\(\int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx\) [853]

Optimal result

Integrand size = 25, antiderivative size = 362 \[ \int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx=-\frac {3}{272} \left (153-31 \sqrt {17}\right ) (2+x)^{2/3}-\frac {3}{272} \left (153+31 \sqrt {17}\right ) (2+x)^{2/3}+\frac {3}{340} \left (85-23 \sqrt {17}\right ) (2+x)^{5/3}+\frac {3}{340} \left (85+23 \sqrt {17}\right ) (2+x)^{5/3}-\frac {1}{8} \sqrt {\frac {3}{17}} \sqrt [3]{96111+23401 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{-1+\sqrt {17}}}}{\sqrt {3}}\right )-\frac {1}{8} \sqrt {\frac {3}{17}} \sqrt [3]{-96111+23401 \sqrt {17}} \arctan \left (\frac {1+\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{1+\sqrt {17}}}}{\sqrt {3}}\right )+\frac {\sqrt [3]{-96111+23401 \sqrt {17}} \log \left (7-\sqrt {17}+4 x\right )}{16 \sqrt {17}}+\frac {\sqrt [3]{96111+23401 \sqrt {17}} \log \left (7+\sqrt {17}+4 x\right )}{16 \sqrt {17}}-\frac {3 \sqrt [3]{-96111+23401 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-2^{2/3} \sqrt [3]{2+x}\right )}{16 \sqrt {17}}-\frac {3 \sqrt [3]{96111+23401 \sqrt {17}} \log \left (\sqrt [3]{-1+\sqrt {17}}+2^{2/3} \sqrt [3]{2+x}\right )}{16 \sqrt {17}} \] Output:

-3/272*(153-31*17^(1/2))*(2+x)^(2/3)-3/272*(153+31*17^(1/2))*(2+x)^(2/3)+3 
/340*(85-23*17^(1/2))*(2+x)^(5/3)+3/340*(85+23*17^(1/2))*(2+x)^(5/3)-1/136 
*51^(1/2)*(96111+23401*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))-1/136*51^(1/2)*(-96111+23401*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))+1/272*( 
-96111+23401*17^(1/2))^(1/3)*ln(7-17^(1/2)+4*x)*17^(1/2)+1/272*(96111+2340 
1*17^(1/2))^(1/3)*ln(7+17^(1/2)+4*x)*17^(1/2)-3/272*(-96111+23401*17^(1/2) 
)^(1/3)*ln((1+17^(1/2))^(1/3)-2^(2/3)*(2+x)^(1/3))*17^(1/2)-3/272*(96111+2 
3401*17^(1/2))^(1/3)*ln((-1+17^(1/2))^(1/3)+2^(2/3)*(2+x)^(1/3))*17^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.25 \[ \int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {3}{8} (2+x)^{2/3} (-9+4 (2+x))+\frac {1}{4} \text {RootSum}\left [-2-\text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-18 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right )+11 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \] Input:

Integrate[((2 + x)^(5/3)*(3 + 5*x))/(4 + 7*x + 2*x^2),x]
 

Output:

(3*(2 + x)^(2/3)*(-9 + 4*(2 + x)))/8 + RootSum[-2 - #1^3 + 2*#1^6 & , (-18 
*Log[(2 + x)^(1/3) - #1] + 11*Log[(2 + x)^(1/3) - #1]*#1^3)/(-#1 + 4*#1^4) 
 & ]/4
 

Rubi [A] (warning: unable to verify)

Time = 1.01 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1196, 25, 1196, 25, 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.

\(\displaystyle \int \frac {(x+2)^{5/3} (5 x+3)}{2 x^2+7 x+4} \, dx\)

\(\Big \downarrow \) 1196

\(\displaystyle \frac {1}{2} \int -\frac {(x+2)^{2/3} (9 x+8)}{2 x^2+7 x+4}dx+\frac {3}{2} (x+2)^{5/3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{2} (x+2)^{5/3}-\frac {1}{2} \int \frac {(x+2)^{2/3} (9 x+8)}{2 x^2+7 x+4}dx\)

\(\Big \downarrow \) 1196

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int -\frac {11 x+4}{\sqrt [3]{x+2} \left (2 x^2+7 x+4\right )}dx-\frac {27}{4} (x+2)^{2/3}\right )+\frac {3}{2} (x+2)^{5/3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {11 x+4}{\sqrt [3]{x+2} \left (2 x^2+7 x+4\right )}dx-\frac {27}{4} (x+2)^{2/3}\right )+\frac {3}{2} (x+2)^{5/3}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \left (\frac {11-\frac {61}{\sqrt {17}}}{\left (4 x-\sqrt {17}+7\right ) \sqrt [3]{x+2}}+\frac {11+\frac {61}{\sqrt {17}}}{\left (4 x+\sqrt {17}+7\right ) \sqrt [3]{x+2}}\right )dx-\frac {27}{4} (x+2)^{2/3}\right )+\frac {3}{2} (x+2)^{5/3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{96111+23401 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{\sqrt {17}-1}}}{\sqrt {3}}\right )-\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{23401 \sqrt {17}-96111} \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{1+\sqrt {17}}}+1}{\sqrt {3}}\right )+\frac {1}{68} \sqrt [3]{6762889-1633887 \sqrt {17}} \log \left (4 x-\sqrt {17}+7\right )+\frac {\sqrt [3]{397817+96111 \sqrt {17}} \log \left (4 x+\sqrt {17}+7\right )}{4\ 17^{2/3}}-\frac {3}{68} \sqrt [3]{6762889-1633887 \sqrt {17}} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{x+2}\right )-\frac {3 \sqrt [3]{397817+96111 \sqrt {17}} \log \left (2 \sqrt [3]{x+2}+\sqrt [3]{2 \left (\sqrt {17}-1\right )}\right )}{4\ 17^{2/3}}\right )-\frac {27}{4} (x+2)^{2/3}\right )+\frac {3}{2} (x+2)^{5/3}\)

Input:

Int[((2 + x)^(5/3)*(3 + 5*x))/(4 + 7*x + 2*x^2),x]
 

Output:

(3*(2 + x)^(5/3))/2 + ((-27*(2 + x)^(2/3))/4 + (-1/2*(Sqrt[3/17]*(96111 + 
23401*Sqrt[17])^(1/3)*ArcTan[(1 - (2*2^(2/3)*(2 + x)^(1/3))/(-1 + Sqrt[17] 
)^(1/3))/Sqrt[3]]) - (Sqrt[3/17]*(-96111 + 23401*Sqrt[17])^(1/3)*ArcTan[(1 
 + (2*2^(2/3)*(2 + x)^(1/3))/(1 + Sqrt[17])^(1/3))/Sqrt[3]])/2 + ((6762889 
 - 1633887*Sqrt[17])^(1/3)*Log[7 - Sqrt[17] + 4*x])/68 + ((397817 + 96111* 
Sqrt[17])^(1/3)*Log[7 + Sqrt[17] + 4*x])/(4*17^(2/3)) - (3*(6762889 - 1633 
887*Sqrt[17])^(1/3)*Log[(2*(1 + Sqrt[17]))^(1/3) - 2*(2 + x)^(1/3)])/68 - 
(3*(397817 + 96111*Sqrt[17])^(1/3)*Log[(2*(-1 + Sqrt[17]))^(1/3) + 2*(2 + 
x)^(1/3)])/(4*17^(2/3)))/2)/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1196

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + 
(c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c   Int 
[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + 
 b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & 
& GtQ[m, 0]
 

rule 1200

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [C] (verified)

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

Time = 22.84 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.18

method	result	size
risch	\(\frac {3 \left (4 x -1\right ) \left (2+x \right )^{\frac {2}{3}}}{8}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (11 \textit {\_R}^{4}-18 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{4}\)	\(65\)
derivativedivides	\(\frac {3 \left (2+x \right )^{\frac {5}{3}}}{2}-\frac {27 \left (2+x \right )^{\frac {2}{3}}}{8}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (11 \textit {\_R}^{4}-18 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{4}\)	\(67\)
default	\(\frac {3 \left (2+x \right )^{\frac {5}{3}}}{2}-\frac {27 \left (2+x \right )^{\frac {2}{3}}}{8}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (11 \textit {\_R}^{4}-18 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{4}\)	\(67\)
trager	\(\text {Expression too large to display}\)	\(7204\)

Input:

int((2+x)^(5/3)*(5*x+3)/(2*x^2+7*x+4),x,method=_RETURNVERBOSE)
 

Output:

3/8*(4*x-1)*(2+x)^(2/3)+1/4*sum((11*_R^4-18*_R)/(4*_R^5-_R^2)*ln((2+x)^(1/ 
3)-_R),_R=RootOf(2*_Z^6-_Z^3-2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.83 \[ \int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx =\text {Too large to display} \] Input:

integrate((2+x)^(5/3)*(3+5*x)/(2*x^2+7*x+4),x, algorithm="fricas")
 

Output:

1/16*(96111/289*sqrt(17) - 23401/17)^(1/3)*(sqrt(-3) - 1)*log(17*(445*sqrt 
(17)*(sqrt(-3) + 1) + 1787*sqrt(-3) + 1787)*(96111/289*sqrt(17) - 23401/17 
)^(2/3) + 346112*(x + 2)^(1/3)) - 1/16*(96111/289*sqrt(17) - 23401/17)^(1/ 
3)*(sqrt(-3) + 1)*log(-17*(445*sqrt(17)*(sqrt(-3) - 1) + 1787*sqrt(-3) - 1 
787)*(96111/289*sqrt(17) - 23401/17)^(2/3) + 346112*(x + 2)^(1/3)) + 1/16* 
(-96111/289*sqrt(17) - 23401/17)^(1/3)*(sqrt(-3) - 1)*log(-17*(445*sqrt(17 
)*(sqrt(-3) + 1) - 1787*sqrt(-3) - 1787)*(-96111/289*sqrt(17) - 23401/17)^ 
(2/3) + 346112*(x + 2)^(1/3)) - 1/16*(-96111/289*sqrt(17) - 23401/17)^(1/3 
)*(sqrt(-3) + 1)*log(17*(445*sqrt(17)*(sqrt(-3) - 1) - 1787*sqrt(-3) + 178 
7)*(-96111/289*sqrt(17) - 23401/17)^(2/3) + 346112*(x + 2)^(1/3)) + 3/8*(4 
*x - 1)*(x + 2)^(2/3) + 1/8*(96111/289*sqrt(17) - 23401/17)^(1/3)*log(-17* 
(445*sqrt(17) + 1787)*(96111/289*sqrt(17) - 23401/17)^(2/3) + 173056*(x + 
2)^(1/3)) + 1/8*(-96111/289*sqrt(17) - 23401/17)^(1/3)*log(17*(445*sqrt(17 
) - 1787)*(-96111/289*sqrt(17) - 23401/17)^(2/3) + 173056*(x + 2)^(1/3))
 

Sympy [F]

\[ \int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\int \frac {\left (x + 2\right )^{\frac {5}{3}} \cdot \left (5 x + 3\right )}{2 x^{2} + 7 x + 4}\, dx \] Input:

integrate((2+x)**(5/3)*(3+5*x)/(2*x**2+7*x+4),x)
 

Output:

Integral((x + 2)**(5/3)*(5*x + 3)/(2*x**2 + 7*x + 4), x)
 

Maxima [F]

\[ \int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\int { \frac {{\left (5 \, x + 3\right )} {\left (x + 2\right )}^{\frac {5}{3}}}{2 \, x^{2} + 7 \, x + 4} \,d x } \] Input:

integrate((2+x)^(5/3)*(3+5*x)/(2*x^2+7*x+4),x, algorithm="maxima")
 

Output:

integrate((5*x + 3)*(x + 2)^(5/3)/(2*x^2 + 7*x + 4), x)
 

Giac [F]

\[ \int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\int { \frac {{\left (5 \, x + 3\right )} {\left (x + 2\right )}^{\frac {5}{3}}}{2 \, x^{2} + 7 \, x + 4} \,d x } \] Input:

integrate((2+x)^(5/3)*(3+5*x)/(2*x^2+7*x+4),x, algorithm="giac")
 

Output:

integrate((5*x + 3)*(x + 2)^(5/3)/(2*x^2 + 7*x + 4), x)
 

Mupad [B] (verification not implemented)

Time = 11.58 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.48 \[ \int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\text {Too large to display} \] Input:

int(((5*x + 3)*(x + 2)^(5/3))/(7*x + 2*x^2 + 4),x)
 

Output:

(3*(x + 2)^(5/3))/2 - (27*(x + 2)^(2/3))/8 + (17^(1/3)*log(- (1026675*(x + 
 2)^(1/3))/1024 - (17^(2/3)*(- 96111*17^(1/2) - 397817)^(2/3)*((17^(1/3)*( 
- 96111*17^(1/2) - 397817)^(1/3)*((4829139*(x + 2)^(1/3))/256 + (243*17^(2 
/3)*(- 96111*17^(1/2) - 397817)^(2/3))/2048))/136 - 150182505/8192))/18496 
)*(- 96111*17^(1/2) - 397817)^(1/3))/136 - (17^(1/3)*log((17^(2/3)*(397817 
 - 96111*17^(1/2))^(2/3)*((17^(1/3)*(397817 - 96111*17^(1/2))^(1/3)*((4829 
139*(x + 2)^(1/3))/256 + (243*17^(2/3)*(397817 - 96111*17^(1/2))^(2/3))/20 
48))/136 + 150182505/8192))/18496 - (1026675*(x + 2)^(1/3))/1024)*(397817 
- 96111*17^(1/2))^(1/3))/136 + (17^(1/3)*log(- (1026675*(x + 2)^(1/3))/102 
4 - (17^(2/3)*(96111*17^(1/2) - 397817)^(2/3)*((17^(1/3)*(96111*17^(1/2) - 
 397817)^(1/3)*((4829139*(x + 2)^(1/3))/256 + (243*17^(2/3)*(96111*17^(1/2 
) - 397817)^(2/3))/2048))/136 - 150182505/8192))/18496)*(96111*17^(1/2) - 
397817)^(1/3))/136 - (17^(1/3)*log((17^(2/3)*(96111*17^(1/2) + 397817)^(2/ 
3)*((17^(1/3)*(96111*17^(1/2) + 397817)^(1/3)*((4829139*(x + 2)^(1/3))/256 
 + (243*17^(2/3)*(96111*17^(1/2) + 397817)^(2/3))/2048))/136 + 150182505/8 
192))/18496 - (1026675*(x + 2)^(1/3))/1024)*(96111*17^(1/2) + 397817)^(1/3 
))/136 - (17^(1/3)*log((17^(2/3)*(3^(1/2)*1i + 1)^2*(- 96111*17^(1/2) - 39 
7817)^(2/3)*((17^(1/3)*(3^(1/2)*1i + 1)*(- 96111*17^(1/2) - 397817)^(1/3)* 
((4829139*(x + 2)^(1/3))/256 + (243*17^(2/3)*(3^(1/2)*1i + 1)^2*(- 96111*1 
7^(1/2) - 397817)^(2/3))/8192))/272 + 150182505/8192))/73984 - (1026675...
 

Reduce [F]

\[ \int \frac {(2+x)^{5/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\int \frac {\left (x +2\right )^{\frac {5}{3}} \left (5 x +3\right )}{2 x^{2}+7 x +4}d x \] Input:

int((2+x)^(5/3)*(3+5*x)/(2*x^2+7*x+4),x)
 

Output:

int((2+x)^(5/3)*(3+5*x)/(2*x^2+7*x+4),x)