3.14.13 \(\int \frac {x^{23/2}}{\sqrt {1+x^5}} \, dx\) [1313]

Optimal. Leaf size=47 \[ -\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{20} \sinh ^{-1}\left (x^{5/2}\right ) \]

[Out]

3/20*arcsinh(x^(5/2))-3/20*x^(5/2)*(x^5+1)^(1/2)+1/10*x^(15/2)*(x^5+1)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {327, 335, 281, 221} \begin {gather*} \frac {3}{20} \sinh ^{-1}\left (x^{5/2}\right )+\frac {1}{10} \sqrt {x^5+1} x^{15/2}-\frac {3}{20} \sqrt {x^5+1} x^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(23/2)/Sqrt[1 + x^5],x]

[Out]

(-3*x^(5/2)*Sqrt[1 + x^5])/20 + (x^(15/2)*Sqrt[1 + x^5])/10 + (3*ArcSinh[x^(5/2)])/20

Rule 221

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

Rule 281

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

Rule 327

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*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

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]

Rubi steps

\begin {align*} \int \frac {x^{23/2}}{\sqrt {1+x^5}} \, dx &=\frac {1}{10} x^{15/2} \sqrt {1+x^5}-\frac {3}{4} \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx\\ &=-\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{8} \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx\\ &=-\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{4} \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{20} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right )\\ &=-\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{20} \sinh ^{-1}\left (x^{5/2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.99, size = 46, normalized size = 0.98 \begin {gather*} \frac {1}{20} x^{5/2} \sqrt {1+x^5} \left (-3+2 x^5\right )+\frac {3}{20} \tanh ^{-1}\left (\frac {x^{5/2}}{\sqrt {1+x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(23/2)/Sqrt[1 + x^5],x]

[Out]

(x^(5/2)*Sqrt[1 + x^5]*(-3 + 2*x^5))/20 + (3*ArcTanh[x^(5/2)/Sqrt[1 + x^5]])/20

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Maple [A]
time = 0.19, size = 38, normalized size = 0.81

method	result	size
meijerg	\(\frac {-\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \left (-10 x^{5}+15\right ) \sqrt {x^{5}+1}}{20}+\frac {3 \sqrt {\pi }\, \arcsinh \left (x^{\frac {5}{2}}\right )}{4}}{5 \sqrt {\pi }}\)	\(38\)
risch	\(\frac {x^{\frac {5}{2}} \left (2 x^{5}-3\right ) \sqrt {x^{5}+1}}{20}+\frac {3 \arcsinh \left (x^{\frac {5}{2}}\right ) \sqrt {x \left (x^{5}+1\right )}}{20 \sqrt {x}\, \sqrt {x^{5}+1}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(23/2)/(x^5+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5/Pi^(1/2)*(-1/20*Pi^(1/2)*x^(5/2)*(-10*x^5+15)*(x^5+1)^(1/2)+3/4*Pi^(1/2)*arcsinh(x^(5/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (31) = 62\).
time = 0.30, size = 86, normalized size = 1.83 \begin {gather*} -\frac {\frac {5 \, \sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} - \frac {3 \, {\left (x^{5} + 1\right )}^{\frac {3}{2}}}{x^{\frac {15}{2}}}}{20 \, {\left (\frac {2 \, {\left (x^{5} + 1\right )}}{x^{5}} - \frac {{\left (x^{5} + 1\right )}^{2}}{x^{10}} - 1\right )}} + \frac {3}{40} \, \log \left (\frac {\sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} + 1\right ) - \frac {3}{40} \, \log \left (\frac {\sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

-1/20*(5*sqrt(x^5 + 1)/x^(5/2) - 3*(x^5 + 1)^(3/2)/x^(15/2))/(2*(x^5 + 1)/x^5 - (x^5 + 1)^2/x^10 - 1) + 3/40*l
og(sqrt(x^5 + 1)/x^(5/2) + 1) - 3/40*log(sqrt(x^5 + 1)/x^(5/2) - 1)

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Fricas [A]
time = 0.40, size = 46, normalized size = 0.98 \begin {gather*} \frac {1}{20} \, {\left (2 \, x^{7} - 3 \, x^{2}\right )} \sqrt {x^{5} + 1} \sqrt {x} + \frac {3}{40} \, \log \left (2 \, x^{5} + 2 \, \sqrt {x^{5} + 1} x^{\frac {5}{2}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

1/20*(2*x^7 - 3*x^2)*sqrt(x^5 + 1)*sqrt(x) + 3/40*log(2*x^5 + 2*sqrt(x^5 + 1)*x^(5/2) + 1)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(23/2)/(x**5+1)**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 1.10, size = 36, normalized size = 0.77 \begin {gather*} \frac {1}{20} \, {\left (2 \, x^{5} - 3\right )} \sqrt {x^{5} + 1} x^{\frac {5}{2}} - \frac {3}{20} \, \log \left (-x^{\frac {5}{2}} + \sqrt {x^{5} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

1/20*(2*x^5 - 3)*sqrt(x^5 + 1)*x^(5/2) - 3/20*log(-x^(5/2) + sqrt(x^5 + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{23/2}}{\sqrt {x^5+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(23/2)/(x^5 + 1)^(1/2),x)

[Out]

int(x^(23/2)/(x^5 + 1)^(1/2), x)

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