\(\int \frac {3 x^2}{2 (1+x^3+\sqrt {1+x^3})} \, dx\) [204]

Optimal result

Integrand size = 23, antiderivative size = 12 \[ \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx=\log \left (1+\sqrt {1+x^3}\right ) \]

[Out]

ln(1+(x^3+1)^(1/2))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 2186, 31} \[ \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx=\log \left (\sqrt {x^3+1}+1\right ) \]

[In]

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

[Out]

Log[1 + Sqrt[1 + x^3]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2186

Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/n, Subst[Int
[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c
- a*d, 0] && IntegerQ[(m + 1)/n]

Rubi steps \begin{align*} \text {integral}& = \frac {3}{2} \int \frac {x^2}{1+x^3+\sqrt {1+x^3}} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x+\sqrt {1+x}} \, dx,x,x^3\right ) \\ & = \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {1+x^3}\right ) \\ & = \log \left (1+\sqrt {1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx=\log \left (1+\sqrt {1+x^3}\right ) \]

[In]

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

[Out]

Log[1 + Sqrt[1 + x^3]]

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.67

method	result	size
trager	\(\frac {\ln \left (-x^{3}-2 \sqrt {x^{3}+1}-2\right )}{2}\)	\(20\)
default	\(-\frac {\ln \left (1+x \right )}{2}+\frac {3 \ln \left (x \right )}{2}-\frac {\ln \left (x^{2}-x +1\right )}{2}+\frac {\ln \left (x^{3}+1\right )}{2}+\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )\)	\(39\)
elliptic	\(\frac {\left (1+\sqrt {x^{3}+1}\right ) \sqrt {x^{3}+1}\, \left (\frac {3 \ln \left (x \right )}{2}+\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )\right )}{1+x^{3}+\sqrt {x^{3}+1}}\)	\(45\)

[In]

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

[Out]

1/2*ln(-x^3-2*(x^3+1)^(1/2)-2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (10) = 20\).

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.42 \[ \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx=\frac {3}{2} \, \log \left (x\right ) + \frac {1}{2} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (10) = 20\).

Time = 49.82 (sec) , antiderivative size = 48, normalized size of antiderivative = 4.00 \[ \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx=- \frac {\log {\left (2 \sqrt {x^{3} + 1} \right )}}{2} + \frac {\log {\left (2 \sqrt {x^{3} + 1} + 2 \right )}}{2} + \frac {\log {\left (3 x^{3} + 3 \sqrt {x^{3} + 1} + 3 \right )}}{2} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (10) = 20\).

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.33 \[ \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx=-\frac {1}{2} \, \log \left (x^{2} - x + 1\right ) + \log \left (\frac {x^{3} + \sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1}{\sqrt {x + 1}}\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx=\log \left (\sqrt {x^{3} + 1} + 1\right ) \]

[In]

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

[Out]

log(sqrt(x^3 + 1) + 1)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 169, normalized size of antiderivative = 14.08 \[ \int \frac {3 x^2}{2 \left (1+x^3+\sqrt {1+x^3}\right )} \, dx=\frac {3\,\ln \left (x\right )}{2}+\frac {3\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int((3*x^2)/(2*((x^3 + 1)^(1/2) + x^3 + 1)),x)

[Out]

(3*log(x))/2 + (3*((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)