Integrand size = 38, antiderivative size = 308 \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {5 \sqrt {3} \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}}{12 \sqrt [3]{10}-2 \sqrt [3]{10} x-2 \sqrt [3]{10} x^2+5 \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}}\right )}{5 \sqrt [3]{10}}+\frac {\log \left (-6 \sqrt [3]{10}+\sqrt [3]{10} x+\sqrt [3]{10} x^2+5 \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (36\ 10^{2/3}-12\ 10^{2/3} x-11\ 10^{2/3} x^2+2\ 10^{2/3} x^3+10^{2/3} x^4+\left (30 \sqrt [3]{10}-5 \sqrt [3]{10} x-5 \sqrt [3]{10} x^2\right ) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}+25 \left (27+27 x+36 x^2+28 x^3+9 x^4+x^5\right )^{2/3}\right )}{10 \sqrt [3]{10}} \] Output:
1/50*3^(1/2)*arctan(5*3^(1/2)*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)/(12* 10^(1/3)-2*10^(1/3)*x-2*10^(1/3)*x^2+5*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^( 1/3)))*10^(2/3)+1/50*ln(-6*10^(1/3)+10^(1/3)*x+10^(1/3)*x^2+5*(x^5+9*x^4+2 8*x^3+36*x^2+27*x+27)^(1/3))*10^(2/3)-1/100*ln(36*10^(2/3)-12*10^(2/3)*x-1 1*10^(2/3)*x^2+2*10^(2/3)*x^3+10^(2/3)*x^4+(30*10^(1/3)-5*10^(1/3)*x-5*10^ (1/3)*x^2)*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3)+25*(x^5+9*x^4+28*x^3+36 *x^2+27*x+27)^(2/3))*10^(2/3)
Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.57 \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=-\frac {(3+x) \sqrt [3]{1+x^2} \left (2 \sqrt {3} \arctan \left (\frac {4 \sqrt [3]{10}-2 \sqrt [3]{10} x+5 \sqrt [3]{1+x^2}}{5 \sqrt {3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 \sqrt [3]{10}+\sqrt [3]{10} x+5 \sqrt [3]{1+x^2}\right )+\log \left (4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2-5 \sqrt [3]{10} (-2+x) \sqrt [3]{1+x^2}+25 \left (1+x^2\right )^{2/3}\right )\right )}{10 \sqrt [3]{10} \sqrt [3]{(3+x)^3 \left (1+x^2\right )}} \] Input:
Integrate[(1 + x)/((1 + 2*x)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^( 1/3)),x]
Output:
-1/10*((3 + x)*(1 + x^2)^(1/3)*(2*Sqrt[3]*ArcTan[(4*10^(1/3) - 2*10^(1/3)* x + 5*(1 + x^2)^(1/3))/(5*Sqrt[3]*(1 + x^2)^(1/3))] - 2*Log[-2*10^(1/3) + 10^(1/3)*x + 5*(1 + x^2)^(1/3)] + Log[4*10^(2/3) - 4*10^(2/3)*x + 10^(2/3) *x^2 - 5*10^(1/3)*(-2 + x)*(1 + x^2)^(1/3) + 25*(1 + x^2)^(2/3)]))/(10^(1/ 3)*((3 + x)^3*(1 + x^2))^(1/3))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.18 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {7239, 7270, 7293, 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.
\(\displaystyle \int \frac {x+1}{(2 x+1) \sqrt [3]{x^5+9 x^4+28 x^3+36 x^2+27 x+27}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {x+1}{(2 x+1) \sqrt [3]{(x+3)^3 \left (x^2+1\right )}}dx\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle \frac {(x+3) \sqrt [3]{x^2+1} \int \frac {x+1}{(x+3) (2 x+1) \sqrt [3]{x^2+1}}dx}{\sqrt [3]{(x+3)^3 \left (x^2+1\right )}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {(x+3) \sqrt [3]{x^2+1} \int \left (\frac {2}{5 (x+3) \sqrt [3]{x^2+1}}+\frac {1}{5 (2 x+1) \sqrt [3]{x^2+1}}\right )dx}{\sqrt [3]{(x+3)^3 \left (x^2+1\right )}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(x+3) \sqrt [3]{x^2+1} \left (\frac {2}{15} x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},\frac {x^2}{9},-x^2\right )+\frac {1}{5} x \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {1}{3},\frac {3}{2},4 x^2,-x^2\right )+\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x^2+1}+\sqrt [3]{5}}{\sqrt {3} \sqrt [3]{5}}\right )}{5 \sqrt [3]{10}}+\frac {\sqrt {3} \arctan \left (\frac {2\ 2^{2/3} \sqrt [3]{x^2+1}+\sqrt [3]{5}}{\sqrt {3} \sqrt [3]{5}}\right )}{10 \sqrt [3]{10}}-\frac {\log \left (1-4 x^2\right )}{20 \sqrt [3]{10}}-\frac {\log \left (9-x^2\right )}{10 \sqrt [3]{10}}+\frac {3 \log \left (\sqrt [3]{10}-2 \sqrt [3]{x^2+1}\right )}{20 \sqrt [3]{10}}+\frac {3 \log \left (\sqrt [3]{10}-\sqrt [3]{x^2+1}\right )}{10 \sqrt [3]{10}}\right )}{\sqrt [3]{(x+3)^3 \left (x^2+1\right )}}\) |
Input:
Int[(1 + x)/((1 + 2*x)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)), x]
Output:
((3 + x)*(1 + x^2)^(1/3)*((2*x*AppellF1[1/2, 1, 1/3, 3/2, x^2/9, -x^2])/15 + (x*AppellF1[1/2, 1, 1/3, 3/2, 4*x^2, -x^2])/5 + (Sqrt[3]*ArcTan[(5^(1/3 ) + 2^(2/3)*(1 + x^2)^(1/3))/(Sqrt[3]*5^(1/3))])/(5*10^(1/3)) + (Sqrt[3]*A rcTan[(5^(1/3) + 2*2^(2/3)*(1 + x^2)^(1/3))/(Sqrt[3]*5^(1/3))])/(10*10^(1/ 3)) - Log[1 - 4*x^2]/(20*10^(1/3)) - Log[9 - x^2]/(10*10^(1/3)) + (3*Log[1 0^(1/3) - 2*(1 + x^2)^(1/3)])/(20*10^(1/3)) + (3*Log[10^(1/3) - (1 + x^2)^ (1/3)])/(10*10^(1/3))))/((3 + x)^3*(1 + x^2))^(1/3)
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 13.29 (sec) , antiderivative size = 2481, normalized size of antiderivative = 8.06
Input:
int((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x,method=_RETURN VERBOSE)
Output:
1/50*RootOf(_Z^3-100)*ln(-(-59749446875*RootOf(81*RootOf(_Z^3-100)^2+225*_ Z*RootOf(_Z^3-100)+625*_Z^2)^2*RootOf(_Z^3-100)^2*x^5+6278650650*RootOf(81 *RootOf(_Z^3-100)^2+225*_Z*RootOf(_Z^3-100)+625*_Z^2)*RootOf(_Z^3-100)^3*x ^5-5377450218750*RootOf(81*RootOf(_Z^3-100)^2+225*_Z*RootOf(_Z^3-100)+625* _Z^2)^2*RootOf(_Z^3-100)^2*x^2+565078558500*RootOf(81*RootOf(_Z^3-100)^2+2 25*_Z*RootOf(_Z^3-100)+625*_Z^2)*RootOf(_Z^3-100)^3*x^2-477995575000*RootO f(81*RootOf(_Z^3-100)^2+225*_Z*RootOf(_Z^3-100)+625*_Z^2)^2*RootOf(_Z^3-10 0)^2*x^4+50229205200*RootOf(81*RootOf(_Z^3-100)^2+225*_Z*RootOf(_Z^3-100)+ 625*_Z^2)*RootOf(_Z^3-100)^3*x^4-1971731746875*RootOf(81*RootOf(_Z^3-100)^ 2+225*_Z*RootOf(_Z^3-100)+625*_Z^2)^2*RootOf(_Z^3-100)^2*x^3+207195471450* RootOf(81*RootOf(_Z^3-100)^2+225*_Z*RootOf(_Z^3-100)+625*_Z^2)*RootOf(_Z^3 -100)^3*x^3-3527607343500*RootOf(81*RootOf(_Z^3-100)^2+225*_Z*RootOf(_Z^3- 100)+625*_Z^2)*x^5+26887251093750*RootOf(81*RootOf(_Z^3-100)^2+225*_Z*Root Of(_Z^3-100)+625*_Z^2)*x^4+214237616715000*RootOf(81*RootOf(_Z^3-100)^2+22 5*_Z*RootOf(_Z^3-100)+625*_Z^2)*x^3+233596437502500*RootOf(81*RootOf(_Z^3- 100)^2+225*_Z*RootOf(_Z^3-100)+625*_Z^2)*x^2-50332934047500*RootOf(81*Root Of(_Z^3-100)^2+225*_Z*RootOf(_Z^3-100)+625*_Z^2)*x+370691534376*RootOf(_Z^ 3-100)*x^5-2825392792500*RootOf(_Z^3-100)*x^4-22512729770640*RootOf(_Z^3-1 00)*x^3-24547012581240*RootOf(_Z^3-100)*x^2+5289135307560*RootOf(_Z^3-100) *x-84207657169800*(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(2/3)+1777529944800...
Exception generated. \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x, algorit hm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int \frac {x + 1}{\sqrt [3]{\left (x + 3\right )^{3} \left (x^{2} + 1\right )} \left (2 x + 1\right )}\, dx \] Input:
integrate((1+x)/(1+2*x)/(x**5+9*x**4+28*x**3+36*x**2+27*x+27)**(1/3),x)
Output:
Integral((x + 1)/(((x + 3)**3*(x**2 + 1))**(1/3)*(2*x + 1)), x)
\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int { \frac {x + 1}{{\left (x^{5} + 9 \, x^{4} + 28 \, x^{3} + 36 \, x^{2} + 27 \, x + 27\right )}^{\frac {1}{3}} {\left (2 \, x + 1\right )}} \,d x } \] Input:
integrate((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x, algorit hm="maxima")
Output:
integrate((x + 1)/((x^5 + 9*x^4 + 28*x^3 + 36*x^2 + 27*x + 27)^(1/3)*(2*x + 1)), x)
\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int { \frac {x + 1}{{\left (x^{5} + 9 \, x^{4} + 28 \, x^{3} + 36 \, x^{2} + 27 \, x + 27\right )}^{\frac {1}{3}} {\left (2 \, x + 1\right )}} \,d x } \] Input:
integrate((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x, algorit hm="giac")
Output:
integrate((x + 1)/((x^5 + 9*x^4 + 28*x^3 + 36*x^2 + 27*x + 27)^(1/3)*(2*x + 1)), x)
Timed out. \[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int \frac {x+1}{\left (2\,x+1\right )\,{\left (x^5+9\,x^4+28\,x^3+36\,x^2+27\,x+27\right )}^{1/3}} \,d x \] Input:
int((x + 1)/((2*x + 1)*(27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5 + 27)^(1/3)), x)
Output:
int((x + 1)/((2*x + 1)*(27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5 + 27)^(1/3)), x)
\[ \int \frac {1+x}{(1+2 x) \sqrt [3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx=\int \frac {x}{2 \left (x^{2}+1\right )^{\frac {1}{3}} x^{2}+7 \left (x^{2}+1\right )^{\frac {1}{3}} x +3 \left (x^{2}+1\right )^{\frac {1}{3}}}d x +\int \frac {1}{2 \left (x^{2}+1\right )^{\frac {1}{3}} x^{2}+7 \left (x^{2}+1\right )^{\frac {1}{3}} x +3 \left (x^{2}+1\right )^{\frac {1}{3}}}d x \] Input:
int((1+x)/(1+2*x)/(x^5+9*x^4+28*x^3+36*x^2+27*x+27)^(1/3),x)
Output:
int(x/(2*(x**2 + 1)**(1/3)*x**2 + 7*(x**2 + 1)**(1/3)*x + 3*(x**2 + 1)**(1 /3)),x) + int(1/(2*(x**2 + 1)**(1/3)*x**2 + 7*(x**2 + 1)**(1/3)*x + 3*(x** 2 + 1)**(1/3)),x)