3.1.58 \(\int \frac {2+5 x^3}{\sqrt {1+x^3} (1+x^2+x^5)} \, dx\)
Optimal. Leaf size=14 \[ 2 \tan ^{-1}\left (x \sqrt {x^3+1}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 0.54, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[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*} \int \frac {2+5 x^3}{\sqrt {1+x^3} \left (1+x^2+x^5\right )} \, dx &=\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 [C] time = 4.92, size = 2691, normalized size = 192.21 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(2 + 5*x^3)/(Sqrt[1 + x^3]*(1 + x^2 + x^5)),x]
[Out]
((-4*I)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) - Root[1 + #1^2 +
#1^5 & , 1, 0]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(Sqrt[1 + x^3]*((-1)^(1/3) -
Root[1 + #1^2 + #1^5 & , 1, 0])*(Root[1 + #1^2 + #1^5 & , 1, 0] - Root[1 + #1^2 + #1^5 & , 2, 0])*(Root[1 + #1
^2 + #1^5 & , 1, 0] - Root[1 + #1^2 + #1^5 & , 3, 0])*(Root[1 + #1^2 + #1^5 & , 1, 0] - Root[1 + #1^2 + #1^5 &
, 4, 0])*(Root[1 + #1^2 + #1^5 & , 1, 0] - Root[1 + #1^2 + #1^5 & , 5, 0])) - ((10*I)*Sqrt[(1 + x)/(1 + (-1)^
(1/3))]*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 1, 0]), ArcSin[Sqrt[(1
+ (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]*Root[1 + #1^2 + #1^5 & , 1, 0]^3)/(Sqrt[1 + x^3]*((-1)^(1/3)
- Root[1 + #1^2 + #1^5 & , 1, 0])*(Root[1 + #1^2 + #1^5 & , 1, 0] - Root[1 + #1^2 + #1^5 & , 2, 0])*(Root[1 +
#1^2 + #1^5 & , 1, 0] - Root[1 + #1^2 + #1^5 & , 3, 0])*(Root[1 + #1^2 + #1^5 & , 1, 0] - Root[1 + #1^2 + #1^5
& , 4, 0])*(Root[1 + #1^2 + #1^5 & , 1, 0] - Root[1 + #1^2 + #1^5 & , 5, 0])) - ((4*I)*Sqrt[(1 + x)/(1 + (-1)
^(1/3))]*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 2, 0]), ArcSin[Sqrt[(
1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(Sqrt[1 + x^3]*((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 2, 0]
)*(-Root[1 + #1^2 + #1^5 & , 1, 0] + Root[1 + #1^2 + #1^5 & , 2, 0])*(Root[1 + #1^2 + #1^5 & , 2, 0] - Root[1
+ #1^2 + #1^5 & , 3, 0])*(Root[1 + #1^2 + #1^5 & , 2, 0] - Root[1 + #1^2 + #1^5 & , 4, 0])*(Root[1 + #1^2 + #1
^5 & , 2, 0] - Root[1 + #1^2 + #1^5 & , 5, 0])) - ((10*I)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*Ell
ipticPi[(I*Sqrt[3])/((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 2, 0]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1
/3))]], (-1)^(1/3)]*Root[1 + #1^2 + #1^5 & , 2, 0]^3)/(Sqrt[1 + x^3]*((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 2,
0])*(-Root[1 + #1^2 + #1^5 & , 1, 0] + Root[1 + #1^2 + #1^5 & , 2, 0])*(Root[1 + #1^2 + #1^5 & , 2, 0] - Root
[1 + #1^2 + #1^5 & , 3, 0])*(Root[1 + #1^2 + #1^5 & , 2, 0] - Root[1 + #1^2 + #1^5 & , 4, 0])*(Root[1 + #1^2 +
#1^5 & , 2, 0] - Root[1 + #1^2 + #1^5 & , 5, 0])) - ((4*I)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*E
llipticPi[(I*Sqrt[3])/((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 3, 0]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^
(1/3))]], (-1)^(1/3)])/(Sqrt[1 + x^3]*((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 3, 0])*(-Root[1 + #1^2 + #1^5 & ,
1, 0] + Root[1 + #1^2 + #1^5 & , 3, 0])*(-Root[1 + #1^2 + #1^5 & , 2, 0] + Root[1 + #1^2 + #1^5 & , 3, 0])*(R
oot[1 + #1^2 + #1^5 & , 3, 0] - Root[1 + #1^2 + #1^5 & , 4, 0])*(Root[1 + #1^2 + #1^5 & , 3, 0] - Root[1 + #1^
2 + #1^5 & , 5, 0])) - ((10*I)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(
1/3) - Root[1 + #1^2 + #1^5 & , 3, 0]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]*Root[1
+ #1^2 + #1^5 & , 3, 0]^3)/(Sqrt[1 + x^3]*((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 3, 0])*(-Root[1 + #1^2 + #1^5
& , 1, 0] + Root[1 + #1^2 + #1^5 & , 3, 0])*(-Root[1 + #1^2 + #1^5 & , 2, 0] + Root[1 + #1^2 + #1^5 & , 3, 0]
)*(Root[1 + #1^2 + #1^5 & , 3, 0] - Root[1 + #1^2 + #1^5 & , 4, 0])*(Root[1 + #1^2 + #1^5 & , 3, 0] - Root[1 +
#1^2 + #1^5 & , 5, 0])) - ((4*I)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1
)^(1/3) - Root[1 + #1^2 + #1^5 & , 4, 0]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(Sq
rt[1 + x^3]*((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 4, 0])*(-Root[1 + #1^2 + #1^5 & , 1, 0] + Root[1 + #1^2 + #
1^5 & , 4, 0])*(-Root[1 + #1^2 + #1^5 & , 2, 0] + Root[1 + #1^2 + #1^5 & , 4, 0])*(-Root[1 + #1^2 + #1^5 & , 3
, 0] + Root[1 + #1^2 + #1^5 & , 4, 0])*(Root[1 + #1^2 + #1^5 & , 4, 0] - Root[1 + #1^2 + #1^5 & , 5, 0])) - ((
10*I)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) - Root[1 + #1^2 + #1
^5 & , 4, 0]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]*Root[1 + #1^2 + #1^5 & , 4, 0]^3
)/(Sqrt[1 + x^3]*((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 4, 0])*(-Root[1 + #1^2 + #1^5 & , 1, 0] + Root[1 + #1^
2 + #1^5 & , 4, 0])*(-Root[1 + #1^2 + #1^5 & , 2, 0] + Root[1 + #1^2 + #1^5 & , 4, 0])*(-Root[1 + #1^2 + #1^5
& , 3, 0] + Root[1 + #1^2 + #1^5 & , 4, 0])*(Root[1 + #1^2 + #1^5 & , 4, 0] - Root[1 + #1^2 + #1^5 & , 5, 0]))
- ((4*I)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) - Root[1 + #1^2
+ #1^5 & , 5, 0]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(Sqrt[1 + x^3]*((-1)^(1/3)
- Root[1 + #1^2 + #1^5 & , 5, 0])*(-Root[1 + #1^2 + #1^5 & , 1, 0] + Root[1 + #1^2 + #1^5 & , 5, 0])*(-Root[1
+ #1^2 + #1^5 & , 2, 0] + Root[1 + #1^2 + #1^5 & , 5, 0])*(-Root[1 + #1^2 + #1^5 & , 3, 0] + Root[1 + #1^2 + #
1^5 & , 5, 0])*(-Root[1 + #1^2 + #1^5 & , 4, 0] + Root[1 + #1^2 + #1^5 & , 5, 0])) - ((10*I)*Sqrt[(1 + x)/(1 +
(-1)^(1/3))]*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) - Root[1 + #1^2 + #1^5 & , 5, 0]), ArcSin[S
qrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]*Root[1 + #1^2 + #1^5 & , 5, 0]^3)/(Sqrt[1 + x^3]*((-1)^
(1/3) - Root[1 + #1^2 + #1^5 & , 5, 0])*(-Root[1 + #1^2 + #1^5 & , 1, 0] + Root[1 + #1^2 + #1^5 & , 5, 0])*(-R
oot[1 + #1^2 + #1^5 & , 2, 0] + Root[1 + #1^2 + #1^5 & , 5, 0])*(-Root[1 + #1^2 + #1^5 & , 3, 0] + Root[1 + #1
^2 + #1^5 & , 5, 0])*(-Root[1 + #1^2 + #1^5 & , 4, 0] + Root[1 + #1^2 + #1^5 & , 5, 0]))
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.66, size = 14, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (x \sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(2 + 5*x^3)/(Sqrt[1 + x^3]*(1 + x^2 + x^5)),x]
[Out]
2*ArcTan[x*Sqrt[1 + x^3]]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 25, normalized size = 1.79 \begin {gather*} \arctan \left (\frac {{\left (x^{5} + x^{2} - 1\right )} \sqrt {x^{3} + 1}}{2 \, {\left (x^{4} + x\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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))
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, x^{3} + 2}{{\left (x^{5} + x^{2} + 1\right )} \sqrt {x^{3} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
maple [C] time = 0.41, size = 60, normalized size = 4.29
| | |
| method |
result |
size |
| | |
| trager |
\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{3}+1}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{5}+x^{2}+1}\right )\) |
\(60\) |
| default |
\(-\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{4}-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{4}}{2}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\) |
\(197\) |
| elliptic |
\(-\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{4}-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{4}}{2}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{2}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\) |
\(197\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[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))
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, x^{3} + 2}{{\left (x^{5} + x^{2} + 1\right )} \sqrt {x^{3} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
mupad [B] time = 1.53, size = 163, normalized size = 11.64 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________