Integrand size = 25, antiderivative size = 391 \[ \int \frac {3+5 x}{(2+x)^{5/3} \left (4+7 x+2 x^2\right )} \, dx=\frac {3 \left (85-23 \sqrt {17}\right )}{34 \left (1+\sqrt {17}\right ) (2+x)^{2/3}}+\frac {3 \left (85+23 \sqrt {17}\right )}{34 \left (1-\sqrt {17}\right ) (2+x)^{2/3}}+\frac {2 \sqrt [3]{2} \sqrt {3} \left (23+5 \sqrt {17}\right ) \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{-1+\sqrt {17}}}}{\sqrt {3}}\right )}{\sqrt {17} \left (-1+\sqrt {17}\right )^{5/3}}+\frac {2 \sqrt [3]{2} \sqrt {3} \left (23-5 \sqrt {17}\right ) \arctan \left (\frac {1+\frac {2\ 2^{2/3} \sqrt [3]{2+x}}{\sqrt [3]{1+\sqrt {17}}}}{\sqrt {3}}\right )}{\sqrt {17} \left (1+\sqrt {17}\right )^{5/3}}-\frac {\sqrt [3]{2} \left (85-23 \sqrt {17}\right ) \log \left (7-\sqrt {17}+4 x\right )}{17 \left (1+\sqrt {17}\right )^{5/3}}+\frac {\sqrt [3]{2} \left (85+23 \sqrt {17}\right ) \log \left (7+\sqrt {17}+4 x\right )}{17 \left (-1+\sqrt {17}\right )^{5/3}}+\frac {3 \sqrt [3]{2} \left (85-23 \sqrt {17}\right ) \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{2+x}\right )}{17 \left (1+\sqrt {17}\right )^{5/3}}-\frac {3 \sqrt [3]{2} \left (85+23 \sqrt {17}\right ) \log \left (\sqrt [3]{2 \left (-1+\sqrt {17}\right )}+2 \sqrt [3]{2+x}\right )}{17 \left (-1+\sqrt {17}\right )^{5/3}} \] Output:
3/34*(85-23*17^(1/2))/(1+17^(1/2))/(2+x)^(2/3)+3/34*(85+23*17^(1/2))/(1-17 ^(1/2))/(2+x)^(2/3)+2/17*2^(1/3)*3^(1/2)*(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))*17^(1/2)/(-1+17^(1/2))^(5 /3)+2/17*2^(1/3)*3^(1/2)*(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))*17^(1/2)/(1+17^(1/2))^(5/3)-1/17*2^(1/3)*( 85-23*17^(1/2))*ln(7-17^(1/2)+4*x)/(1+17^(1/2))^(5/3)+1/17*2^(1/3)*(85+23* 17^(1/2))*ln(7+17^(1/2)+4*x)/(-1+17^(1/2))^(5/3)+3/17*2^(1/3)*(85-23*17^(1 /2))*ln((2+2*17^(1/2))^(1/3)-2*(2+x)^(1/3))/(1+17^(1/2))^(5/3)-3/17*2^(1/3 )*(85+23*17^(1/2))*ln((-2+2*17^(1/2))^(1/3)+2*(2+x)^(1/3))/(-1+17^(1/2))^( 5/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.21 \[ \int \frac {3+5 x}{(2+x)^{5/3} \left (4+7 x+2 x^2\right )} \, dx=-\frac {21}{4 (2+x)^{2/3}}-\frac {1}{2} \text {RootSum}\left [-2-\text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-17 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right )+14 \log \left (\sqrt [3]{2+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(3 + 5*x)/((2 + x)^(5/3)*(4 + 7*x + 2*x^2)),x]
Output:
-21/(4*(2 + x)^(2/3)) - RootSum[-2 - #1^3 + 2*#1^6 & , (-17*Log[(2 + x)^(1 /3) - #1] + 14*Log[(2 + x)^(1/3) - #1]*#1^3)/(-#1^2 + 4*#1^5) & ]/2
Time = 0.92 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.75, 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.
| \(\displaystyle \int \frac {5 x+3}{(x+2)^{5/3} \left (2 x^2+7 x+4\right )} \, dx\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle -\frac {1}{2} \int \frac {14 x+11}{(x+2)^{2/3} \left (2 x^2+7 x+4\right )}dx-\frac {21}{4 (x+2)^{2/3}}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {14-\frac {54}{\sqrt {17}}}{\left (4 x-\sqrt {17}+7\right ) (x+2)^{2/3}}+\frac {14+\frac {54}{\sqrt {17}}}{\left (4 x+\sqrt {17}+7\right ) (x+2)^{2/3}}\right )dx-\frac {21}{4 (x+2)^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{142973+34677 \sqrt {17}} \arctan \left (\frac {1-\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{\sqrt {17}-1}}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {\frac {3}{17}} \sqrt [3]{34677 \sqrt {17}-142973} \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{x+2}}{\sqrt [3]{1+\sqrt {17}}}+1}{\sqrt {3}}\right )+\frac {\sqrt [3]{589509-142973 \sqrt {17}} \log \left (4 x-\sqrt {17}+7\right )}{4\ 17^{2/3}}+\frac {\sqrt [3]{589509+142973 \sqrt {17}} \log \left (4 x+\sqrt {17}+7\right )}{4\ 17^{2/3}}-\frac {3 \sqrt [3]{589509-142973 \sqrt {17}} \log \left (\sqrt [3]{2 \left (1+\sqrt {17}\right )}-2 \sqrt [3]{x+2}\right )}{4\ 17^{2/3}}-\frac {3 \sqrt [3]{589509+142973 \sqrt {17}} \log \left (2 \sqrt [3]{x+2}+\sqrt [3]{2 \left (\sqrt {17}-1\right )}\right )}{4\ 17^{2/3}}\right )-\frac {21}{4 (x+2)^{2/3}}\) |
Input:
Int[(3 + 5*x)/((2 + x)^(5/3)*(4 + 7*x + 2*x^2)),x]
Output:
-21/(4*(2 + x)^(2/3)) + ((Sqrt[3/17]*(142973 + 34677*Sqrt[17])^(1/3)*ArcTa n[(1 - (2*2^(2/3)*(2 + x)^(1/3))/(-1 + Sqrt[17])^(1/3))/Sqrt[3]])/2 + (Sqr t[3/17]*(-142973 + 34677*Sqrt[17])^(1/3)*ArcTan[(1 + (2*2^(2/3)*(2 + x)^(1 /3))/(1 + Sqrt[17])^(1/3))/Sqrt[3]])/2 + ((589509 - 142973*Sqrt[17])^(1/3) *Log[7 - Sqrt[17] + 4*x])/(4*17^(2/3)) + ((589509 + 142973*Sqrt[17])^(1/3) *Log[7 + Sqrt[17] + 4*x])/(4*17^(2/3)) - (3*(589509 - 142973*Sqrt[17])^(1/ 3)*Log[(2*(1 + Sqrt[17]))^(1/3) - 2*(2 + x)^(1/3)])/(4*17^(2/3)) - (3*(589 509 + 142973*Sqrt[17])^(1/3)*Log[(2*(-1 + Sqrt[17]))^(1/3) + 2*(2 + x)^(1/ 3)])/(4*17^(2/3)))/2
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 ]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 72.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.15
| method | result | size |
| derivativedivides | \(-\frac {21}{4 \left (2+x \right )^{\frac {2}{3}}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (-14 \textit {\_R}^{3}+17\right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\) | \(58\) |
| default | \(-\frac {21}{4 \left (2+x \right )^{\frac {2}{3}}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-\textit {\_Z}^{3}-2\right )}{\sum }\frac {\left (-14 \textit {\_R}^{3}+17\right ) \ln \left (\left (2+x \right )^{\frac {1}{3}}-\textit {\_R} \right )}{4 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\) | \(58\) |
| trager | \(\text {Expression too large to display}\) | \(6264\) |
| risch | \(\text {Expression too large to display}\) | \(18975\) |
Input:
int((5*x+3)/(2+x)^(5/3)/(2*x^2+7*x+4),x,method=_RETURNVERBOSE)
Output:
-21/4/(2+x)^(2/3)+1/2*sum((-14*_R^3+17)/(4*_R^5-_R^2)*ln((2+x)^(1/3)-_R),_ R=RootOf(2*_Z^6-_Z^3-2))
Time = 0.08 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.84 \[ \int \frac {3+5 x}{(2+x)^{5/3} \left (4+7 x+2 x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((3+5*x)/(2+x)^(5/3)/(2*x^2+7*x+4),x, algorithm="fricas")
Output:
-1/8*((sqrt(-3)*(x + 2) + x + 2)*(142973/289*sqrt(17) - 34677/17)^(1/3)*lo g(-(73*sqrt(17)*(sqrt(-3) + 1) + 289*sqrt(-3) + 289)*(142973/289*sqrt(17) - 34677/17)^(1/3) + 416*(x + 2)^(1/3)) - (sqrt(-3)*(x + 2) - x - 2)*(14297 3/289*sqrt(17) - 34677/17)^(1/3)*log((73*sqrt(17)*(sqrt(-3) - 1) + 289*sqr t(-3) - 289)*(142973/289*sqrt(17) - 34677/17)^(1/3) + 416*(x + 2)^(1/3)) - 2*(x + 2)*(142973/289*sqrt(17) - 34677/17)^(1/3)*log((142973/289*sqrt(17) - 34677/17)^(1/3)*(73*sqrt(17) + 289) + 208*(x + 2)^(1/3)) + (sqrt(-3)*(x + 2) + x + 2)*(-142973/289*sqrt(17) - 34677/17)^(1/3)*log((73*sqrt(17)*(s qrt(-3) + 1) - 289*sqrt(-3) - 289)*(-142973/289*sqrt(17) - 34677/17)^(1/3) + 416*(x + 2)^(1/3)) - (sqrt(-3)*(x + 2) - x - 2)*(-142973/289*sqrt(17) - 34677/17)^(1/3)*log(-(73*sqrt(17)*(sqrt(-3) - 1) - 289*sqrt(-3) + 289)*(- 142973/289*sqrt(17) - 34677/17)^(1/3) + 416*(x + 2)^(1/3)) - 2*(x + 2)*(-1 42973/289*sqrt(17) - 34677/17)^(1/3)*log(-(73*sqrt(17) - 289)*(-142973/289 *sqrt(17) - 34677/17)^(1/3) + 208*(x + 2)^(1/3)) + 42*(x + 2)^(1/3))/(x + 2)
\[ \int \frac {3+5 x}{(2+x)^{5/3} \left (4+7 x+2 x^2\right )} \, dx=\int \frac {5 x + 3}{\left (x + 2\right )^{\frac {5}{3}} \cdot \left (2 x^{2} + 7 x + 4\right )}\, dx \] Input:
integrate((3+5*x)/(2+x)**(5/3)/(2*x**2+7*x+4),x)
Output:
Integral((5*x + 3)/((x + 2)**(5/3)*(2*x**2 + 7*x + 4)), x)
\[ \int \frac {3+5 x}{(2+x)^{5/3} \left (4+7 x+2 x^2\right )} \, dx=\int { \frac {5 \, x + 3}{{\left (2 \, x^{2} + 7 \, x + 4\right )} {\left (x + 2\right )}^{\frac {5}{3}}} \,d x } \] Input:
integrate((3+5*x)/(2+x)^(5/3)/(2*x^2+7*x+4),x, algorithm="maxima")
Output:
integrate((5*x + 3)/((2*x^2 + 7*x + 4)*(x + 2)^(5/3)), x)
\[ \int \frac {3+5 x}{(2+x)^{5/3} \left (4+7 x+2 x^2\right )} \, dx=\int { \frac {5 \, x + 3}{{\left (2 \, x^{2} + 7 \, x + 4\right )} {\left (x + 2\right )}^{\frac {5}{3}}} \,d x } \] Input:
integrate((3+5*x)/(2+x)^(5/3)/(2*x^2+7*x+4),x, algorithm="giac")
Output:
integrate((5*x + 3)/((2*x^2 + 7*x + 4)*(x + 2)^(5/3)), x)
Time = 11.75 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.35 \[ \int \frac {3+5 x}{(2+x)^{5/3} \left (4+7 x+2 x^2\right )} \, dx=\text {Too large to display} \] Input:
int((5*x + 3)/((x + 2)^(5/3)*(7*x + 2*x^2 + 4)),x)
Output:
(17^(1/3)*log((2467179*(x + 2)^(1/3))/128 + (17^(1/3)*(- 142973*17^(1/2) - 589509)^(1/3)*((17^(2/3)*(- 142973*17^(1/2) - 589509)^(2/3)*((3160215*(x + 2)^(1/3))/128 - (4131*17^(1/3)*(- 142973*17^(1/2) - 589509)^(1/3))/128)) /4624 + 55921347/256))/68)*(- 142973*17^(1/2) - 589509)^(1/3))/68 - 21/(4* (x + 2)^(2/3)) - (17^(1/3)*log((2467179*(x + 2)^(1/3))/128 - (17^(1/3)*(58 9509 - 142973*17^(1/2))^(1/3)*((17^(2/3)*(589509 - 142973*17^(1/2))^(2/3)* ((3160215*(x + 2)^(1/3))/128 + (4131*17^(1/3)*(589509 - 142973*17^(1/2))^( 1/3))/128))/4624 + 55921347/256))/68)*(589509 - 142973*17^(1/2))^(1/3))/68 + (17^(1/3)*log((2467179*(x + 2)^(1/3))/128 + (17^(1/3)*(142973*17^(1/2) - 589509)^(1/3)*((17^(2/3)*(142973*17^(1/2) - 589509)^(2/3)*((3160215*(x + 2)^(1/3))/128 - (4131*17^(1/3)*(142973*17^(1/2) - 589509)^(1/3))/128))/46 24 + 55921347/256))/68)*(142973*17^(1/2) - 589509)^(1/3))/68 - (17^(1/3)*l og((2467179*(x + 2)^(1/3))/128 - (17^(1/3)*(142973*17^(1/2) + 589509)^(1/3 )*((17^(2/3)*(142973*17^(1/2) + 589509)^(2/3)*((3160215*(x + 2)^(1/3))/128 + (4131*17^(1/3)*(142973*17^(1/2) + 589509)^(1/3))/128))/4624 + 55921347/ 256))/68)*(142973*17^(1/2) + 589509)^(1/3))/68 - (17^(1/3)*log((2467179*(x + 2)^(1/3))/128 - (17^(1/3)*(3^(1/2)*1i + 1)*(- 142973*17^(1/2) - 589509) ^(1/3)*((17^(2/3)*(3^(1/2)*1i + 1)^2*(- 142973*17^(1/2) - 589509)^(2/3)*(( 3160215*(x + 2)^(1/3))/128 + (4131*17^(1/3)*(3^(1/2)*1i + 1)*(- 142973*17^ (1/2) - 589509)^(1/3))/256))/18496 + 55921347/256))/136)*(3^(1/2)*1i + ...
\[ \int \frac {3+5 x}{(2+x)^{5/3} \left (4+7 x+2 x^2\right )} \, dx=\int \frac {5 x +3}{\left (x +2\right )^{\frac {5}{3}} \left (2 x^{2}+7 x +4\right )}d x \] Input:
int((3+5*x)/(2+x)^(5/3)/(2*x^2+7*x+4),x)
Output:
int((3+5*x)/(2+x)^(5/3)/(2*x^2+7*x+4),x)