3.8.43 \(\int \frac {(-3+2 x^5) (1+2 x^5+x^6+x^{10})}{x^6 (1-x^3+x^5) \sqrt [4]{x+x^6}} \, dx\) [743]

3.8.43.1 Optimal result

Integrand size = 45, antiderivative size = 57 \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (3+7 x^3+3 x^5\right ) \left (x+x^6\right )^{3/4}}{21 x^6}-4 \arctan \left (\frac {x}{\sqrt [4]{x+x^6}}\right )-4 \text {arctanh}\left (\frac {x}{\sqrt [4]{x+x^6}}\right ) \]

output

4/21*(3*x^5+7*x^3+3)*(x^6+x)^(3/4)/x^6-4*arctan(x/(x^6+x)^(1/4))-4*arctanh 
(x/(x^6+x)^(1/4))
 

3.8.43.2 Mathematica [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \]

input

Integrate[((-3 + 2*x^5)*(1 + 2*x^5 + x^6 + x^10))/(x^6*(1 - x^3 + x^5)*(x 
+ x^6)^(1/4)),x]
 

output

Integrate[((-3 + 2*x^5)*(1 + 2*x^5 + x^6 + x^10))/(x^6*(1 - x^3 + x^5)*(x 
+ x^6)^(1/4)), x]
 

3.8.43.3 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.

\(\displaystyle \int \frac {\left (2 x^5-3\right ) \left (x^{10}+x^6+2 x^5+1\right )}{x^6 \left (x^5-x^3+1\right ) \sqrt [4]{x^6+x}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{x^5+1} \int -\frac {\left (3-2 x^5\right ) \left (x^{10}+x^6+2 x^5+1\right )}{x^{25/4} \sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}dx}{\sqrt [4]{x^6+x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{x^5+1} \int \frac {\left (3-2 x^5\right ) \left (x^{10}+x^6+2 x^5+1\right )}{x^{25/4} \sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}dx}{\sqrt [4]{x^6+x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \int \frac {\left (3-2 x^5\right ) \left (x^{10}+x^6+2 x^5+1\right )}{x^{11/2} \sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}d\sqrt [4]{x}}{\sqrt [4]{x^6+x}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \int \left (-\frac {2 x^{9/2}}{\sqrt [4]{x^5+1}}-\frac {2 x^{5/2}}{\sqrt [4]{x^5+1}}-\frac {4 \sqrt {x}}{\sqrt [4]{x^5+1}}+\frac {2 \left (5-2 x^3\right ) \sqrt {x}}{\sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}+\frac {1}{\sqrt [4]{x^5+1} \sqrt {x}}+\frac {3}{\sqrt [4]{x^5+1} x^{5/2}}+\frac {3}{\sqrt [4]{x^5+1} x^{11/2}}\right )d\sqrt [4]{x}}{\sqrt [4]{x^6+x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{x^5+1} \left (10 \int \frac {\sqrt {x}}{\sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}d\sqrt [4]{x}-4 \int \frac {x^{7/2}}{\sqrt [4]{x^5+1} \left (x^5-x^3+1\right )}d\sqrt [4]{x}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{20},\frac {1}{4},\frac {19}{20},-x^5\right )}{\sqrt [4]{x}}-\frac {2}{19} x^{19/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )-\frac {2}{11} x^{11/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )-\frac {4}{3} x^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{20},\frac {1}{4},\frac {23}{20},-x^5\right )-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {9}{20},\frac {1}{4},\frac {11}{20},-x^5\right )}{3 x^{9/4}}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {21}{20},\frac {1}{4},-\frac {1}{20},-x^5\right )}{7 x^{21/4}}\right )}{\sqrt [4]{x^6+x}}\)

input

Int[((-3 + 2*x^5)*(1 + 2*x^5 + x^6 + x^10))/(x^6*(1 - x^3 + x^5)*(x + x^6) 
^(1/4)),x]
 

output

$Aborted
 

3.8.43.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.8.43.4 Maple [A] (verified)

Time = 7.46 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.39

method	result	size
pseudoelliptic	\(\frac {\left (12 x^{5}+28 x^{3}+12\right ) \left (x^{6}+x \right )^{\frac {3}{4}}+42 x^{6} \left (\ln \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}}-x}{x}\right )-\ln \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{6}+x \right )^{\frac {1}{4}}}{x}\right )\right )}{21 x^{6}}\)	\(79\)
trager	\(\frac {4 \left (3 x^{5}+7 x^{3}+3\right ) \left (x^{6}+x \right )^{\frac {3}{4}}}{21 x^{6}}+2 \ln \left (-\frac {-x^{5}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{6}+x}+2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{5}-x^{3}+1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{6}+x \right )^{\frac {3}{4}}+2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{3}+1}\right )\)	\(176\)
risch	\(\frac {\frac {4}{7} x^{10}+\frac {8}{7} x^{5}+\frac {4}{7}+\frac {4}{3} x^{8}+\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{5}+1\right )\right )}^{\frac {1}{4}}}+2 \ln \left (-\frac {-x^{5}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{6}+x}+2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{5}-x^{3}+1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-2 \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{3}+1}\right )\)	\(189\)

input

int((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x,method=_R 
ETURNVERBOSE)
 

output

1/21*((12*x^5+28*x^3+12)*(x^6+x)^(3/4)+42*x^6*(ln(((x^6+x)^(1/4)-x)/x)-ln( 
((x^6+x)^(1/4)+x)/x)+2*arctan(1/x*(x^6+x)^(1/4))))/x^6
 

3.8.43.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).

Time = 48.02 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.18 \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=-\frac {2 \, {\left (21 \, x^{6} \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + {\left (x^{6} + x\right )}^{\frac {3}{4}}\right )}}{x^{5} - x^{3} + 1}\right ) - 21 \, x^{6} \log \left (\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} + x} x - 2 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1}{x^{5} - x^{3} + 1}\right ) - 2 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{5} + 7 \, x^{3} + 3\right )}\right )}}{21 \, x^{6}} \]

input

integrate((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x, al 
gorithm="fricas")
 

output

-2/21*(21*x^6*arctan(2*((x^6 + x)^(1/4)*x^2 + (x^6 + x)^(3/4))/(x^5 - x^3 
+ 1)) - 21*x^6*log((x^5 + x^3 - 2*(x^6 + x)^(1/4)*x^2 + 2*sqrt(x^6 + x)*x 
- 2*(x^6 + x)^(3/4) + 1)/(x^5 - x^3 + 1)) - 2*(x^6 + x)^(3/4)*(3*x^5 + 7*x 
^3 + 3))/x^6
 

3.8.43.6 Sympy [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2 x^{5} - 3\right ) \left (x^{10} + x^{6} + 2 x^{5} + 1\right )}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (x^{5} - x^{3} + 1\right )}\, dx \]

input

integrate((2*x**5-3)*(x**10+x**6+2*x**5+1)/x**6/(x**5-x**3+1)/(x**6+x)**(1 
/4),x)
 

output

Integral((2*x**5 - 3)*(x**10 + x**6 + 2*x**5 + 1)/(x**6*(x*(x + 1)*(x**4 - 
 x**3 + x**2 - x + 1))**(1/4)*(x**5 - x**3 + 1)), x)
 

3.8.43.7 Maxima [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (x^{10} + x^{6} + 2 \, x^{5} + 1\right )} {\left (2 \, x^{5} - 3\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} - x^{3} + 1\right )} x^{6}} \,d x } \]

input

integrate((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x, al 
gorithm="maxima")
 

output

integrate((x^10 + x^6 + 2*x^5 + 1)*(2*x^5 - 3)/((x^6 + x)^(1/4)*(x^5 - x^3 
 + 1)*x^6), x)
 

3.8.43.8 Giac [F]

\[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (x^{10} + x^{6} + 2 \, x^{5} + 1\right )} {\left (2 \, x^{5} - 3\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} - x^{3} + 1\right )} x^{6}} \,d x } \]

input

integrate((2*x^5-3)*(x^10+x^6+2*x^5+1)/x^6/(x^5-x^3+1)/(x^6+x)^(1/4),x, al 
gorithm="giac")
 

output

integrate((x^10 + x^6 + 2*x^5 + 1)*(2*x^5 - 3)/((x^6 + x)^(1/4)*(x^5 - x^3 
 + 1)*x^6), x)
 

3.8.43.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-3+2 x^5\right ) \left (1+2 x^5+x^6+x^{10}\right )}{x^6 \left (1-x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2\,x^5-3\right )\,\left (x^{10}+x^6+2\,x^5+1\right )}{x^6\,{\left (x^6+x\right )}^{1/4}\,\left (x^5-x^3+1\right )} \,d x \]

input

int(((2*x^5 - 3)*(2*x^5 + x^6 + x^10 + 1))/(x^6*(x + x^6)^(1/4)*(x^5 - x^3 
 + 1)),x)
 

output

int(((2*x^5 - 3)*(2*x^5 + x^6 + x^10 + 1))/(x^6*(x + x^6)^(1/4)*(x^5 - x^3 
 + 1)), x)