\(\int \frac {3+5 x}{(2+x)^{4/3} (4+7 x+2 x^2)} \, dx\) [859]

Optimal result

Integrand size = 25, antiderivative size = 389 \[ \int \frac {3+5 x}{(2+x)^{4/3} \left (4+7 x+2 x^2\right )} \, dx=\frac {3 \left (85-23 \sqrt {17}\right )}{17 \left (1+\sqrt {17}\right ) \sqrt [3]{2+x}}+\frac {3 \left (85+23 \sqrt {17}\right )}{17 \left (1-\sqrt {17}\right ) \sqrt [3]{2+x}}+\frac {\sqrt {\frac {3}{17}} 2^{2/3} \left (23+5 \sqrt {17}\right ) \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{-1+\sqrt {17}}}}{\sqrt {3}}\right )}{\left (-1+\sqrt {17}\right )^{4/3}}-\frac {\sqrt {\frac {3}{17}} 2^{2/3} \left (23-5 \sqrt {17}\right ) \arctan \left (\frac {1+\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{1+\sqrt {17}}}}{\sqrt {3}}\right )}{\left (1+\sqrt {17}\right )^{4/3}}-\frac {\left (85-23 \sqrt {17}\right ) \log \left (7-\sqrt {17}+4 x\right )}{17 \sqrt [3]{2} \left (1+\sqrt {17}\right )^{4/3}}-\frac {\left (85+23 \sqrt {17}\right ) \log \left (7+\sqrt {17}+4 x\right )}{17 \sqrt [3]{2} \left (-1+\sqrt {17}\right )^{4/3}}+\frac {3 \left (85-23 \sqrt {17}\right ) \log \left (\sqrt [3]{1+\sqrt {17}}-2^{2/3} \sqrt [3]{2+x}\right )}{17 \sqrt [3]{2} \left (1+\sqrt {17}\right )^{4/3}}+\frac {3 \left (85+23 \sqrt {17}\right ) \log \left (\sqrt [3]{-1+\sqrt {17}}+2^{2/3} \sqrt [3]{2+x}\right )}{17 \sqrt [3]{2} \left (-1+\sqrt {17}\right )^{4/3}} \] Output:

3/17*(85-23*17^(1/2))/(1+17^(1/2))/(2+x)^(1/3)+3/17*(85+23*17^(1/2))/(1-17 
^(1/2))/(2+x)^(1/3)+1/17*51^(1/2)*2^(2/3)*(23+5*17^(1/2))*arctan(1/3*(1-2* 
2^(2/3)*(2+x)^(1/3)/(-1+17^(1/2))^(1/3))*3^(1/2))/(-1+17^(1/2))^(4/3)-1/17 
*51^(1/2)*2^(2/3)*(23-5*17^(1/2))*arctan(1/3*(1+2*2^(2/3)*(2+x)^(1/3)/(1+1 
7^(1/2))^(1/3))*3^(1/2))/(1+17^(1/2))^(4/3)-1/34*(85-23*17^(1/2))*ln(7-17^ 
(1/2)+4*x)*2^(2/3)/(1+17^(1/2))^(4/3)-1/34*(85+23*17^(1/2))*ln(7+17^(1/2)+ 
4*x)*2^(2/3)/(-1+17^(1/2))^(4/3)+3/34*(85-23*17^(1/2))*ln((1+17^(1/2))^(1/ 
3)-2^(2/3)*(2+x)^(1/3))*2^(2/3)/(1+17^(1/2))^(4/3)+3/34*(85+23*17^(1/2))*l 
n((-1+17^(1/2))^(1/3)+2^(2/3)*(2+x)^(1/3))*2^(2/3)/(-1+17^(1/2))^(4/3)
 

Mathematica [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.21 \[ \int \frac {3+5 x}{(2+x)^{4/3} \left (4+7 x+2 x^2\right )} \, dx=-\frac {21}{2 \sqrt [3]{2+x}}-\frac {1}{2} \text {RootSum}\left [-2-\text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-17 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right )+14 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \] Input:

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

Output:

-21/(2*(2 + x)^(1/3)) - RootSum[-2 - #1^3 + 2*#1^6 & , (-17*Log[(2 + x)^(1 
/3) - #1] + 14*Log[(2 + x)^(1/3) - #1]*#1^3)/(-#1 + 4*#1^4) & ]/2
 

Rubi [A] (warning: unable to verify)

Time = 0.91 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1198, 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.

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

\(\Big \downarrow \) 1198

\(\displaystyle -\frac {1}{2} \int \frac {14 x+11}{\sqrt [3]{x+2} \left (2 x^2+7 x+4\right )}dx-\frac {21}{2 \sqrt [3]{x+2}}\)

\(\Big \downarrow \) 1200

\(\displaystyle -\frac {1}{2} \int \left (\frac {14-\frac {54}{\sqrt {17}}}{\left (4 x-\sqrt {17}+7\right ) \sqrt [3]{x+2}}+\frac {14+\frac {54}{\sqrt {17}}}{\left (4 x+\sqrt {17}+7\right ) \sqrt [3]{x+2}}\right )dx-\frac {21}{2 \sqrt [3]{x+2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {\frac {3}{17}} \sqrt [3]{55817+13537 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{\sqrt {17}-1}}}{\sqrt {3}}\right )}{2^{2/3}}-\frac {\sqrt {\frac {3}{17}} \sqrt [3]{55817-13537 \sqrt {17}} \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{1+\sqrt {17}}}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\sqrt [3]{55817 \sqrt {17}-230129} \log \left (4 x-\sqrt {17}+7\right )}{2\ 34^{2/3}}-\frac {\sqrt [3]{230129+55817 \sqrt {17}} \log \left (4 x+\sqrt {17}+7\right )}{2\ 34^{2/3}}-\frac {3 \sqrt [3]{55817 \sqrt {17}-230129} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{x+2}\right )}{2\ 34^{2/3}}+\frac {3 \sqrt [3]{230129+55817 \sqrt {17}} \log \left (2 \sqrt [3]{x+2}+\sqrt [3]{2 \left (\sqrt {17}-1\right )}\right )}{2\ 34^{2/3}}\right )-\frac {21}{2 \sqrt [3]{x+2}}\)

Input:

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

Output:

-21/(2*(2 + x)^(1/3)) + ((Sqrt[3/17]*(55817 + 13537*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]*(55817 - 13537*Sqrt[17])^(1/3)*ArcTan[(1 + (2*2^(2/3)*(2 + x) 
^(1/3))/(1 + Sqrt[17])^(1/3))/Sqrt[3]])/2^(2/3) + ((-230129 + 55817*Sqrt[1 
7])^(1/3)*Log[7 - Sqrt[17] + 4*x])/(2*34^(2/3)) - ((230129 + 55817*Sqrt[17 
])^(1/3)*Log[7 + Sqrt[17] + 4*x])/(2*34^(2/3)) - (3*(-230129 + 55817*Sqrt[ 
17])^(1/3)*Log[(2*(1 + Sqrt[17]))^(1/3) - 2*(2 + x)^(1/3)])/(2*34^(2/3)) + 
 (3*(230129 + 55817*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 1198

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

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 = 34.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.15

method	result	size
derivativedivides	\(-\frac {21}{2 \left (2+x \right )^{\frac {1}{3}}}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (14 \textit {\_R}^{4}-17 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(60\)
default	\(-\frac {21}{2 \left (2+x \right )^{\frac {1}{3}}}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (14 \textit {\_R}^{4}-17 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(60\)
risch	\(-\frac {21}{2 \left (2+x \right )^{\frac {1}{3}}}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (14 \textit {\_R}^{4}-17 \textit {\_R} \right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\)	\(60\)
trager	\(\text {Expression too large to display}\)	\(8082\)

Input:

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

Output:

-21/2/(2+x)^(1/3)-1/2*sum((14*_R^4-17*_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 = 331, normalized size of antiderivative = 0.85 \[ \int \frac {3+5 x}{(2+x)^{4/3} \left (4+7 x+2 x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*((sqrt(-3)*(x + 2) - x - 2)*(55817/1156*sqrt(17) + 13537/68)^(1/3)*log 
(-17*(485*sqrt(17)*(sqrt(-3) + 1) - 1997*sqrt(-3) - 1997)*(55817/1156*sqrt 
(17) + 13537/68)^(2/3) + 5408*(x + 2)^(1/3)) - (sqrt(-3)*(x + 2) + x + 2)* 
(55817/1156*sqrt(17) + 13537/68)^(1/3)*log(17*(485*sqrt(17)*(sqrt(-3) - 1) 
 - 1997*sqrt(-3) + 1997)*(55817/1156*sqrt(17) + 13537/68)^(2/3) + 5408*(x 
+ 2)^(1/3)) + 2*(x + 2)*(55817/1156*sqrt(17) + 13537/68)^(1/3)*log(17*(485 
*sqrt(17) - 1997)*(55817/1156*sqrt(17) + 13537/68)^(2/3) + 2704*(x + 2)^(1 
/3)) + (sqrt(-3)*(x + 2) - x - 2)*(-55817/1156*sqrt(17) + 13537/68)^(1/3)* 
log(17*(485*sqrt(17)*(sqrt(-3) + 1) + 1997*sqrt(-3) + 1997)*(-55817/1156*s 
qrt(17) + 13537/68)^(2/3) + 5408*(x + 2)^(1/3)) - (sqrt(-3)*(x + 2) + x + 
2)*(-55817/1156*sqrt(17) + 13537/68)^(1/3)*log(-17*(485*sqrt(17)*(sqrt(-3) 
 - 1) + 1997*sqrt(-3) - 1997)*(-55817/1156*sqrt(17) + 13537/68)^(2/3) + 54 
08*(x + 2)^(1/3)) + 2*(x + 2)*(-55817/1156*sqrt(17) + 13537/68)^(1/3)*log( 
-17*(485*sqrt(17) + 1997)*(-55817/1156*sqrt(17) + 13537/68)^(2/3) + 2704*( 
x + 2)^(1/3)) - 42*(x + 2)^(2/3))/(x + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 11.59 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.46 \[ \int \frac {3+5 x}{(2+x)^{4/3} \left (4+7 x+2 x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(34^(1/3)*log((205335*(x + 2)^(1/3))/128 - (34^(2/3)*(230129 - 55817*17^(1 
/2))^(2/3)*((34^(1/3)*(230129 - 55817*17^(1/2))^(1/3)*((21154851*(x + 2)^( 
1/3))/256 + (243*34^(2/3)*(230129 - 55817*17^(1/2))^(2/3))/512))/68 + 2183 
2335/128))/4624)*(230129 - 55817*17^(1/2))^(1/3))/68 - 21/(2*(x + 2)^(1/3) 
) - (34^(1/3)*log((205335*(x + 2)^(1/3))/128 + (34^(2/3)*(55817*17^(1/2) - 
 230129)^(2/3)*((34^(1/3)*(55817*17^(1/2) - 230129)^(1/3)*((21154851*(x + 
2)^(1/3))/256 + (243*34^(2/3)*(55817*17^(1/2) - 230129)^(2/3))/512))/68 - 
21832335/128))/4624)*(55817*17^(1/2) - 230129)^(1/3))/68 + (34^(1/3)*log(( 
205335*(x + 2)^(1/3))/128 - (34^(2/3)*(55817*17^(1/2) + 230129)^(2/3)*((34 
^(1/3)*(55817*17^(1/2) + 230129)^(1/3)*((21154851*(x + 2)^(1/3))/256 + (24 
3*34^(2/3)*(55817*17^(1/2) + 230129)^(2/3))/512))/68 + 21832335/128))/4624 
)*(55817*17^(1/2) + 230129)^(1/3))/68 + (34^(1/3)*log((205335*(x + 2)^(1/3 
))/128 - (34^(2/3)*(3^(1/2)*1i - 1)^2*(55817*17^(1/2) + 230129)^(2/3)*((34 
^(1/3)*(3^(1/2)*1i - 1)*(55817*17^(1/2) + 230129)^(1/3)*((21154851*(x + 2) 
^(1/3))/256 + (243*34^(2/3)*(3^(1/2)*1i - 1)^2*(55817*17^(1/2) + 230129)^( 
2/3))/2048))/136 + 21832335/128))/18496)*(3^(1/2)*1i - 1)*(55817*17^(1/2) 
+ 230129)^(1/3))/136 - (34^(1/3)*log((205335*(x + 2)^(1/3))/128 + (34^(2/3 
)*(3^(1/2)*1i + 1)^2*(230129 - 55817*17^(1/2))^(2/3)*((34^(1/3)*(3^(1/2)*1 
i + 1)*(230129 - 55817*17^(1/2))^(1/3)*((21154851*(x + 2)^(1/3))/256 + (24 
3*34^(2/3)*(3^(1/2)*1i + 1)^2*(230129 - 55817*17^(1/2))^(2/3))/2048))/1...
 

Reduce [F]

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

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

Output:

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