\(\int \frac {3+5 x}{\sqrt [3]{2+x} (4+7 x+2 x^2)} \, dx\) [857]

Optimal result

Integrand size = 25, antiderivative size = 345 \[ \int \frac {3+5 x}{\sqrt [3]{2+x} \left (4+7 x+2 x^2\right )} \, dx=-\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{6641+1611 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{-1+\sqrt {17}}}}{\sqrt {3}}\right )-\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{-6641+1611 \sqrt {17}} \arctan \left (\frac {1+\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{1+\sqrt {17}}}}{\sqrt {3}}\right )-\frac {\sqrt [3]{27387-6641 \sqrt {17}} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{2+x}\right )}{2\ 17^{2/3}}-\frac {\sqrt [3]{27387+6641 \sqrt {17}} \log \left (\sqrt [3]{2 \left (-1+\sqrt {17}\right )}+2 \sqrt [3]{2+x}\right )}{2\ 17^{2/3}}+\frac {\sqrt [3]{27387+6641 \sqrt {17}} \log \left (\left (2 \left (-1+\sqrt {17}\right )\right )^{2/3}-2 \sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [3]{2+x}+4 (2+x)^{2/3}\right )}{4\ 17^{2/3}}+\frac {\sqrt [3]{27387-6641 \sqrt {17}} \log \left (\left (2 \left (1+\sqrt {17}\right )\right )^{2/3}+2 \sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [3]{2+x}+4 (2+x)^{2/3}\right )}{4\ 17^{2/3}} \] Output:

-1/34*51^(1/2)*(6641+1611*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)*(-6641+1611*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*(2 
7387-6641*17^(1/2))^(1/3)*ln((2+2*17^(1/2))^(1/3)-2*(2+x)^(1/3))*17^(1/3)- 
1/34*(27387+6641*17^(1/2))^(1/3)*ln((-2+2*17^(1/2))^(1/3)+2*(2+x)^(1/3))*1 
7^(1/3)+1/68*(27387+6641*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*(27387-6641*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)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.19 \[ \int \frac {3+5 x}{\sqrt [3]{2+x} \left (4+7 x+2 x^2\right )} \, dx=\text {RootSum}\left [-2-\text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-7 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right )+5 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \] Input:

Integrate[(3 + 5*x)/((2 + x)^(1/3)*(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 + 4*#1^4) & ]
 

Rubi [A] (warning: unable to verify)

Time = 0.86 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.77, 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.

\(\displaystyle \int \frac {5 x+3}{\sqrt [3]{x+2} \left (2 x^2+7 x+4\right )} \, dx\)

\(\Big \downarrow \) 1200

\(\displaystyle \int \left (\frac {5-\frac {23}{\sqrt {17}}}{\left (4 x-\sqrt {17}+7\right ) \sqrt [3]{x+2}}+\frac {5+\frac {23}{\sqrt {17}}}{\left (4 x+\sqrt {17}+7\right ) \sqrt [3]{x+2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{6641+1611 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{\sqrt {17}-1}}}{\sqrt {3}}\right )-\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{1611 \sqrt {17}-6641} \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{1+\sqrt {17}}}+1}{\sqrt {3}}\right )+\frac {1}{68} \sqrt [3]{465579-112897 \sqrt {17}} \log \left (4 x-\sqrt {17}+7\right )+\frac {\sqrt [3]{27387+6641 \sqrt {17}} \log \left (4 x+\sqrt {17}+7\right )}{4\ 17^{2/3}}-\frac {3}{68} \sqrt [3]{465579-112897 \sqrt {17}} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{x+2}\right )-\frac {3 \sqrt [3]{27387+6641 \sqrt {17}} \log \left (2 \sqrt [3]{x+2}+\sqrt [3]{2 \left (\sqrt {17}-1\right )}\right )}{4\ 17^{2/3}}\)

Input:

Int[(3 + 5*x)/((2 + x)^(1/3)*(4 + 7*x + 2*x^2)),x]
 

Output:

-1/2*(Sqrt[3/17]*(6641 + 1611*Sqrt[17])^(1/3)*ArcTan[(1 - (2*2^(2/3)*(2 + 
x)^(1/3))/(-1 + Sqrt[17])^(1/3))/Sqrt[3]]) - (Sqrt[3/17]*(-6641 + 1611*Sqr 
t[17])^(1/3)*ArcTan[(1 + (2*2^(2/3)*(2 + x)^(1/3))/(1 + Sqrt[17])^(1/3))/S 
qrt[3]])/2 + ((465579 - 112897*Sqrt[17])^(1/3)*Log[7 - Sqrt[17] + 4*x])/68 
 + ((27387 + 6641*Sqrt[17])^(1/3)*Log[7 + Sqrt[17] + 4*x])/(4*17^(2/3)) - 
(3*(465579 - 112897*Sqrt[17])^(1/3)*Log[(2*(1 + Sqrt[17]))^(1/3) - 2*(2 + 
x)^(1/3)])/68 - (3*(27387 + 6641*Sqrt[17])^(1/3)*Log[(2*(-1 + Sqrt[17]))^( 
1/3) + 2*(2 + x)^(1/3)])/(4*17^(2/3))
 

Defintions of rubi rules used

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 = 39.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.14

method	result	size
derivativedivides	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (5 \textit {\_R}^{4}-7 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\)	\(50\)
default	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (5 \textit {\_R}^{4}-7 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\)	\(50\)
trager	\(\text {Expression too large to display}\)	\(9814\)

Input:

int((5*x+3)/(2+x)^(1/3)/(2*x^2+7*x+4),x,method=_RETURNVERBOSE)
 

Output:

sum((5*_R^4-7*_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 = 0.09 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.84 \[ \int \frac {3+5 x}{\sqrt [3]{2+x} \left (4+7 x+2 x^2\right )} \, dx=\frac {1}{4} \, {\left (\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (17 \, {\left (37 \, \sqrt {17} {\left (\sqrt {-3} + 1\right )} + 152 \, \sqrt {-3} + 152\right )} {\left (\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {2}{3}} + 676 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) - \frac {1}{4} \, {\left (\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (-17 \, {\left (37 \, \sqrt {17} {\left (\sqrt {-3} - 1\right )} + 152 \, \sqrt {-3} - 152\right )} {\left (\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {2}{3}} + 676 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (-\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (-17 \, {\left (37 \, \sqrt {17} {\left (\sqrt {-3} + 1\right )} - 152 \, \sqrt {-3} - 152\right )} {\left (-\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {2}{3}} + 676 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) - \frac {1}{4} \, {\left (-\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (17 \, {\left (37 \, \sqrt {17} {\left (\sqrt {-3} - 1\right )} - 152 \, \sqrt {-3} + 152\right )} {\left (-\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {2}{3}} + 676 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\left (\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {1}{3}} \log \left (-17 \, {\left (37 \, \sqrt {17} + 152\right )} {\left (\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {2}{3}} + 338 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\left (-\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {1}{3}} \log \left (17 \, {\left (37 \, \sqrt {17} - 152\right )} {\left (-\frac {6641}{289} \, \sqrt {17} - \frac {1611}{17}\right )}^{\frac {2}{3}} + 338 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) \] Input:

integrate((3+5*x)/(2+x)^(1/3)/(2*x^2+7*x+4),x, algorithm="fricas")
 

Output:

1/4*(6641/289*sqrt(17) - 1611/17)^(1/3)*(sqrt(-3) - 1)*log(17*(37*sqrt(17) 
*(sqrt(-3) + 1) + 152*sqrt(-3) + 152)*(6641/289*sqrt(17) - 1611/17)^(2/3) 
+ 676*(x + 2)^(1/3)) - 1/4*(6641/289*sqrt(17) - 1611/17)^(1/3)*(sqrt(-3) + 
 1)*log(-17*(37*sqrt(17)*(sqrt(-3) - 1) + 152*sqrt(-3) - 152)*(6641/289*sq 
rt(17) - 1611/17)^(2/3) + 676*(x + 2)^(1/3)) + 1/4*(-6641/289*sqrt(17) - 1 
611/17)^(1/3)*(sqrt(-3) - 1)*log(-17*(37*sqrt(17)*(sqrt(-3) + 1) - 152*sqr 
t(-3) - 152)*(-6641/289*sqrt(17) - 1611/17)^(2/3) + 676*(x + 2)^(1/3)) - 1 
/4*(-6641/289*sqrt(17) - 1611/17)^(1/3)*(sqrt(-3) + 1)*log(17*(37*sqrt(17) 
*(sqrt(-3) - 1) - 152*sqrt(-3) + 152)*(-6641/289*sqrt(17) - 1611/17)^(2/3) 
 + 676*(x + 2)^(1/3)) + 1/2*(6641/289*sqrt(17) - 1611/17)^(1/3)*log(-17*(3 
7*sqrt(17) + 152)*(6641/289*sqrt(17) - 1611/17)^(2/3) + 338*(x + 2)^(1/3)) 
 + 1/2*(-6641/289*sqrt(17) - 1611/17)^(1/3)*log(17*(37*sqrt(17) - 152)*(-6 
641/289*sqrt(17) - 1611/17)^(2/3) + 338*(x + 2)^(1/3))
 

Sympy [F]

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

integrate((3+5*x)/(2+x)**(1/3)/(2*x**2+7*x+4),x)
 

Output:

Integral((5*x + 3)/((x + 2)**(1/3)*(2*x**2 + 7*x + 4)), x)
 

Maxima [F]

\[ \int \frac {3+5 x}{\sqrt [3]{2+x} \left (4+7 x+2 x^2\right )} \, dx=\int { \frac {5 \, x + 3}{{\left (2 \, x^{2} + 7 \, x + 4\right )} {\left (x + 2\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((3+5*x)/(2+x)^(1/3)/(2*x^2+7*x+4),x, algorithm="maxima")
 

Output:

integrate((5*x + 3)/((2*x^2 + 7*x + 4)*(x + 2)^(1/3)), x)
 

Giac [F]

\[ \int \frac {3+5 x}{\sqrt [3]{2+x} \left (4+7 x+2 x^2\right )} \, dx=\int { \frac {5 \, x + 3}{{\left (2 \, x^{2} + 7 \, x + 4\right )} {\left (x + 2\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((3+5*x)/(2+x)^(1/3)/(2*x^2+7*x+4),x, algorithm="giac")
 

Output:

integrate((5*x + 3)/((2*x^2 + 7*x + 4)*(x + 2)^(1/3)), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 11.71 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.52 \[ \int \frac {3+5 x}{\sqrt [3]{2+x} \left (4+7 x+2 x^2\right )} \, dx =\text {Too large to display} \] Input:

int((5*x + 3)/((x + 2)^(1/3)*(7*x + 2*x^2 + 4)),x)
 

Output:

(17^(1/3)*log(- (369603*(x + 2)^(1/3))/256 - (17^(2/3)*(- 6641*17^(1/2) - 
27387)^(2/3)*((17^(1/3)*(- 6641*17^(1/2) - 27387)^(1/3)*((3226311*(x + 2)^ 
(1/3))/64 + (243*17^(2/3)*(- 6641*17^(1/2) - 27387)^(2/3))/128))/34 - 1038 
9465/128))/1156)*(- 6641*17^(1/2) - 27387)^(1/3))/34 - (17^(1/3)*log((17^( 
2/3)*(27387 - 6641*17^(1/2))^(2/3)*((17^(1/3)*(27387 - 6641*17^(1/2))^(1/3 
)*((3226311*(x + 2)^(1/3))/64 + (243*17^(2/3)*(27387 - 6641*17^(1/2))^(2/3 
))/128))/34 + 10389465/128))/1156 - (369603*(x + 2)^(1/3))/256)*(27387 - 6 
641*17^(1/2))^(1/3))/34 + (17^(1/3)*log(- (369603*(x + 2)^(1/3))/256 - (17 
^(2/3)*(6641*17^(1/2) - 27387)^(2/3)*((17^(1/3)*(6641*17^(1/2) - 27387)^(1 
/3)*((3226311*(x + 2)^(1/3))/64 + (243*17^(2/3)*(6641*17^(1/2) - 27387)^(2 
/3))/128))/34 - 10389465/128))/1156)*(6641*17^(1/2) - 27387)^(1/3))/34 - ( 
17^(1/3)*log((17^(2/3)*(6641*17^(1/2) + 27387)^(2/3)*((17^(1/3)*(6641*17^( 
1/2) + 27387)^(1/3)*((3226311*(x + 2)^(1/3))/64 + (243*17^(2/3)*(6641*17^( 
1/2) + 27387)^(2/3))/128))/34 + 10389465/128))/1156 - (369603*(x + 2)^(1/3 
))/256)*(6641*17^(1/2) + 27387)^(1/3))/34 - (17^(1/3)*log((17^(2/3)*(3^(1/ 
2)*1i + 1)^2*(- 6641*17^(1/2) - 27387)^(2/3)*((17^(1/3)*(3^(1/2)*1i + 1)*( 
- 6641*17^(1/2) - 27387)^(1/3)*((3226311*(x + 2)^(1/3))/64 + (243*17^(2/3) 
*(3^(1/2)*1i + 1)^2*(- 6641*17^(1/2) - 27387)^(2/3))/512))/68 + 10389465/1 
28))/4624 - (369603*(x + 2)^(1/3))/256)*(3^(1/2)*1i + 1)*(- 6641*17^(1/2) 
- 27387)^(1/3))/68 - (17^(1/3)*log((17^(2/3)*(3^(1/2)*1i + 1)^2*(6641*1...
 

Reduce [F]

\[ \int \frac {3+5 x}{\sqrt [3]{2+x} \left (4+7 x+2 x^2\right )} \, dx=\int \frac {5 x +3}{\left (x +2\right )^{\frac {1}{3}} \left (2 x^{2}+7 x +4\right )}d x \] Input:

int((3+5*x)/(2+x)^(1/3)/(2*x^2+7*x+4),x)
 

Output:

int((3+5*x)/(2+x)^(1/3)/(2*x^2+7*x+4),x)