\(\int \frac {(2+x)^{2/3} (3+5 x)}{4+7 x+2 x^2} \, dx\) [855]

Optimal result

Integrand size = 25, antiderivative size = 322 \[ \int \frac {(2+x)^{2/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {3}{136} \left (85-23 \sqrt {17}\right ) (2+x)^{2/3}+\frac {3}{136} \left (85+23 \sqrt {17}\right ) (2+x)^{2/3}+\frac {1}{4} \sqrt {\frac {3}{17}} \sqrt [3]{25301+6131 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{-1+\sqrt {17}}}}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {\frac {3}{17}} \sqrt [3]{25301-6131 \sqrt {17}} \arctan \left (\frac {1+\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{1+\sqrt {17}}}}{\sqrt {3}}\right )+\frac {\sqrt [3]{25301-6131 \sqrt {17}} \log \left (7-\sqrt {17}+4 x\right )}{8 \sqrt {17}}-\frac {\sqrt [3]{25301+6131 \sqrt {17}} \log \left (7+\sqrt {17}+4 x\right )}{8 \sqrt {17}}-\frac {3 \sqrt [3]{25301-6131 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-2^{2/3} \sqrt [3]{2+x}\right )}{8 \sqrt {17}}+\frac {3 \sqrt [3]{25301+6131 \sqrt {17}} \log \left (\sqrt [3]{-1+\sqrt {17}}+2^{2/3} \sqrt [3]{2+x}\right )}{8 \sqrt {17}} \] Output:

3/136*(85-23*17^(1/2))*(2+x)^(2/3)+3/136*(85+23*17^(1/2))*(2+x)^(2/3)+1/68 
*51^(1/2)*(25301+6131*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))-1/68*51^(1/2)*(25301-6131*17^(1/2))^(1/3)*ar 
ctan(1/3*(1+2*2^(2/3)*(2+x)^(1/3)/(1+17^(1/2))^(1/3))*3^(1/2))+1/136*(2530 
1-6131*17^(1/2))^(1/3)*ln(7-17^(1/2)+4*x)*17^(1/2)-1/136*(25301+6131*17^(1 
/2))^(1/3)*ln(7+17^(1/2)+4*x)*17^(1/2)-3/136*(25301-6131*17^(1/2))^(1/3)*l 
n((1+17^(1/2))^(1/3)-2^(2/3)*(2+x)^(1/3))*17^(1/2)+3/136*(25301+6131*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 = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.25 \[ \int \frac {(2+x)^{2/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {15}{4} (2+x)^{2/3}-\frac {1}{2} \text {RootSum}\left [-2-\text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-10 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right )+9 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \] Input:

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

Output:

(15*(2 + x)^(2/3))/4 - RootSum[-2 - #1^3 + 2*#1^6 & , (-10*Log[(2 + x)^(1/ 
3) - #1] + 9*Log[(2 + x)^(1/3) - #1]*#1^3)/(-#1 + 4*#1^4) & ]/2
 

Rubi [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {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.

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

\(\Big \downarrow \) 1196

\(\displaystyle \frac {1}{2} \int -\frac {9 x+8}{\sqrt [3]{x+2} \left (2 x^2+7 x+4\right )}dx+\frac {15}{4} (x+2)^{2/3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {15}{4} (x+2)^{2/3}-\frac {1}{2} \int \frac {9 x+8}{\sqrt [3]{x+2} \left (2 x^2+7 x+4\right )}dx\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {15}{4} (x+2)^{2/3}-\frac {1}{2} \int \left (\frac {9-\frac {31}{\sqrt {17}}}{\left (4 x-\sqrt {17}+7\right ) \sqrt [3]{x+2}}+\frac {9+\frac {31}{\sqrt {17}}}{\left (4 x+\sqrt {17}+7\right ) \sqrt [3]{x+2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{25301+6131 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{\sqrt {17}-1}}}{\sqrt {3}}\right )-\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{25301-6131 \sqrt {17}} \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{1+\sqrt {17}}}+1}{\sqrt {3}}\right )+\frac {1}{68} \sqrt [3]{430117 \sqrt {17}-1771859} \log \left (4 x-\sqrt {17}+7\right )-\frac {1}{68} \sqrt [3]{1771859+430117 \sqrt {17}} \log \left (4 x+\sqrt {17}+7\right )-\frac {3}{68} \sqrt [3]{430117 \sqrt {17}-1771859} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{x+2}\right )+\frac {3}{68} \sqrt [3]{1771859+430117 \sqrt {17}} \log \left (2 \sqrt [3]{x+2}+\sqrt [3]{2 \left (\sqrt {17}-1\right )}\right )\right )+\frac {15}{4} (x+2)^{2/3}\)

Input:

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

Output:

(15*(2 + x)^(2/3))/4 + ((Sqrt[3/17]*(25301 + 6131*Sqrt[17])^(1/3)*ArcTan[( 
1 - (2*2^(2/3)*(2 + x)^(1/3))/(-1 + Sqrt[17])^(1/3))/Sqrt[3]])/2 - (Sqrt[3 
/17]*(25301 - 6131*Sqrt[17])^(1/3)*ArcTan[(1 + (2*2^(2/3)*(2 + x)^(1/3))/( 
1 + Sqrt[17])^(1/3))/Sqrt[3]])/2 + ((-1771859 + 430117*Sqrt[17])^(1/3)*Log 
[7 - Sqrt[17] + 4*x])/68 - ((1771859 + 430117*Sqrt[17])^(1/3)*Log[7 + Sqrt 
[17] + 4*x])/68 - (3*(-1771859 + 430117*Sqrt[17])^(1/3)*Log[(2*(1 + Sqrt[1 
7]))^(1/3) - 2*(2 + x)^(1/3)])/68 + (3*(1771859 + 430117*Sqrt[17])^(1/3)*L 
og[(2*(-1 + Sqrt[17]))^(1/3) + 2*(2 + x)^(1/3)])/68)/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 = 21.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.19

method	result	size
derivativedivides	\(\frac {15 \left (2+x \right )^{\frac {2}{3}}}{4}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (9 \textit {\_R}^{4}-10 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(60\)
default	\(\frac {15 \left (2+x \right )^{\frac {2}{3}}}{4}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (9 \textit {\_R}^{4}-10 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(60\)
risch	\(\frac {15 \left (2+x \right )^{\frac {2}{3}}}{4}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (9 \textit {\_R}^{4}-10 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(60\)
trager	\(\text {Expression too large to display}\)	\(7200\)

Input:

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

Output:

15/4*(2+x)^(2/3)-1/2*sum((9*_R^4-10*_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 = 0.10 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.92 \[ \int \frac {(2+x)^{2/3} (3+5 x)}{4+7 x+2 x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*(25301/289*sqrt(17) + 6131/17)^(1/3)*(sqrt(-3) - 1)*log(-17*(181*sqrt( 
17)*(sqrt(-3) + 1) - 739*sqrt(-3) - 739)*(25301/289*sqrt(17) + 6131/17)^(2 
/3) + 21632*(x + 2)^(1/3)) - 1/8*(25301/289*sqrt(17) + 6131/17)^(1/3)*(sqr 
t(-3) + 1)*log(17*(181*sqrt(17)*(sqrt(-3) - 1) - 739*sqrt(-3) + 739)*(2530 
1/289*sqrt(17) + 6131/17)^(2/3) + 21632*(x + 2)^(1/3)) + 1/8*(-25301/289*s 
qrt(17) + 6131/17)^(1/3)*(sqrt(-3) - 1)*log(17*(181*sqrt(17)*(sqrt(-3) + 1 
) + 739*sqrt(-3) + 739)*(-25301/289*sqrt(17) + 6131/17)^(2/3) + 21632*(x + 
 2)^(1/3)) - 1/8*(-25301/289*sqrt(17) + 6131/17)^(1/3)*(sqrt(-3) + 1)*log( 
-17*(181*sqrt(17)*(sqrt(-3) - 1) + 739*sqrt(-3) - 739)*(-25301/289*sqrt(17 
) + 6131/17)^(2/3) + 21632*(x + 2)^(1/3)) + 1/4*(25301/289*sqrt(17) + 6131 
/17)^(1/3)*log(17*(181*sqrt(17) - 739)*(25301/289*sqrt(17) + 6131/17)^(2/3 
) + 10816*(x + 2)^(1/3)) + 1/4*(-25301/289*sqrt(17) + 6131/17)^(1/3)*log(- 
17*(181*sqrt(17) + 739)*(-25301/289*sqrt(17) + 6131/17)^(2/3) + 10816*(x + 
 2)^(1/3)) + 15/4*(x + 2)^(2/3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 11.76 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.76 \[ \int \frac {(2+x)^{2/3} (3+5 x)}{4+7 x+2 x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(15*(x + 2)^(2/3))/4 + (17^(1/3)*log((451737*(x + 2)^(1/3))/512 - (17^(2/3 
)*(104227 - 25301*17^(1/2))^(2/3)*((17^(1/3)*(104227 - 25301*17^(1/2))^(1/ 
3)*((1970487*(x + 2)^(1/3))/64 + (243*17^(2/3)*(104227 - 25301*17^(1/2))^( 
2/3))/512))/68 + 39595635/1024))/4624)*(104227 - 25301*17^(1/2))^(1/3))/68 
 - (17^(1/3)*log((451737*(x + 2)^(1/3))/512 + (17^(2/3)*(25301*17^(1/2) - 
104227)^(2/3)*((17^(1/3)*(25301*17^(1/2) - 104227)^(1/3)*((1970487*(x + 2) 
^(1/3))/64 + (243*17^(2/3)*(25301*17^(1/2) - 104227)^(2/3))/512))/68 - 395 
95635/1024))/4624)*(25301*17^(1/2) - 104227)^(1/3))/68 + (17^(1/3)*log((45 
1737*(x + 2)^(1/3))/512 - (17^(2/3)*(25301*17^(1/2) + 104227)^(2/3)*((17^( 
1/3)*(25301*17^(1/2) + 104227)^(1/3)*((1970487*(x + 2)^(1/3))/64 + (243*17 
^(2/3)*(25301*17^(1/2) + 104227)^(2/3))/512))/68 + 39595635/1024))/4624)*( 
25301*17^(1/2) + 104227)^(1/3))/68 + (17^(1/3)*log((451737*(x + 2)^(1/3))/ 
512 - (17^(2/3)*(3^(1/2)*1i - 1)^2*(25301*17^(1/2) + 104227)^(2/3)*((17^(1 
/3)*(3^(1/2)*1i - 1)*(25301*17^(1/2) + 104227)^(1/3)*((1970487*(x + 2)^(1/ 
3))/64 + (243*17^(2/3)*(3^(1/2)*1i - 1)^2*(25301*17^(1/2) + 104227)^(2/3)) 
/2048))/136 + 39595635/1024))/18496)*(3^(1/2)*1i - 1)*(25301*17^(1/2) + 10 
4227)^(1/3))/136 - (17^(1/3)*log((451737*(x + 2)^(1/3))/512 + (17^(2/3)*(3 
^(1/2)*1i + 1)^2*(104227 - 25301*17^(1/2))^(2/3)*((17^(1/3)*(3^(1/2)*1i + 
1)*(104227 - 25301*17^(1/2))^(1/3)*((1970487*(x + 2)^(1/3))/64 + (243*17^( 
2/3)*(3^(1/2)*1i + 1)^2*(104227 - 25301*17^(1/2))^(2/3))/2048))/136 - 3...
 

Reduce [F]

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

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

Output:

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