log(-4*sqrt(2)*sqrt(x
) + 4*x + 4) - 8*(45*x^4 + 81*x^2 + 32)*sqrt(x))/(x^5 + 2*x^3 + x)

________________________________________________________________________________________

giac [A]  time = 0.63, size = 99, normalized size = 0.72 \begin {gather*} -\frac {45}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \frac {45}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {45}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {45}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {2}{\sqrt {x}} - \frac {13 \, x^{\frac {7}{2}} + 17 \, x^{\frac {3}{2}}}{16 \, {\left (x^{2} + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(x^2+1)^3,x, algorithm="giac")

[Out]

-45/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 45/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt
(x))) + 45/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 45/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 2/sqrt(x)
- 1/16*(13*x^(7/2) + 17*x^(3/2))/(x^2 + 1)^2

________________________________________________________________________________________

maple [A]  time = 0.01, size = 87, normalized size = 0.63 \begin {gather*} -\frac {45 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{64}-\frac {45 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{64}-\frac {45 \sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{128}-\frac {2}{\sqrt {x}}-\frac {2 \left (\frac {13 x^{\frac {7}{2}}}{32}+\frac {17 x^{\frac {3}{2}}}{32}\right )}{\left (x^{2}+1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(x^2+1)^3,x)

[Out]

-2*(13/32*x^(7/2)+17/32*x^(3/2))/(x^2+1)^2-45/64*2^(1/2)*arctan(2^(1/2)*x^(1/2)-1)-45/128*2^(1/2)*ln((x-2^(1/2
)*x^(1/2)+1)/(x+2^(1/2)*x^(1/2)+1))-45/64*2^(1/2)*arctan(2^(1/2)*x^(1/2)+1)-2/x^(1/2)

________________________________________________________________________________________

maxima [A]  time = 2.98, size = 102, normalized size = 0.74 \begin {gather*} -\frac {45}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \frac {45}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {45}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {45}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {45 \, x^{4} + 81 \, x^{2} + 32}{16 \, {\left (x^{\frac {9}{2}} + 2 \, x^{\frac {5}{2}} + \sqrt {x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(x^2+1)^3,x, algorithm="maxima")

[Out]

-45/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 45/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt
(x))) + 45/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 45/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 1/16*(45*
x^4 + 81*x^2 + 32)/(x^(9/2) + 2*x^(5/2) + sqrt(x))

________________________________________________________________________________________

mupad [B]  time = 4.72, size = 65, normalized size = 0.47 \begin {gather*} -\frac {\frac {45\,x^4}{16}+\frac {81\,x^2}{16}+2}{\sqrt {x}+2\,x^{5/2}+x^{9/2}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {45}{64}+\frac {45}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {45}{64}-\frac {45}{64}{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(x^2 + 1)^3),x)

[Out]

- 2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 - 1i/2))*(45/64 - 45i/64) - 2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 + 1i/2))*(45
/64 + 45i/64) - ((81*x^2)/16 + (45*x^4)/16 + 2)/(x^(1/2) + 2*x^(5/2) + x^(9/2))

________________________________________________________________________________________

sympy [B]  time = 12.63, size = 653, normalized size = 4.73 \begin {gather*} - \frac {45 \sqrt {2} x^{\frac {9}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} + \frac {45 \sqrt {2} x^{\frac {9}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} x^{\frac {5}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} + \frac {90 \sqrt {2} x^{\frac {5}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {180 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {180 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {45 \sqrt {2} \sqrt {x} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} + \frac {45 \sqrt {2} \sqrt {x} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} \sqrt {x} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} \sqrt {x} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {360 x^{4}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {648 x^{2}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {256}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(x**2+1)**3,x)

[Out]

-45*sqrt(2)*x**(9/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) + 45*sqrt(2
)*x**(9/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*x**(9/2)*
atan(sqrt(2)*sqrt(x) - 1)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*x**(9/2)*atan(sqrt(2)*sqrt(
x) + 1)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*x**(5/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(1
28*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) + 90*sqrt(2)*x**(5/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2)
+ 256*x**(5/2) + 128*sqrt(x)) - 180*sqrt(2)*x**(5/2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**(9/2) + 256*x**(5/2) +
128*sqrt(x)) - 180*sqrt(2)*x**(5/2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 4
5*sqrt(2)*sqrt(x)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) + 45*sqrt(2)*s
qrt(x)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*sqrt(x)*atan(
sqrt(2)*sqrt(x) - 1)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*sqrt(x)*atan(sqrt(2)*sqrt(x) + 1
)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 360*x**4/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 648*x**
2/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 256/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x))

________________________________________________________________________________________