\(\int \frac {\sqrt [3]{2+x} (3+5 x)}{4+7 x+2 x^2} \, dx\) [856]

Optimal result

Integrand size = 25, antiderivative size = 317 \[ \int \frac {\sqrt [3]{2+x} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {3}{68} \left (85-23 \sqrt {17}\right ) \sqrt [3]{2+x}+\frac {3}{68} \left (85+23 \sqrt {17}\right ) \sqrt [3]{2+x}+\frac {1}{8} \sqrt {\frac {3}{17}} \sqrt [3]{259056+62864 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{-1+\sqrt {17}}}}{\sqrt {3}}\right )+\frac {1}{8} \sqrt {\frac {3}{17}} \sqrt [3]{-259056+62864 \sqrt {17}} \arctan \left (\frac {1+\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{1+\sqrt {17}}}}{\sqrt {3}}\right )+\frac {\sqrt [3]{-259056+62864 \sqrt {17}} \log \left (7-\sqrt {17}+4 x\right )}{16 \sqrt {17}}+\frac {\sqrt [3]{259056+62864 \sqrt {17}} \log \left (7+\sqrt {17}+4 x\right )}{16 \sqrt {17}}-\frac {3 \sqrt [3]{-259056+62864 \sqrt {17}} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{2+x}\right )}{16 \sqrt {17}}-\frac {3 \sqrt [3]{259056+62864 \sqrt {17}} \log \left (\sqrt [3]{2 \left (-1+\sqrt {17}\right )}+2 \sqrt [3]{2+x}\right )}{16 \sqrt {17}} \] Output:

3/68*(85-23*17^(1/2))*(2+x)^(1/3)+3/68*(85+23*17^(1/2))*(2+x)^(1/3)+1/136* 
51^(1/2)*(259056+62864*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)*(-259056+62864*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* 
(-259056+62864*17^(1/2))^(1/3)*ln(7-17^(1/2)+4*x)*17^(1/2)+1/272*(259056+6 
2864*17^(1/2))^(1/3)*ln(7+17^(1/2)+4*x)*17^(1/2)-3/272*(-259056+62864*17^( 
1/2))^(1/3)*ln((2+2*17^(1/2))^(1/3)-2*(2+x)^(1/3))*17^(1/2)-3/272*(259056+ 
62864*17^(1/2))^(1/3)*ln((-2+2*17^(1/2))^(1/3)+2*(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 = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt [3]{2+x} (3+5 x)}{4+7 x+2 x^2} \, dx=\frac {15 \sqrt [3]{2+x}}{2}-\frac {1}{2} \text {RootSum}\left [-2-\text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-10 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right )+9 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \] Input:

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

Output:

(15*(2 + x)^(1/3))/2 - RootSum[-2 - #1^3 + 2*#1^6 & , (-10*Log[(2 + x)^(1/ 
3) - #1] + 9*Log[(2 + x)^(1/3) - #1]*#1^3)/(-#1^2 + 4*#1^5) & ]/2
 

Rubi [A] (warning: unable to verify)

Time = 0.96 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.94, 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.

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

\(\Big \downarrow \) 1196

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1200

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {\frac {3}{17}} \sqrt [3]{16191+3929 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{\sqrt {17}-1}}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\sqrt {\frac {3}{17}} \sqrt [3]{3929 \sqrt {17}-16191} \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{1+\sqrt {17}}}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\sqrt [3]{66793-16191 \sqrt {17}} \log \left (4 x-\sqrt {17}+7\right )}{2\ 34^{2/3}}+\frac {\sqrt [3]{66793+16191 \sqrt {17}} \log \left (4 x+\sqrt {17}+7\right )}{2\ 34^{2/3}}-\frac {3 \sqrt [3]{66793-16191 \sqrt {17}} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{x+2}\right )}{2\ 34^{2/3}}-\frac {3 \sqrt [3]{66793+16191 \sqrt {17}} \log \left (2 \sqrt [3]{x+2}+\sqrt [3]{2 \left (\sqrt {17}-1\right )}\right )}{2\ 34^{2/3}}\right )+\frac {15 \sqrt [3]{x+2}}{2}\)

Input:

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

Output:

(15*(2 + x)^(1/3))/2 + ((Sqrt[3/17]*(16191 + 3929*Sqrt[17])^(1/3)*ArcTan[( 
1 - (2*2^(2/3)*(2 + x)^(1/3))/(-1 + Sqrt[17])^(1/3))/Sqrt[3]])/2^(2/3) + ( 
Sqrt[3/17]*(-16191 + 3929*Sqrt[17])^(1/3)*ArcTan[(1 + (2*2^(2/3)*(2 + x)^( 
1/3))/(1 + Sqrt[17])^(1/3))/Sqrt[3]])/2^(2/3) + ((66793 - 16191*Sqrt[17])^ 
(1/3)*Log[7 - Sqrt[17] + 4*x])/(2*34^(2/3)) + ((66793 + 16191*Sqrt[17])^(1 
/3)*Log[7 + Sqrt[17] + 4*x])/(2*34^(2/3)) - (3*(66793 - 16191*Sqrt[17])^(1 
/3)*Log[(2*(1 + Sqrt[17]))^(1/3) - 2*(2 + x)^(1/3)])/(2*34^(2/3)) - (3*(66 
793 + 16191*Sqrt[17])^(1/3)*Log[(2*(-1 + Sqrt[17]))^(1/3) + 2*(2 + x)^(1/3 
)])/(2*34^(2/3)))/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 = 64.81 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.18

method	result	size
derivativedivides	\(\frac {15 \left (2+x \right )^{\frac {1}{3}}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (-9 \textit {\_R}^{3}+10\right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(58\)
default	\(\frac {15 \left (2+x \right )^{\frac {1}{3}}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (-9 \textit {\_R}^{3}+10\right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(58\)
trager	\(\text {Expression too large to display}\)	\(7988\)
risch	\(\text {Expression too large to display}\)	\(16743\)

Input:

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

Output:

15/2*(2+x)^(1/3)+1/2*sum((-9*_R^3+10)/(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 = 293, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt [3]{2+x} (3+5 x)}{4+7 x+2 x^2} \, dx=-\frac {1}{4} \, {\left (\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (-{\left (23 \, \sqrt {17} {\left (\sqrt {-3} + 1\right )} + 85 \, \sqrt {-3} + 85\right )} {\left (\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} + 104 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left ({\left (23 \, \sqrt {17} {\left (\sqrt {-3} - 1\right )} + 85 \, \sqrt {-3} - 85\right )} {\left (\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} + 104 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) - \frac {1}{4} \, {\left (-\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left ({\left (23 \, \sqrt {17} {\left (\sqrt {-3} + 1\right )} - 85 \, \sqrt {-3} - 85\right )} {\left (-\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} + 104 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (-\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (-{\left (23 \, \sqrt {17} {\left (\sqrt {-3} - 1\right )} - 85 \, \sqrt {-3} + 85\right )} {\left (-\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} + 104 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\left (\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} \log \left ({\left (23 \, \sqrt {17} + 85\right )} {\left (\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} + 52 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\left (-\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} \log \left (-{\left (23 \, \sqrt {17} - 85\right )} {\left (-\frac {16191}{1156} \, \sqrt {17} - \frac {3929}{68}\right )}^{\frac {1}{3}} + 52 \, {\left (x + 2\right )}^{\frac {1}{3}}\right ) + \frac {15}{2} \, {\left (x + 2\right )}^{\frac {1}{3}} \] Input:

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

Output:

-1/4*(16191/1156*sqrt(17) - 3929/68)^(1/3)*(sqrt(-3) + 1)*log(-(23*sqrt(17 
)*(sqrt(-3) + 1) + 85*sqrt(-3) + 85)*(16191/1156*sqrt(17) - 3929/68)^(1/3) 
 + 104*(x + 2)^(1/3)) + 1/4*(16191/1156*sqrt(17) - 3929/68)^(1/3)*(sqrt(-3 
) - 1)*log((23*sqrt(17)*(sqrt(-3) - 1) + 85*sqrt(-3) - 85)*(16191/1156*sqr 
t(17) - 3929/68)^(1/3) + 104*(x + 2)^(1/3)) - 1/4*(-16191/1156*sqrt(17) - 
3929/68)^(1/3)*(sqrt(-3) + 1)*log((23*sqrt(17)*(sqrt(-3) + 1) - 85*sqrt(-3 
) - 85)*(-16191/1156*sqrt(17) - 3929/68)^(1/3) + 104*(x + 2)^(1/3)) + 1/4* 
(-16191/1156*sqrt(17) - 3929/68)^(1/3)*(sqrt(-3) - 1)*log(-(23*sqrt(17)*(s 
qrt(-3) - 1) - 85*sqrt(-3) + 85)*(-16191/1156*sqrt(17) - 3929/68)^(1/3) + 
104*(x + 2)^(1/3)) + 1/2*(16191/1156*sqrt(17) - 3929/68)^(1/3)*log((23*sqr 
t(17) + 85)*(16191/1156*sqrt(17) - 3929/68)^(1/3) + 52*(x + 2)^(1/3)) + 1/ 
2*(-16191/1156*sqrt(17) - 3929/68)^(1/3)*log(-(23*sqrt(17) - 85)*(-16191/1 
156*sqrt(17) - 3929/68)^(1/3) + 52*(x + 2)^(1/3)) + 15/2*(x + 2)^(1/3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 11.61 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.66 \[ \int \frac {\sqrt [3]{2+x} (3+5 x)}{4+7 x+2 x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*(x + 2)^(1/3))/2 + (34^(1/3)*log((3692871*(x + 2)^(1/3))/512 + (34^(1/ 
3)*(- 16191*17^(1/2) - 66793)^(1/3)*((34^(2/3)*(- 16191*17^(1/2) - 66793)^ 
(2/3)*((491589*(x + 2)^(1/3))/32 - (4131*34^(1/3)*(- 16191*17^(1/2) - 6679 
3)^(1/3))/128))/4624 + 25327161/512))/68)*(- 16191*17^(1/2) - 66793)^(1/3) 
)/68 - (34^(1/3)*log((3692871*(x + 2)^(1/3))/512 - (34^(1/3)*(66793 - 1619 
1*17^(1/2))^(1/3)*((34^(2/3)*(66793 - 16191*17^(1/2))^(2/3)*((491589*(x + 
2)^(1/3))/32 + (4131*34^(1/3)*(66793 - 16191*17^(1/2))^(1/3))/128))/4624 + 
 25327161/512))/68)*(66793 - 16191*17^(1/2))^(1/3))/68 + (34^(1/3)*log((36 
92871*(x + 2)^(1/3))/512 + (34^(1/3)*(16191*17^(1/2) - 66793)^(1/3)*((34^( 
2/3)*(16191*17^(1/2) - 66793)^(2/3)*((491589*(x + 2)^(1/3))/32 - (4131*34^ 
(1/3)*(16191*17^(1/2) - 66793)^(1/3))/128))/4624 + 25327161/512))/68)*(161 
91*17^(1/2) - 66793)^(1/3))/68 - (34^(1/3)*log((3692871*(x + 2)^(1/3))/512 
 - (34^(1/3)*(16191*17^(1/2) + 66793)^(1/3)*((34^(2/3)*(16191*17^(1/2) + 6 
6793)^(2/3)*((491589*(x + 2)^(1/3))/32 + (4131*34^(1/3)*(16191*17^(1/2) + 
66793)^(1/3))/128))/4624 + 25327161/512))/68)*(16191*17^(1/2) + 66793)^(1/ 
3))/68 - (34^(1/3)*log((3692871*(x + 2)^(1/3))/512 - (34^(1/3)*(3^(1/2)*1i 
 + 1)*(- 16191*17^(1/2) - 66793)^(1/3)*((34^(2/3)*(3^(1/2)*1i + 1)^2*(- 16 
191*17^(1/2) - 66793)^(2/3)*((491589*(x + 2)^(1/3))/32 + (4131*34^(1/3)*(3 
^(1/2)*1i + 1)*(- 16191*17^(1/2) - 66793)^(1/3))/256))/18496 + 25327161/51 
2))/136)*(3^(1/2)*1i + 1)*(- 16191*17^(1/2) - 66793)^(1/3))/136 - (34^(...
 

Reduce [F]

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

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

Output:

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