\(\int \frac {2+5 x^3}{\sqrt {1+x^3} (1+x^2+x^5)} \, dx\) [58]

Optimal result

Integrand size = 27, antiderivative size = 14 \[ \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx=2 \arctan \left (x \sqrt {1+x^3}\right ) \]

[Out]

2*arctan(x*(x^3+1)^(1/2))

Rubi [F]

\[ \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx=\int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx \]

[In]

Int[(2 + 5*x^3)/(Sqrt[1 + x^3]*(1 + x^2 + x^5)),x]

[Out]

2*Defer[Int][1/(Sqrt[1 + x^3]*(1 + x^2 + x^5)), x] + 5*Defer[Int][x^3/(Sqrt[1 + x^3]*(1 + x^2 + x^5)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {1+x^3} \left (1+x^2+x^5\right )}+\frac {5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx+5 \int \frac {x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx=2 \arctan \left (x \sqrt {1+x^3}\right ) \]

[In]

Integrate[(2 + 5*x^3)/(Sqrt[1 + x^3]*(1 + x^2 + x^5)),x]

[Out]

2*ArcTan[x*Sqrt[1 + x^3]]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 4.29

method	result	size
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{3}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{5}+x^{2}+1}\right )\)	\(60\)
default	\(-\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{5}+\textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{4}-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{4}}{2}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\)	\(197\)
elliptic	\(-\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{5}+\textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{4}-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{4}}{2}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\)	\(197\)

[In]

int((5*x^3+2)/(x^3+1)^(1/2)/(x^5+x^2+1),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^2+1)*ln(-(-RootOf(_Z^2+1)*x^5-RootOf(_Z^2+1)*x^2+2*x*(x^3+1)^(1/2)+RootOf(_Z^2+1))/(x^5+x^2+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx=\arctan \left (\frac {{\left (x^{5} + x^{2} - 1\right )} \sqrt {x^{3} + 1}}{2 \, {\left (x^{4} + x\right )}}\right ) \]

[In]

integrate((5*x^3+2)/(x^3+1)^(1/2)/(x^5+x^2+1),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx=\int \frac {5 x^{3} + 2}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{5} + x^{2} + 1\right )}\, dx \]

[In]

integrate((5*x**3+2)/(x**3+1)**(1/2)/(x**5+x**2+1),x)

[Out]

Integral((5*x**3 + 2)/(sqrt((x + 1)*(x**2 - x + 1))*(x**5 + x**2 + 1)), x)

Maxima [F]

\[ \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx=\int { \frac {5 \, x^{3} + 2}{{\left (x^{5} + x^{2} + 1\right )} \sqrt {x^{3} + 1}} \,d x } \]

[In]

integrate((5*x^3+2)/(x^3+1)^(1/2)/(x^5+x^2+1),x, algorithm="maxima")

[Out]

integrate((5*x^3 + 2)/((x^5 + x^2 + 1)*sqrt(x^3 + 1)), x)

Giac [F]

\[ \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx=\int { \frac {5 \, x^{3} + 2}{{\left (x^{5} + x^{2} + 1\right )} \sqrt {x^{3} + 1}} \,d x } \]

[In]

integrate((5*x^3+2)/(x^3+1)^(1/2)/(x^5+x^2+1),x, algorithm="giac")

[Out]

integrate((5*x^3 + 2)/((x^5 + x^2 + 1)*sqrt(x^3 + 1)), x)

Mupad [B] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 163, normalized size of antiderivative = 11.64 \[ \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx=\sum _{k=1}^5\left (-\frac {\sqrt {6}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (x+1\right )}\,\Pi \left (\frac {3+\sqrt {3}\,1{}\mathrm {i}}{2\,\left (\mathrm {root}\left (z^5+z^2+1,z,k\right )+1\right )};\mathrm {asin}\left (\frac {\sqrt {6}\,\sqrt {-\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (x+1\right )}}{6}\right )\middle |\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {3-3\,x+\sqrt {3}\,x\,1{}\mathrm {i}+\sqrt {3}\,1{}\mathrm {i}}\,\sqrt {3-3\,x-\sqrt {3}\,x\,1{}\mathrm {i}-\sqrt {3}\,1{}\mathrm {i}}}{18\,\sqrt {x^3+1}\,\left (\mathrm {root}\left (z^5+z^2+1,z,k\right )+1\right )\,\mathrm {root}\left (z^5+z^2+1,z,k\right )}\right ) \]

[In]

int((5*x^3 + 2)/((x^3 + 1)^(1/2)*(x^2 + x^5 + 1)),x)

[Out]

symsum(-(6^(1/2)*((3^(1/2)*1i)/2 + 3/2)*(-(3^(1/2)*1i - 3)*(x + 1))^(1/2)*ellipticPi((3^(1/2)*1i + 3)/(2*(root
(z^5 + z^2 + 1, z, k) + 1)), asin((6^(1/2)*(-(3^(1/2)*1i - 3)*(x + 1))^(1/2))/6), (3^(1/2)*1i)/2 + 1/2)*(3^(1/
2)*x*1i - 3*x + 3^(1/2)*1i + 3)^(1/2)*(3 - 3^(1/2)*x*1i - 3^(1/2)*1i - 3*x)^(1/2))/(18*(x^3 + 1)^(1/2)*(root(z
^5 + z^2 + 1, z, k) + 1)*root(z^5 + z^2 + 1, z, k)), k, 1, 5)