3.6.43 \(\int \frac {1}{x^{10} \sqrt {1+x^3}} \, dx\)

Optimal. Leaf size=43 \[ \frac {5}{24} \tanh ^{-1}\left (\sqrt {x^3+1}\right )+\frac {\sqrt {x^3+1} \left (-15 x^6+10 x^3-8\right )}{72 x^9} \]

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Rubi [A]  time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \begin {gather*} -\frac {5 \sqrt {x^3+1}}{24 x^3}+\frac {5}{24} \tanh ^{-1}\left (\sqrt {x^3+1}\right )-\frac {\sqrt {x^3+1}}{9 x^9}+\frac {5 \sqrt {x^3+1}}{36 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^10*Sqrt[1 + x^3]),x]

[Out]

-1/9*Sqrt[1 + x^3]/x^9 + (5*Sqrt[1 + x^3])/(36*x^6) - (5*Sqrt[1 + x^3])/(24*x^3) + (5*ArcTanh[Sqrt[1 + x^3]])/
24

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

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

Rule 266

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

Rubi steps

\begin {align*} \int \frac {1}{x^{10} \sqrt {1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}-\frac {5}{18} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}+\frac {5 \sqrt {1+x^3}}{36 x^6}+\frac {5}{24} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}+\frac {5 \sqrt {1+x^3}}{36 x^6}-\frac {5 \sqrt {1+x^3}}{24 x^3}-\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}+\frac {5 \sqrt {1+x^3}}{36 x^6}-\frac {5 \sqrt {1+x^3}}{24 x^3}-\frac {5}{24} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=-\frac {\sqrt {1+x^3}}{9 x^9}+\frac {5 \sqrt {1+x^3}}{36 x^6}-\frac {5 \sqrt {1+x^3}}{24 x^3}+\frac {5}{24} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 26, normalized size = 0.60 \begin {gather*} \frac {2}{3} \sqrt {x^3+1} \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};x^3+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*Sqrt[1 + x^3]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^3} \left (-8+10 x^3-15 x^6\right )}{72 x^9}+\frac {5}{24} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^10*Sqrt[1 + x^3]),x]

[Out]

(Sqrt[1 + x^3]*(-8 + 10*x^3 - 15*x^6))/(72*x^9) + (5*ArcTanh[Sqrt[1 + x^3]])/24

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fricas [A]  time = 0.53, size = 57, normalized size = 1.33 \begin {gather*} \frac {15 \, x^{9} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 15 \, x^{9} \log \left (\sqrt {x^{3} + 1} - 1\right ) - 2 \, {\left (15 \, x^{6} - 10 \, x^{3} + 8\right )} \sqrt {x^{3} + 1}}{144 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

1/144*(15*x^9*log(sqrt(x^3 + 1) + 1) - 15*x^9*log(sqrt(x^3 + 1) - 1) - 2*(15*x^6 - 10*x^3 + 8)*sqrt(x^3 + 1))/
x^9

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giac [A]  time = 0.30, size = 59, normalized size = 1.37 \begin {gather*} -\frac {15 \, {\left (x^{3} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{3} + 1}}{72 \, x^{9}} + \frac {5}{48} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {5}{48} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

-1/72*(15*(x^3 + 1)^(5/2) - 40*(x^3 + 1)^(3/2) + 33*sqrt(x^3 + 1))/x^9 + 5/48*log(sqrt(x^3 + 1) + 1) - 5/48*lo
g(abs(sqrt(x^3 + 1) - 1))

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maple [A]  time = 0.23, size = 41, normalized size = 0.95

method	result	size
risch	\(-\frac {15 x^{9}+5 x^{6}-2 x^{3}+8}{72 x^{9} \sqrt {x^{3}+1}}+\frac {5 \arctanh \left (\sqrt {x^{3}+1}\right )}{24}\)	\(41\)
default	\(-\frac {\sqrt {x^{3}+1}}{9 x^{9}}+\frac {5 \sqrt {x^{3}+1}}{36 x^{6}}-\frac {5 \sqrt {x^{3}+1}}{24 x^{3}}+\frac {5 \arctanh \left (\sqrt {x^{3}+1}\right )}{24}\)	\(48\)
elliptic	\(-\frac {\sqrt {x^{3}+1}}{9 x^{9}}+\frac {5 \sqrt {x^{3}+1}}{36 x^{6}}-\frac {5 \sqrt {x^{3}+1}}{24 x^{3}}+\frac {5 \arctanh \left (\sqrt {x^{3}+1}\right )}{24}\)	\(48\)
trager	\(-\frac {\left (15 x^{6}-10 x^{3}+8\right ) \sqrt {x^{3}+1}}{72 x^{9}}-\frac {5 \ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{48}\)	\(50\)
meijerg	\(\frac {-\frac {\sqrt {\pi }}{3 x^{9}}+\frac {\sqrt {\pi }}{4 x^{6}}-\frac {3 \sqrt {\pi }}{8 x^{3}}-\frac {5 \left (\frac {37}{30}-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }}{16}+\frac {\sqrt {\pi }\, \left (148 x^{9}+144 x^{6}-96 x^{3}+128\right )}{384 x^{9}}-\frac {\sqrt {\pi }\, \left (240 x^{6}-160 x^{3}+128\right ) \sqrt {x^{3}+1}}{384 x^{9}}+\frac {5 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{8}}{3 \sqrt {\pi }}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/72*(15*x^9+5*x^6-2*x^3+8)/x^9/(x^3+1)^(1/2)+5/24*arctanh((x^3+1)^(1/2))

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maxima [B]  time = 0.40, size = 80, normalized size = 1.86 \begin {gather*} -\frac {15 \, {\left (x^{3} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{3} + 1}}{72 \, {\left ({\left (x^{3} + 1\right )}^{3} + 3 \, x^{3} - 3 \, {\left (x^{3} + 1\right )}^{2} + 2\right )}} + \frac {5}{48} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {5}{48} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

-1/72*(15*(x^3 + 1)^(5/2) - 40*(x^3 + 1)^(3/2) + 33*sqrt(x^3 + 1))/((x^3 + 1)^3 + 3*x^3 - 3*(x^3 + 1)^2 + 2) +
5/48*log(sqrt(x^3 + 1) + 1) - 5/48*log(sqrt(x^3 + 1) - 1)

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mupad [B]  time = 0.05, size = 201, normalized size = 4.67 \begin {gather*} \frac {5\,\sqrt {x^3+1}}{36\,x^6}-\frac {5\,\sqrt {x^3+1}}{24\,x^3}-\frac {\sqrt {x^3+1}}{9\,x^9}+\frac {5\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{8\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^10*(x^3 + 1)^(1/2)),x)

[Out]

(5*(x^3 + 1)^(1/2))/(36*x^6) - (5*(x^3 + 1)^(1/2))/(24*x^3) - (x^3 + 1)^(1/2)/(9*x^9) + (5*((3^(1/2)*1i)/2 + 3
/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2
)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3^(1/2)*1i)/2 + 3/2, asin(((x + 1)/((3^(1/2)*1i)/
2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(8*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/
2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))

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sympy [B]  time = 3.93, size = 85, normalized size = 1.98 \begin {gather*} \frac {5 \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} - \frac {5}{24 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {5}{72 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} + \frac {1}{36 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {1}{9 x^{\frac {21}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(x**3+1)**(1/2),x)

[Out]

5*asinh(x**(-3/2))/24 - 5/(24*x**(3/2)*sqrt(1 + x**(-3))) - 5/(72*x**(9/2)*sqrt(1 + x**(-3))) + 1/(36*x**(15/2
)*sqrt(1 + x**(-3))) - 1/(9*x**(21/2)*sqrt(1 + x**(-3)))

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