3.31.70 \(\int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx\) [3070]

3.31.70.1 Optimal result

Integrand size = 48, antiderivative size = 496 \[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {5 \sqrt {3} \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}{12 \sqrt [3]{10}+22 \sqrt [3]{10} x-6 \sqrt [3]{10} x^2-4 \sqrt [3]{10} x^3+5 \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (3+7 x+2 x^2\right )}{5 \sqrt [3]{10}}+\frac {\log \left (9+42 x+61 x^2+28 x^3+4 x^4\right )}{10 \sqrt [3]{10}}+\frac {\log \left (-6 \sqrt [3]{10}-11 \sqrt [3]{10} x+3 \sqrt [3]{10} x^2+2 \sqrt [3]{10} x^3+5 \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (36\ 10^{2/3}+132\ 10^{2/3} x+85\ 10^{2/3} x^2-90\ 10^{2/3} x^3-35\ 10^{2/3} x^4+12\ 10^{2/3} x^5+4\ 10^{2/3} x^6+\left (30 \sqrt [3]{10}+55 \sqrt [3]{10} x-15 \sqrt [3]{10} x^2-10 \sqrt [3]{10} x^3\right ) \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}+25 \left (27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8\right )^{2/3}\right )}{10 \sqrt [3]{10}} \]

output

1/50*3^(1/2)*arctan(5*3^(1/2)*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^ 
3+522*x^2+189*x+27)^(1/3)/(12*10^(1/3)+22*10^(1/3)*x-6*10^(1/3)*x^2-4*10^( 
1/3)*x^3+5*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27) 
^(1/3)))*10^(2/3)-1/50*ln(2*x^2+7*x+3)*10^(2/3)+1/100*ln(4*x^4+28*x^3+61*x 
^2+42*x+9)*10^(2/3)+1/50*ln(-6*10^(1/3)-11*10^(1/3)*x+3*10^(1/3)*x^2+2*10^ 
(1/3)*x^3+5*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27 
)^(1/3))*10^(2/3)-1/100*ln(36*10^(2/3)+132*10^(2/3)*x+85*10^(2/3)*x^2-90*1 
0^(2/3)*x^3-35*10^(2/3)*x^4+12*10^(2/3)*x^5+4*10^(2/3)*x^6+(30*10^(1/3)+55 
*10^(1/3)*x-15*10^(1/3)*x^2-10*10^(1/3)*x^3)*(8*x^8+84*x^7+338*x^6+679*x^5 
+825*x^4+784*x^3+522*x^2+189*x+27)^(1/3)+25*(8*x^8+84*x^7+338*x^6+679*x^5+ 
825*x^4+784*x^3+522*x^2+189*x+27)^(2/3))*10^(2/3)
 

3.31.70.2 Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.38 \[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx=-\frac {\sqrt [3]{1+x^2} \left (3+7 x+2 x^2\right ) \left (2 \sqrt {3} \arctan \left (\frac {4 \sqrt [3]{10}-2 \sqrt [3]{10} x+5 \sqrt [3]{1+x^2}}{5 \sqrt {3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 \sqrt [3]{10}+\sqrt [3]{10} x+5 \sqrt [3]{1+x^2}\right )+\log \left (4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2-5 \sqrt [3]{10} (-2+x) \sqrt [3]{1+x^2}+25 \left (1+x^2\right )^{2/3}\right )\right )}{10 \sqrt [3]{10} \sqrt [3]{\left (1+x^2\right ) \left (3+7 x+2 x^2\right )^3}} \]

input

Integrate[(1 + x)/(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 33 
8*x^6 + 84*x^7 + 8*x^8)^(1/3),x]
 

output

-1/10*((1 + x^2)^(1/3)*(3 + 7*x + 2*x^2)*(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*1 
0^(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)*((1 + x^2)*(3 + 7*x + 2*x^2)^3)^(1/3))
 

3.31.70.3 Rubi [F]

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.

\(\displaystyle \int \frac {x+1}{\sqrt [3]{8 x^8+84 x^7+338 x^6+679 x^5+825 x^4+784 x^3+522 x^2+189 x+27}} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7270

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

\(\Big \downarrow \) 1376

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

input

Int[(1 + x)/(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 
+ 84*x^7 + 8*x^8)^(1/3),x]
 

output

$Aborted
 

3.31.70.3.1 Defintions of rubi rules used

rule 1376

Int[((g_.) + (h_.)*(x_))*((a_.) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + 
(f_.)*(x_)^2)^(q_), x_Symbol] :> Unintegrable[(g + h*x)*(a + c*x^2)^p*(d + 
e*x + f*x^2)^q, x] /; FreeQ[{a, c, d, e, f, g, h, p, q}, x]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7270

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]
 

3.31.70.4 Maple [C] (warning: unable to verify)

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

Time = 11.60 (sec) , antiderivative size = 5455, normalized size of antiderivative = 11.00

method	result	size
trager	\(\text {Expression too large to display}\)	\(5455\)

input

int((1+x)/(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^ 
(1/3),x,method=_RETURNVERBOSE)
 

output

result too large to display
 

3.31.70.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((1+x)/(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189* 
x+27)^(1/3),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (residue poly has multiple non-linear fac 
tors)
 

3.31.70.6 Sympy [F]

\[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx=\int \frac {x + 1}{\sqrt [3]{\left (x + 3\right )^{3} \left (2 x + 1\right )^{3} \left (x^{2} + 1\right )}}\, dx \]

input

integrate((1+x)/(8*x**8+84*x**7+338*x**6+679*x**5+825*x**4+784*x**3+522*x* 
*2+189*x+27)**(1/3),x)
 

output

Integral((x + 1)/((x + 3)**3*(2*x + 1)**3*(x**2 + 1))**(1/3), x)
 

3.31.70.7 Maxima [F]

\[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx=\int { \frac {x + 1}{{\left (8 \, x^{8} + 84 \, x^{7} + 338 \, x^{6} + 679 \, x^{5} + 825 \, x^{4} + 784 \, x^{3} + 522 \, x^{2} + 189 \, x + 27\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate((1+x)/(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189* 
x+27)^(1/3),x, algorithm="maxima")
 

output

integrate((x + 1)/(8*x^8 + 84*x^7 + 338*x^6 + 679*x^5 + 825*x^4 + 784*x^3 
+ 522*x^2 + 189*x + 27)^(1/3), x)
 

3.31.70.8 Giac [F]

\[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx=\int { \frac {x + 1}{{\left (8 \, x^{8} + 84 \, x^{7} + 338 \, x^{6} + 679 \, x^{5} + 825 \, x^{4} + 784 \, x^{3} + 522 \, x^{2} + 189 \, x + 27\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate((1+x)/(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189* 
x+27)^(1/3),x, algorithm="giac")
 

output

integrate((x + 1)/(8*x^8 + 84*x^7 + 338*x^6 + 679*x^5 + 825*x^4 + 784*x^3 
+ 522*x^2 + 189*x + 27)^(1/3), x)
 

3.31.70.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx=\int \frac {x+1}{{\left (8\,x^8+84\,x^7+338\,x^6+679\,x^5+825\,x^4+784\,x^3+522\,x^2+189\,x+27\right )}^{1/3}} \,d x \]

input

int((x + 1)/(189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84* 
x^7 + 8*x^8 + 27)^(1/3),x)
 

output

int((x + 1)/(189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84* 
x^7 + 8*x^8 + 27)^(1/3), x)