Integrand size = 25, antiderivative size = 345 \[ \int \frac {3+5 x}{(2+x)^{2/3} \left (4+7 x+2 x^2\right )} \, dx=-\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{8507+2063 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{-1+\sqrt {17}}}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{8507-2063 \sqrt {17}} \arctan \left (\frac {1+\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{1+\sqrt {17}}}}{\sqrt {3}}\right )-\frac {\sqrt [3]{8507-2063 \sqrt {17}} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{2+x}\right )}{2 \sqrt {17}}+\frac {\sqrt [3]{8507+2063 \sqrt {17}} \log \left (\sqrt [3]{2 \left (-1+\sqrt {17}\right )}+2 \sqrt [3]{2+x}\right )}{2 \sqrt {17}}-\frac {\sqrt [3]{35071+8507 \sqrt {17}} \log \left (\left (2 \left (-1+\sqrt {17}\right )\right )^{2/3}-2 \sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [3]{2+x}+4 (2+x)^{2/3}\right )}{4\ 17^{2/3}}+\frac {\sqrt [3]{-35071+8507 \sqrt {17}} \log \left (\left (2 \left (1+\sqrt {17}\right )\right )^{2/3}+2 \sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [3]{2+x}+4 (2+x)^{2/3}\right )}{4\ 17^{2/3}} \] Output:
-1/34*51^(1/2)*(8507+2063*17^(1/2))^(1/3)*arctan(1/3*(1-2*2^(2/3)*(2+x)^(1 /3)/(-1+17^(1/2))^(1/3))*3^(1/2))+1/34*51^(1/2)*(8507-2063*17^(1/2))^(1/3) *arctan(1/3*(1+2*2^(2/3)*(2+x)^(1/3)/(1+17^(1/2))^(1/3))*3^(1/2))-1/34*(85 07-2063*17^(1/2))^(1/3)*ln((2+2*17^(1/2))^(1/3)-2*(2+x)^(1/3))*17^(1/2)+1/ 34*(8507+2063*17^(1/2))^(1/3)*ln((-2+2*17^(1/2))^(1/3)+2*(2+x)^(1/3))*17^( 1/2)-1/68*(35071+8507*17^(1/2))^(1/3)*ln((-2+2*17^(1/2))^(2/3)-2*(-2+2*17^ (1/2))^(1/3)*(2+x)^(1/3)+4*(2+x)^(2/3))*17^(1/3)+1/68*(-35071+8507*17^(1/2 ))^(1/3)*ln((2+2*17^(1/2))^(2/3)+2*(2+2*17^(1/2))^(1/3)*(2+x)^(1/3)+4*(2+x )^(2/3))*17^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.20 \[ \int \frac {3+5 x}{(2+x)^{2/3} \left (4+7 x+2 x^2\right )} \, dx=\text {RootSum}\left [-2-\text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-7 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right )+5 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(3 + 5*x)/((2 + x)^(2/3)*(4 + 7*x + 2*x^2)),x]
Output:
RootSum[-2 - #1^3 + 2*#1^6 & , (-7*Log[(2 + x)^(1/3) - #1] + 5*Log[(2 + x) ^(1/3) - #1]*#1^3)/(-#1^2 + 4*#1^5) & ]
Time = 0.87 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1200, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {5 x+3}{(x+2)^{2/3} \left (2 x^2+7 x+4\right )} \, dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \left (\frac {5-\frac {23}{\sqrt {17}}}{\left (4 x-\sqrt {17}+7\right ) (x+2)^{2/3}}+\frac {5+\frac {23}{\sqrt {17}}}{\left (4 x+\sqrt {17}+7\right ) (x+2)^{2/3}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{8507+2063 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{\sqrt {17}-1}}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{8507-2063 \sqrt {17}} \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{1+\sqrt {17}}}+1}{\sqrt {3}}\right )+\frac {\sqrt [3]{8507 \sqrt {17}-35071} \log \left (4 x-\sqrt {17}+7\right )}{4\ 17^{2/3}}-\frac {\sqrt [3]{35071+8507 \sqrt {17}} \log \left (4 x+\sqrt {17}+7\right )}{4\ 17^{2/3}}-\frac {3 \sqrt [3]{8507 \sqrt {17}-35071} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{x+2}\right )}{4\ 17^{2/3}}+\frac {3 \sqrt [3]{35071+8507 \sqrt {17}} \log \left (2 \sqrt [3]{x+2}+\sqrt [3]{2 \left (\sqrt {17}-1\right )}\right )}{4\ 17^{2/3}}\) |
Input:
Int[(3 + 5*x)/((2 + x)^(2/3)*(4 + 7*x + 2*x^2)),x]
Output:
-1/2*(Sqrt[3/17]*(8507 + 2063*Sqrt[17])^(1/3)*ArcTan[(1 - (2*2^(2/3)*(2 + x)^(1/3))/(-1 + Sqrt[17])^(1/3))/Sqrt[3]]) + (Sqrt[3/17]*(8507 - 2063*Sqrt [17])^(1/3)*ArcTan[(1 + (2*2^(2/3)*(2 + x)^(1/3))/(1 + Sqrt[17])^(1/3))/Sq rt[3]])/2 + ((-35071 + 8507*Sqrt[17])^(1/3)*Log[7 - Sqrt[17] + 4*x])/(4*17 ^(2/3)) - ((35071 + 8507*Sqrt[17])^(1/3)*Log[7 + Sqrt[17] + 4*x])/(4*17^(2 /3)) - (3*(-35071 + 8507*Sqrt[17])^(1/3)*Log[(2*(1 + Sqrt[17]))^(1/3) - 2* (2 + x)^(1/3)])/(4*17^(2/3)) + (3*(35071 + 8507*Sqrt[17])^(1/3)*Log[(2*(-1 + Sqrt[17]))^(1/3) + 2*(2 + x)^(1/3)])/(4*17^(2/3))
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 36.78 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.14
| method | result | size |
| derivativedivides | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (5 \textit {\_R}^{3}-7\right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\) | \(48\) |
| default | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (5 \textit {\_R}^{3}-7\right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\) | \(48\) |
| trager | \(\text {Expression too large to display}\) | \(10949\) |
Input:
int((5*x+3)/(2+x)^(2/3)/(2*x^2+7*x+4),x,method=_RETURNVERBOSE)
Output:
sum((5*_R^3-7)/(4*_R^5-_R^2)*ln((2+x)^(1/3)-_R),_R=RootOf(2*_Z^6-_Z^3-2))
Time = 0.10 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.83 \[ \int \frac {3+5 x}{(2+x)^{2/3} \left (4+7 x+2 x^2\right )} \, dx=-\frac {1}{4} \, {\left (\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left ({\left (27 \, \sqrt {17} {\left (\sqrt {-3} + 1\right )} - 119 \, \sqrt {-3} - 119\right )} {\left (\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} + 104 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (-{\left (27 \, \sqrt {17} {\left (\sqrt {-3} - 1\right )} - 119 \, \sqrt {-3} + 119\right )} {\left (\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} + 104 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) - \frac {1}{4} \, {\left (-\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (-{\left (27 \, \sqrt {17} {\left (\sqrt {-3} + 1\right )} + 119 \, \sqrt {-3} + 119\right )} {\left (-\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} + 104 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (-\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left ({\left (27 \, \sqrt {17} {\left (\sqrt {-3} - 1\right )} + 119 \, \sqrt {-3} - 119\right )} {\left (-\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} + 104 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\left (\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} \log \left (-{\left (\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} {\left (27 \, \sqrt {17} - 119\right )} + 52 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\left (-\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} \log \left ({\left (27 \, \sqrt {17} + 119\right )} {\left (-\frac {8507}{289} \, \sqrt {17} + \frac {2063}{17}\right )}^{\frac {1}{3}} + 52 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) \] Input:
integrate((3+5*x)/(2+x)^(2/3)/(2*x^2+7*x+4),x, algorithm="fricas")
Output:
-1/4*(8507/289*sqrt(17) + 2063/17)^(1/3)*(sqrt(-3) + 1)*log((27*sqrt(17)*( sqrt(-3) + 1) - 119*sqrt(-3) - 119)*(8507/289*sqrt(17) + 2063/17)^(1/3) + 104*(x + 2)^(1/3)) + 1/4*(8507/289*sqrt(17) + 2063/17)^(1/3)*(sqrt(-3) - 1 )*log(-(27*sqrt(17)*(sqrt(-3) - 1) - 119*sqrt(-3) + 119)*(8507/289*sqrt(17 ) + 2063/17)^(1/3) + 104*(x + 2)^(1/3)) - 1/4*(-8507/289*sqrt(17) + 2063/1 7)^(1/3)*(sqrt(-3) + 1)*log(-(27*sqrt(17)*(sqrt(-3) + 1) + 119*sqrt(-3) + 119)*(-8507/289*sqrt(17) + 2063/17)^(1/3) + 104*(x + 2)^(1/3)) + 1/4*(-850 7/289*sqrt(17) + 2063/17)^(1/3)*(sqrt(-3) - 1)*log((27*sqrt(17)*(sqrt(-3) - 1) + 119*sqrt(-3) - 119)*(-8507/289*sqrt(17) + 2063/17)^(1/3) + 104*(x + 2)^(1/3)) + 1/2*(8507/289*sqrt(17) + 2063/17)^(1/3)*log(-(8507/289*sqrt(1 7) + 2063/17)^(1/3)*(27*sqrt(17) - 119) + 52*(x + 2)^(1/3)) + 1/2*(-8507/2 89*sqrt(17) + 2063/17)^(1/3)*log((27*sqrt(17) + 119)*(-8507/289*sqrt(17) + 2063/17)^(1/3) + 52*(x + 2)^(1/3))
\[ \int \frac {3+5 x}{(2+x)^{2/3} \left (4+7 x+2 x^2\right )} \, dx=\int \frac {5 x + 3}{\left (x + 2\right )^{\frac {2}{3}} \cdot \left (2 x^{2} + 7 x + 4\right )}\, dx \] Input:
integrate((3+5*x)/(2+x)**(2/3)/(2*x**2+7*x+4),x)
Output:
Integral((5*x + 3)/((x + 2)**(2/3)*(2*x**2 + 7*x + 4)), x)
\[ \int \frac {3+5 x}{(2+x)^{2/3} \left (4+7 x+2 x^2\right )} \, dx=\int { \frac {5 \, x + 3}{{\left (2 \, x^{2} + 7 \, x + 4\right )} {\left (x + 2\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((3+5*x)/(2+x)^(2/3)/(2*x^2+7*x+4),x, algorithm="maxima")
Output:
integrate((5*x + 3)/((2*x^2 + 7*x + 4)*(x + 2)^(2/3)), x)
\[ \int \frac {3+5 x}{(2+x)^{2/3} \left (4+7 x+2 x^2\right )} \, dx=\int { \frac {5 \, x + 3}{{\left (2 \, x^{2} + 7 \, x + 4\right )} {\left (x + 2\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((3+5*x)/(2+x)^(2/3)/(2*x^2+7*x+4),x, algorithm="giac")
Output:
integrate((5*x + 3)/((2*x^2 + 7*x + 4)*(x + 2)^(2/3)), x)
Time = 0.19 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.60 \[ \int \frac {3+5 x}{(2+x)^{2/3} \left (4+7 x+2 x^2\right )} \, dx=\text {Too large to display} \] Input:
int((5*x + 3)/((x + 2)^(2/3)*(7*x + 2*x^2 + 4)),x)
Output:
(17^(1/3)*log((1506843*(x + 2)^(1/3))/128 + (17^(1/3)*(35071 - 8507*17^(1/ 2))^(1/3)*((17^(2/3)*(35071 - 8507*17^(1/2))^(2/3)*((1193859*(x + 2)^(1/3) )/64 + (4131*17^(1/3)*(35071 - 8507*17^(1/2))^(1/3))/64))/1156 + 6655041/6 4))/34)*(35071 - 8507*17^(1/2))^(1/3))/34 - (17^(1/3)*log((1506843*(x + 2) ^(1/3))/128 - (17^(1/3)*(8507*17^(1/2) - 35071)^(1/3)*((17^(2/3)*(8507*17^ (1/2) - 35071)^(2/3)*((1193859*(x + 2)^(1/3))/64 - (4131*17^(1/3)*(8507*17 ^(1/2) - 35071)^(1/3))/64))/1156 + 6655041/64))/34)*(8507*17^(1/2) - 35071 )^(1/3))/34 + (17^(1/3)*log((1506843*(x + 2)^(1/3))/128 + (17^(1/3)*(8507* 17^(1/2) + 35071)^(1/3)*((17^(2/3)*(8507*17^(1/2) + 35071)^(2/3)*((1193859 *(x + 2)^(1/3))/64 + (4131*17^(1/3)*(8507*17^(1/2) + 35071)^(1/3))/64))/11 56 + 6655041/64))/34)*(8507*17^(1/2) + 35071)^(1/3))/34 + (17^(1/3)*log((1 506843*(x + 2)^(1/3))/128 + (17^(1/3)*(3^(1/2)*1i - 1)*(8507*17^(1/2) + 35 071)^(1/3)*((17^(2/3)*(3^(1/2)*1i - 1)^2*(8507*17^(1/2) + 35071)^(2/3)*((1 193859*(x + 2)^(1/3))/64 + (4131*17^(1/3)*(3^(1/2)*1i - 1)*(8507*17^(1/2) + 35071)^(1/3))/128))/4624 + 6655041/64))/68)*(3^(1/2)*1i - 1)*(8507*17^(1 /2) + 35071)^(1/3))/68 - (17^(1/3)*log((1506843*(x + 2)^(1/3))/128 - (17^( 1/3)*(3^(1/2)*1i + 1)*(35071 - 8507*17^(1/2))^(1/3)*((17^(2/3)*(3^(1/2)*1i + 1)^2*(35071 - 8507*17^(1/2))^(2/3)*((1193859*(x + 2)^(1/3))/64 - (4131* 17^(1/3)*(3^(1/2)*1i + 1)*(35071 - 8507*17^(1/2))^(1/3))/128))/4624 + 6655 041/64))/68)*(3^(1/2)*1i + 1)*(35071 - 8507*17^(1/2))^(1/3))/68 - (17^(...
\[ \int \frac {3+5 x}{(2+x)^{2/3} \left (4+7 x+2 x^2\right )} \, dx=\int \frac {5 x +3}{\left (x +2\right )^{\frac {2}{3}} \left (2 x^{2}+7 x +4\right )}d x \] Input:
int((3+5*x)/(2+x)^(2/3)/(2*x^2+7*x+4),x)
Output:
int((3+5*x)/(2+x)^(2/3)/(2*x^2+7*x+4),x)