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

Optimal. Leaf size=138 \[ \frac{11}{16 x^{3/2} \left (x^2+1\right )}-\frac{77}{48 x^{3/2}}+\frac{1}{4 x^{3/2} \left (x^2+1\right )^2}+\frac{77 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{77 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{77 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{77 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

[Out]

-77/(48*x^(3/2)) + 1/(4*x^(3/2)*(1 + x^2)^2) + 11/(16*x^(3/2)*(1 + x^2)) + (77*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(3
2*Sqrt[2]) - (77*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (77*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) -
(77*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])

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Rubi [A]  time = 0.070822, antiderivative size = 138, normalized size of antiderivative = 1., 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, 211, 1165, 628, 1162, 617, 204} \[ \frac{11}{16 x^{3/2} \left (x^2+1\right )}-\frac{77}{48 x^{3/2}}+\frac{1}{4 x^{3/2} \left (x^2+1\right )^2}+\frac{77 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{77 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{77 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{77 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(1 + x^2)^3),x]

[Out]

-77/(48*x^(3/2)) + 1/(4*x^(3/2)*(1 + x^2)^2) + 11/(16*x^(3/2)*(1 + x^2)) + (77*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(3
2*Sqrt[2]) - (77*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (77*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) -
(77*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])

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 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 211

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

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]

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])

Rubi steps

\begin{align*} \int \frac{1}{x^{5/2} \left (1+x^2\right )^3} \, dx &=\frac{1}{4 x^{3/2} \left (1+x^2\right )^2}+\frac{11}{8} \int \frac{1}{x^{5/2} \left (1+x^2\right )^2} \, dx\\ &=\frac{1}{4 x^{3/2} \left (1+x^2\right )^2}+\frac{11}{16 x^{3/2} \left (1+x^2\right )}+\frac{77}{32} \int \frac{1}{x^{5/2} \left (1+x^2\right )} \, dx\\ &=-\frac{77}{48 x^{3/2}}+\frac{1}{4 x^{3/2} \left (1+x^2\right )^2}+\frac{11}{16 x^{3/2} \left (1+x^2\right )}-\frac{77}{32} \int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx\\ &=-\frac{77}{48 x^{3/2}}+\frac{1}{4 x^{3/2} \left (1+x^2\right )^2}+\frac{11}{16 x^{3/2} \left (1+x^2\right )}-\frac{77}{16} \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{77}{48 x^{3/2}}+\frac{1}{4 x^{3/2} \left (1+x^2\right )^2}+\frac{11}{16 x^{3/2} \left (1+x^2\right )}-\frac{77}{32} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )-\frac{77}{32} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{77}{48 x^{3/2}}+\frac{1}{4 x^{3/2} \left (1+x^2\right )^2}+\frac{11}{16 x^{3/2} \left (1+x^2\right )}-\frac{77}{64} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )-\frac{77}{64} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{77 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}+\frac{77 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}\\ &=-\frac{77}{48 x^{3/2}}+\frac{1}{4 x^{3/2} \left (1+x^2\right )^2}+\frac{11}{16 x^{3/2} \left (1+x^2\right )}+\frac{77 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}-\frac{77 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}-\frac{77 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{77 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}\\ &=-\frac{77}{48 x^{3/2}}+\frac{1}{4 x^{3/2} \left (1+x^2\right )^2}+\frac{11}{16 x^{3/2} \left (1+x^2\right )}+\frac{77 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{77 \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{77 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}-\frac{77 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0047892, size = 22, normalized size = 0.16 \[ -\frac{2 \, _2F_1\left (-\frac{3}{4},3;\frac{1}{4};-x^2\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(1 + x^2)^3),x]

[Out]

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

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Maple [A]  time = 0.01, size = 87, normalized size = 0.6 \begin{align*} -{\frac{2}{3}{x}^{-{\frac{3}{2}}}}-2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ({\frac{15\,{x}^{5/2}}{32}}+{\frac{19\,\sqrt{x}}{32}} \right ) }-{\frac{77\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }-{\frac{77\,\sqrt{2}}{64}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }-{\frac{77\,\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(1/x^(5/2)/(x^2+1)^3,x)

[Out]

-2/3/x^(3/2)-2*(15/32*x^(5/2)+19/32*x^(1/2))/(x^2+1)^2-77/64*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)-77/64*arctan(-1
+2^(1/2)*x^(1/2))*2^(1/2)-77/128*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 = 3.77732, size = 138, normalized size = 1. \begin{align*} -\frac{77}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{77}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{77}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{77}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{77 \, x^{4} + 121 \, x^{2} + 32}{48 \,{\left (x^{\frac{11}{2}} + 2 \, x^{\frac{7}{2}} + x^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-77/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 77/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt
(x))) - 77/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 77/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 1/48*(77*
x^4 + 121*x^2 + 32)/(x^(11/2) + 2*x^(7/2) + x^(3/2))

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Fricas [B]  time = 1.41728, size = 555, normalized size = 4.02 \begin{align*} \frac{924 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) + 924 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4} - \sqrt{2} \sqrt{x} + 1\right ) - 231 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) + 231 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 8 \,{\left (77 \, x^{4} + 121 \, x^{2} + 32\right )} \sqrt{x}}{384 \,{\left (x^{6} + 2 \, x^{4} + x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(924*sqrt(2)*(x^6 + 2*x^4 + x^2)*arctan(sqrt(2)*sqrt(sqrt(2)*sqrt(x) + x + 1) - sqrt(2)*sqrt(x) - 1) + 9
24*sqrt(2)*(x^6 + 2*x^4 + x^2)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*sqrt(x) + 4*x + 4) - sqrt(2)*sqrt(x) + 1) -
231*sqrt(2)*(x^6 + 2*x^4 + x^2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4) + 231*sqrt(2)*(x^6 + 2*x^4 + x^2)*log(-4*sqrt
(2)*sqrt(x) + 4*x + 4) - 8*(77*x^4 + 121*x^2 + 32)*sqrt(x))/(x^6 + 2*x^4 + x^2)

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Sympy [B]  time = 43.6697, size = 653, normalized size = 4.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

231*sqrt(2)*x**(11/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) - 231*sq
rt(2)*x**(11/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) - 462*sqrt(2)*x
**(11/2)*atan(sqrt(2)*sqrt(x) - 1)/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) - 462*sqrt(2)*x**(11/2)*atan(
sqrt(2)*sqrt(x) + 1)/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) + 462*sqrt(2)*x**(7/2)*log(-4*sqrt(2)*sqrt(
x) + 4*x + 4)/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) - 462*sqrt(2)*x**(7/2)*log(4*sqrt(2)*sqrt(x) + 4*x
+ 4)/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) - 924*sqrt(2)*x**(7/2)*atan(sqrt(2)*sqrt(x) - 1)/(384*x**(
11/2) + 768*x**(7/2) + 384*x**(3/2)) - 924*sqrt(2)*x**(7/2)*atan(sqrt(2)*sqrt(x) + 1)/(384*x**(11/2) + 768*x**
(7/2) + 384*x**(3/2)) + 231*sqrt(2)*x**(3/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(384*x**(11/2) + 768*x**(7/2) +
384*x**(3/2)) - 231*sqrt(2)*x**(3/2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(384*x**(11/2) + 768*x**(7/2) + 384*x**
(3/2)) - 462*sqrt(2)*x**(3/2)*atan(sqrt(2)*sqrt(x) - 1)/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) - 462*sq
rt(2)*x**(3/2)*atan(sqrt(2)*sqrt(x) + 1)/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) - 616*x**4/(384*x**(11/
2) + 768*x**(7/2) + 384*x**(3/2)) - 968*x**2/(384*x**(11/2) + 768*x**(7/2) + 384*x**(3/2)) - 256/(384*x**(11/2
) + 768*x**(7/2) + 384*x**(3/2))

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Giac [A]  time = 2.67007, size = 134, normalized size = 0.97 \begin{align*} -\frac{77}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{77}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{77}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{77}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{15 \, x^{\frac{5}{2}} + 19 \, \sqrt{x}}{16 \,{\left (x^{2} + 1\right )}^{2}} - \frac{2}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-77/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 77/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt
(x))) - 77/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 77/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 1/16*(15*
x^(5/2) + 19*sqrt(x))/(x^2 + 1)^2 - 2/3/x^(3/2)