3.330 \(\int \frac{x^{5/2}}{(1+x^2)^3} \, dx\)

Optimal. Leaf size=129 \[ \frac{3 x^{3/2}}{16 \left (x^2+1\right )}-\frac{x^{3/2}}{4 \left (x^2+1\right )^2}+\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

[Out]

-x^(3/2)/(4*(1 + x^2)^2) + (3*x^(3/2))/(16*(1 + x^2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (3*ArcT
an[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (3*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) - (3*Log[1 + Sqrt[2]*Sqr
t[x] + x])/(64*Sqrt[2])

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Rubi [A]  time = 0.0698324, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {288, 290, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{3 x^{3/2}}{16 \left (x^2+1\right )}-\frac{x^{3/2}}{4 \left (x^2+1\right )^2}+\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/(1 + x^2)^3,x]

[Out]

-x^(3/2)/(4*(1 + x^2)^2) + (3*x^(3/2))/(16*(1 + x^2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (3*ArcT
an[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (3*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) - (3*Log[1 + Sqrt[2]*Sqr
t[x] + x])/(64*Sqrt[2])

Rule 288

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

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 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 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 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 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 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 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]

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]

Rubi steps

\begin{align*} \int \frac{x^{5/2}}{\left (1+x^2\right )^3} \, dx &=-\frac{x^{3/2}}{4 \left (1+x^2\right )^2}+\frac{3}{8} \int \frac{\sqrt{x}}{\left (1+x^2\right )^2} \, dx\\ &=-\frac{x^{3/2}}{4 \left (1+x^2\right )^2}+\frac{3 x^{3/2}}{16 \left (1+x^2\right )}+\frac{3}{32} \int \frac{\sqrt{x}}{1+x^2} \, dx\\ &=-\frac{x^{3/2}}{4 \left (1+x^2\right )^2}+\frac{3 x^{3/2}}{16 \left (1+x^2\right )}+\frac{3}{16} \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{x^{3/2}}{4 \left (1+x^2\right )^2}+\frac{3 x^{3/2}}{16 \left (1+x^2\right )}-\frac{3}{32} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{x^{3/2}}{4 \left (1+x^2\right )^2}+\frac{3 x^{3/2}}{16 \left (1+x^2\right )}+\frac{3}{64} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{3}{64} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}\\ &=-\frac{x^{3/2}}{4 \left (1+x^2\right )^2}+\frac{3 x^{3/2}}{16 \left (1+x^2\right )}+\frac{3 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}-\frac{3 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}\\ &=-\frac{x^{3/2}}{4 \left (1+x^2\right )^2}+\frac{3 x^{3/2}}{16 \left (1+x^2\right )}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}-\frac{3 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0122691, size = 32, normalized size = 0.25 \[ \frac{2}{5} x^{3/2} \left (\, _2F_1\left (\frac{3}{4},3;\frac{7}{4};-x^2\right )-\frac{1}{\left (x^2+1\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/(1 + x^2)^3,x]

[Out]

(2*x^(3/2)*(-(1 + x^2)^(-2) + Hypergeometric2F1[3/4, 3, 7/4, -x^2]))/5

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Maple [A]  time = 0.009, size = 82, normalized size = 0.6 \begin{align*} 2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ({\frac{3\,{x}^{7/2}}{32}}-1/32\,{x}^{3/2} \right ) }+{\frac{3\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{3\,\sqrt{2}}{64}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }+{\frac{3\,\sqrt{2}}{128}\ln \left ({ \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(x^2+1)^3,x)

[Out]

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

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Maxima [A]  time = 2.54206, size = 134, normalized size = 1.04 \begin{align*} \frac{3}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{3}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{3}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{3 \, x^{\frac{7}{2}} - x^{\frac{3}{2}}}{16 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(x^2+1)^3,x, algorithm="maxima")

[Out]

3/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 3/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x)
)) - 3/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 3/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(3*x^(7/2
) - x^(3/2))/(x^4 + 2*x^2 + 1)

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Fricas [A]  time = 1.6743, size = 518, normalized size = 4.02 \begin{align*} -\frac{12 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) + 12 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4} - \sqrt{2} \sqrt{x} + 1\right ) + 3 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 3 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 8 \,{\left (3 \, x^{3} - x\right )} \sqrt{x}}{128 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(x^2+1)^3,x, algorithm="fricas")

[Out]

-1/128*(12*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(sqrt(2)*sqrt(sqrt(2)*sqrt(x) + x + 1) - sqrt(2)*sqrt(x) - 1) + 12*
sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*sqrt(x) + 4*x + 4) - sqrt(2)*sqrt(x) + 1) + 3*sqr
t(2)*(x^4 + 2*x^2 + 1)*log(4*sqrt(2)*sqrt(x) + 4*x + 4) - 3*sqrt(2)*(x^4 + 2*x^2 + 1)*log(-4*sqrt(2)*sqrt(x) +
 4*x + 4) - 8*(3*x^3 - x)*sqrt(x))/(x^4 + 2*x^2 + 1)

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Sympy [B]  time = 29.0798, size = 481, normalized size = 3.73 \begin{align*} \frac{24 x^{\frac{7}{2}}}{128 x^{4} + 256 x^{2} + 128} - \frac{8 x^{\frac{3}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac{3 \sqrt{2} x^{4} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{3 \sqrt{2} x^{4} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{6 \sqrt{2} x^{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{12 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{12 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{3 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{3 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)/(x**2+1)**3,x)

[Out]

24*x**(7/2)/(128*x**4 + 256*x**2 + 128) - 8*x**(3/2)/(128*x**4 + 256*x**2 + 128) + 3*sqrt(2)*x**4*log(-4*sqrt(
2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) - 3*sqrt(2)*x**4*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4
+ 256*x**2 + 128) + 6*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 6*sqrt(2)*x**4*atan
(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128) + 6*sqrt(2)*x**2*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4
 + 256*x**2 + 128) - 6*sqrt(2)*x**2*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 12*sqrt(2)*
x**2*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 12*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) + 1)/(128*x*
*4 + 256*x**2 + 128) + 3*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) - 3*sqrt(2)*log
(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 6*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 25
6*x**2 + 128) + 6*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128)

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Giac [A]  time = 1.57622, size = 127, normalized size = 0.98 \begin{align*} \frac{3}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{3}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{3}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{3 \, x^{\frac{7}{2}} - x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(x^2+1)^3,x, algorithm="giac")

[Out]

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