3.6.32 \(\int \frac {(2+x^3) (1+x^3+x^6)}{x^4 \sqrt {1+x^3}} \, dx\)

Optimal. Leaf size=41 \[ \frac {2 \sqrt {x^3+1} \left (x^6+7 x^3-3\right )}{9 x^3}-\frac {4}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 57, normalized size of antiderivative = 1.39, number of steps used = 8, 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 {2}{9} \left (x^3+1\right )^{3/2}-\frac {2 \sqrt {x^3+1}}{3 x^3}+\frac {4 \sqrt {x^3+1}}{3}-\frac {4}{3} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + x^3)*(1 + x^3 + x^6))/(x^4*Sqrt[1 + x^3]),x]

[Out]

(4*Sqrt[1 + x^3])/3 - (2*Sqrt[1 + x^3])/(3*x^3) + (2*(1 + x^3)^(3/2))/9 - (4*ArcTanh[Sqrt[1 + x^3]])/3

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^4 \sqrt {1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(2+x) \left (1+x+x^2\right )}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {2}{\sqrt {1+x}}+\frac {2}{x^2 \sqrt {1+x}}+\frac {3}{x \sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,x^3\right )\\ &=\frac {4 \sqrt {1+x^3}}{3}+\frac {2}{9} \left (1+x^3\right )^{3/2}+\frac {2}{3} \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 {4 \sqrt {1+x^3}}{3}-\frac {2 \sqrt {1+x^3}}{3 x^3}+\frac {2}{9} \left (1+x^3\right )^{3/2}-\frac {1}{3} \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 {4 \sqrt {1+x^3}}{3}-\frac {2 \sqrt {1+x^3}}{3 x^3}+\frac {2}{9} \left (1+x^3\right )^{3/2}-2 \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=\frac {4 \sqrt {1+x^3}}{3}-\frac {2 \sqrt {1+x^3}}{3 x^3}+\frac {2}{9} \left (1+x^3\right )^{3/2}-\frac {4}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 40, normalized size = 0.98 \begin {gather*} \frac {2}{9} \left (\frac {\sqrt {x^3+1} \left (x^6+7 x^3-3\right )}{x^3}-6 \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^4*Sqrt[1 + x^3]),x]

[Out]

(2*((Sqrt[1 + x^3]*(-3 + 7*x^3 + x^6))/x^3 - 6*ArcTanh[Sqrt[1 + x^3]]))/9

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IntegrateAlgebraic [A]  time = 0.05, size = 41, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+x^3} \left (-3+7 x^3+x^6\right )}{9 x^3}-\frac {4}{3} \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^4*Sqrt[1 + x^3]),x]

[Out]

(2*Sqrt[1 + x^3]*(-3 + 7*x^3 + x^6))/(9*x^3) - (4*ArcTanh[Sqrt[1 + x^3]])/3

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fricas [A]  time = 0.47, size = 55, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (3 \, x^{3} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 3 \, x^{3} \log \left (\sqrt {x^{3} + 1} - 1\right ) - {\left (x^{6} + 7 \, x^{3} - 3\right )} \sqrt {x^{3} + 1}\right )}}{9 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+2)*(x^6+x^3+1)/x^4/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

-2/9*(3*x^3*log(sqrt(x^3 + 1) + 1) - 3*x^3*log(sqrt(x^3 + 1) - 1) - (x^6 + 7*x^3 - 3)*sqrt(x^3 + 1))/x^3

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giac [A]  time = 0.40, size = 56, normalized size = 1.37 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + \frac {4}{3} \, \sqrt {x^{3} + 1} - \frac {2 \, \sqrt {x^{3} + 1}}{3 \, x^{3}} - \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {2}{3} \, \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^4/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

2/9*(x^3 + 1)^(3/2) + 4/3*sqrt(x^3 + 1) - 2/3*sqrt(x^3 + 1)/x^3 - 2/3*log(sqrt(x^3 + 1) + 1) + 2/3*log(abs(sqr
t(x^3 + 1) - 1))

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maple [A]  time = 0.25, size = 45, normalized size = 1.10

method	result	size
default	\(\frac {2 \sqrt {x^{3}+1}\, x^{3}}{9}+\frac {14 \sqrt {x^{3}+1}}{9}-\frac {4 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}-\frac {2 \sqrt {x^{3}+1}}{3 x^{3}}\)	\(45\)
risch	\(\frac {2 \sqrt {x^{3}+1}\, x^{3}}{9}+\frac {14 \sqrt {x^{3}+1}}{9}-\frac {4 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}-\frac {2 \sqrt {x^{3}+1}}{3 x^{3}}\)	\(45\)
elliptic	\(\frac {2 \sqrt {x^{3}+1}\, x^{3}}{9}+\frac {14 \sqrt {x^{3}+1}}{9}-\frac {4 \arctanh \left (\sqrt {x^{3}+1}\right )}{3}-\frac {2 \sqrt {x^{3}+1}}{3 x^{3}}\)	\(45\)
trager	\(\frac {2 \sqrt {x^{3}+1}\, \left (x^{6}+7 x^{3}-3\right )}{9 x^{3}}-\frac {2 \ln \left (-\frac {x^{3}+2 \sqrt {x^{3}+1}+2}{x^{3}}\right )}{3}\)	\(46\)
meijerg	\(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right ) \sqrt {x^{3}+1}}{6}}{3 \sqrt {\pi }}+\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{3}+1}}{\sqrt {\pi }}+\frac {\left (-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{\sqrt {\pi }}+\frac {-\frac {2 \sqrt {\pi }}{3 x^{3}}-\frac {\left (1-2 \ln \relax (2)+3 \ln \relax (x )\right ) \sqrt {\pi }}{3}+\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right )}{12 x^{3}}-\frac {2 \sqrt {\pi }\, \sqrt {x^{3}+1}}{3 x^{3}}+\frac {2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right ) \sqrt {\pi }}{3}}{\sqrt {\pi }}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+2)*(x^6+x^3+1)/x^4/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9*(x^3+1)^(1/2)*x^3+14/9*(x^3+1)^(1/2)-4/3*arctanh((x^3+1)^(1/2))-2/3*(x^3+1)^(1/2)/x^3

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maxima [A]  time = 0.67, size = 55, normalized size = 1.34 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + \frac {4}{3} \, \sqrt {x^{3} + 1} - \frac {2 \, \sqrt {x^{3} + 1}}{3 \, x^{3}} - \frac {2}{3} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {2}{3} \, \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^4/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 198, normalized size = 4.83 \begin {gather*} \frac {14\,\sqrt {x^3+1}}{9}-\frac {2\,\sqrt {x^3+1}}{3\,x^3}+\frac {2\,x^3\,\sqrt {x^3+1}}{9}-\frac {4\,\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 )}{\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^4*(x^3 + 1)^(1/2)),x)

[Out]

(14*(x^3 + 1)^(1/2))/9 - (2*(x^3 + 1)^(1/2))/(3*x^3) + (2*x^3*(x^3 + 1)^(1/2))/9 - (4*((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)))/(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 = 74.24, size = 83, normalized size = 2.02 \begin {gather*} \frac {2 \left (x^{3} + 1\right )^{\frac {3}{2}}}{9} + \frac {4 \sqrt {x^{3} + 1}}{3} + \frac {2 \log {\left (-1 + \frac {1}{\sqrt {x^{3} + 1}} \right )}}{3} - \frac {2 \log {\left (1 + \frac {1}{\sqrt {x^{3} + 1}} \right )}}{3} + \frac {1}{3 \left (1 + \frac {1}{\sqrt {x^{3} + 1}}\right )} + \frac {1}{3 \left (-1 + \frac {1}{\sqrt {x^{3} + 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**4/(x**3+1)**(1/2),x)

[Out]

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

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