3.6.59 \(\int \frac {(2+x^3) (1+x^3+x^6)}{x^7 \sqrt {1+x^3}} \, dx\)

Optimal. Leaf size=43 \[ \frac {\sqrt {x^3+1} \left (4 x^6-3 x^3-2\right )}{6 x^6}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 60, normalized size of antiderivative = 1.40, number of steps used = 12, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1821, 1612, 51, 63, 207} \begin {gather*} -\frac {\sqrt {x^3+1}}{2 x^3}+\frac {2 \sqrt {x^3+1}}{3}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {x^3+1}\right )-\frac {\sqrt {x^3+1}}{3 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + x^3)*(1 + x^3 + x^6))/(x^7*Sqrt[1 + x^3]),x]

[Out]

(2*Sqrt[1 + x^3])/3 - Sqrt[1 + x^3]/(3*x^6) - Sqrt[1 + x^3]/(2*x^3) - (3*ArcTanh[Sqrt[1 + x^3]])/2

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 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

Rule 1821

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

Rubi steps

\begin {align*} \int \frac {\left (2+x^3\right ) \left (1+x^3+x^6\right )}{x^7 \sqrt {1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(2+x) \left (1+x+x^2\right )}{x^3 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1+x}}+\frac {2}{x^3 \sqrt {1+x}}+\frac {3}{x^2 \sqrt {1+x}}+\frac {3}{x \sqrt {1+x}}\right ) \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,x^3\right )+\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )+\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {\sqrt {1+x^3}}{3 x^6}-\frac {\sqrt {1+x^3}}{x^3}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )+2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {\sqrt {1+x^3}}{3 x^6}-\frac {\sqrt {1+x^3}}{2 x^3}-2 \tanh ^{-1}\left (\sqrt {1+x^3}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )-\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {\sqrt {1+x^3}}{3 x^6}-\frac {\sqrt {1+x^3}}{2 x^3}-\tanh ^{-1}\left (\sqrt {1+x^3}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {\sqrt {1+x^3}}{3 x^6}-\frac {\sqrt {1+x^3}}{2 x^3}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 61, normalized size = 1.42 \begin {gather*} \frac {1}{3} \left (-4 \sqrt {x^3+1} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};x^3+1\right )+\frac {\sqrt {x^3+1} \left (2 x^3-3\right )}{x^3}-3 \tanh ^{-1}\left (\sqrt {x^3+1}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + x^3)*(1 + x^3 + x^6))/(x^7*Sqrt[1 + x^3]),x]

[Out]

((Sqrt[1 + x^3]*(-3 + 2*x^3))/x^3 - 3*ArcTanh[Sqrt[1 + x^3]] - 4*Sqrt[1 + x^3]*Hypergeometric2F1[1/2, 3, 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 (-2-3 x^3+4 x^6\right )}{6 x^6}-\frac {3}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + x^3)*(1 + x^3 + x^6))/(x^7*Sqrt[1 + x^3]),x]

[Out]

(Sqrt[1 + x^3]*(-2 - 3*x^3 + 4*x^6))/(6*x^6) - (3*ArcTanh[Sqrt[1 + x^3]])/2

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fricas [A]  time = 0.44, size = 57, normalized size = 1.33 \begin {gather*} -\frac {9 \, x^{6} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 9 \, x^{6} \log \left (\sqrt {x^{3} + 1} - 1\right ) - 2 \, {\left (4 \, x^{6} - 3 \, x^{3} - 2\right )} \sqrt {x^{3} + 1}}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^6+x^3+1)/x^7/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

-1/12*(9*x^6*log(sqrt(x^3 + 1) + 1) - 9*x^6*log(sqrt(x^3 + 1) - 1) - 2*(4*x^6 - 3*x^3 - 2)*sqrt(x^3 + 1))/x^6

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giac [A]  time = 0.34, size = 59, normalized size = 1.37 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {3 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{3} + 1}}{6 \, x^{6}} - \frac {3}{4} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {3}{4} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^6+x^3+1)/x^7/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt(x^3 + 1) - 1/6*(3*(x^3 + 1)^(3/2) - sqrt(x^3 + 1))/x^6 - 3/4*log(sqrt(x^3 + 1) + 1) + 3/4*log(abs(sqr
t(x^3 + 1) - 1))

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maple [A]  time = 0.25, size = 45, normalized size = 1.05

method	result	size
default	\(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {3 \arctanh \left (\sqrt {x^{3}+1}\right )}{2}-\frac {\sqrt {x^{3}+1}}{3 x^{6}}-\frac {\sqrt {x^{3}+1}}{2 x^{3}}\)	\(45\)
risch	\(-\frac {3 x^{6}+5 x^{3}+2}{6 x^{6} \sqrt {x^{3}+1}}+\frac {2 \sqrt {x^{3}+1}}{3}-\frac {3 \arctanh \left (\sqrt {x^{3}+1}\right )}{2}\)	\(45\)
elliptic	\(\frac {2 \sqrt {x^{3}+1}}{3}-\frac {3 \arctanh \left (\sqrt {x^{3}+1}\right )}{2}-\frac {\sqrt {x^{3}+1}}{3 x^{6}}-\frac {\sqrt {x^{3}+1}}{2 x^{3}}\)	\(45\)
trager	\(\frac {\sqrt {x^{3}+1}\, \left (4 x^{6}-3 x^{3}-2\right )}{6 x^{6}}-\frac {3 \ln \left (-\frac {x^{3}+2 \sqrt {x^{3}+1}+2}{x^{3}}\right )}{4}\)	\(48\)
meijerg	\(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{3}+1}}{3 \sqrt {\pi }}+\frac {\left (-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{\sqrt {\pi }}+\frac {-\frac {\sqrt {\pi }}{x^{3}}-\frac {\left (1-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right )}{8 x^{3}}-\frac {\sqrt {\pi }\, \sqrt {x^{3}+1}}{x^{3}}+\ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{\sqrt {\pi }}+\frac {-\frac {\sqrt {\pi }}{3 x^{6}}+\frac {\sqrt {\pi }}{3 x^{3}}+\frac {\left (\frac {7}{6}-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }\, \left (-7 x^{6}-8 x^{3}+8\right )}{24 x^{6}}-\frac {\sqrt {\pi }\, \left (-12 x^{3}+8\right ) \sqrt {x^{3}+1}}{24 x^{6}}-\frac {\ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{2}}{\sqrt {\pi }}\)	\(230\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+2)*(x^6+x^3+1)/x^7/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(x^3+1)^(1/2)-3/2*arctanh((x^3+1)^(1/2))-1/3/x^6*(x^3+1)^(1/2)-1/2*(x^3+1)^(1/2)/x^3

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maxima [B]  time = 0.50, size = 85, normalized size = 1.98 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} + 1} - \frac {3 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x^{3} + 1}}{6 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} - \frac {\sqrt {x^{3} + 1}}{x^{3}} - \frac {3}{4} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {3}{4} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^6+x^3+1)/x^7/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

2/3*sqrt(x^3 + 1) - 1/6*(3*(x^3 + 1)^(3/2) - 5*sqrt(x^3 + 1))/(2*x^3 - (x^3 + 1)^2 + 1) - sqrt(x^3 + 1)/x^3 -
3/4*log(sqrt(x^3 + 1) + 1) + 3/4*log(sqrt(x^3 + 1) - 1)

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mupad [B]  time = 0.24, size = 198, normalized size = 4.60 \begin {gather*} \frac {2\,\sqrt {x^3+1}}{3}-\frac {\sqrt {x^3+1}}{2\,x^3}-\frac {\sqrt {x^3+1}}{3\,x^6}-\frac {9\,\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 )}{2\,\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(((x^3 + 2)*(x^3 + x^6 + 1))/(x^7*(x^3 + 1)^(1/2)),x)

[Out]

(2*(x^3 + 1)^(1/2))/3 - (x^3 + 1)^(1/2)/(2*x^3) - (x^3 + 1)^(1/2)/(3*x^6) - (9*((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)))/(2*(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 = 151.83, size = 105, normalized size = 2.44 \begin {gather*} \frac {2 \sqrt {x^{3} + 1}}{3} + \frac {3 \log {\left (-1 + \frac {1}{\sqrt {x^{3} + 1}} \right )}}{4} - \frac {3 \log {\left (1 + \frac {1}{\sqrt {x^{3} + 1}} \right )}}{4} + \frac {1}{12 \left (1 + \frac {1}{\sqrt {x^{3} + 1}}\right )} + \frac {1}{12 \left (1 + \frac {1}{\sqrt {x^{3} + 1}}\right )^{2}} + \frac {1}{12 \left (-1 + \frac {1}{\sqrt {x^{3} + 1}}\right )} - \frac {1}{12 \left (-1 + \frac {1}{\sqrt {x^{3} + 1}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+2)*(x**6+x**3+1)/x**7/(x**3+1)**(1/2),x)

[Out]

2*sqrt(x**3 + 1)/3 + 3*log(-1 + 1/sqrt(x**3 + 1))/4 - 3*log(1 + 1/sqrt(x**3 + 1))/4 + 1/(12*(1 + 1/sqrt(x**3 +
1))) + 1/(12*(1 + 1/sqrt(x**3 + 1))**2) + 1/(12*(-1 + 1/sqrt(x**3 + 1))) - 1/(12*(-1 + 1/sqrt(x**3 + 1))**2)

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