\(\int \frac {\sqrt [4]{-x^3+x^4}}{x^2} \, dx\) [708]

Optimal result

Integrand size = 17, antiderivative size = 55 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2} \, dx=-\frac {4 \sqrt [4]{-x^3+x^4}}{x}-2 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right ) \] Output:

-4*(x^4-x^3)^(1/4)/x-2*arctan(x/(x^4-x^3)^(1/4))+2*arctanh(x/(x^4-x^3)^(1/ 
4))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2} \, dx=-\frac {2 (-1+x)^{3/4} x^2 \left (2 \sqrt [4]{-1+x}+\sqrt [4]{x} \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\sqrt [4]{x} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{\left ((-1+x) x^3\right )^{3/4}} \] Input:

Integrate[(-x^3 + x^4)^(1/4)/x^2,x]
 

Output:

(-2*(-1 + x)^(3/4)*x^2*(2*(-1 + x)^(1/4) + x^(1/4)*ArcTan[((-1 + x)/x)^(-1 
/4)] - x^(1/4)*ArcTanh[((-1 + x)/x)^(-1/4)]))/((-1 + x)*x^3)^(3/4)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.51, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {1926, 1938, 73, 770, 756, 216, 219}

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.

\(\displaystyle \int \frac {\sqrt [4]{x^4-x^3}}{x^2} \, dx\)

\(\Big \downarrow \) 1926

\(\displaystyle \int \frac {x^2}{\left (x^4-x^3\right )^{3/4}}dx-\frac {4 \sqrt [4]{x^4-x^3}}{x}\)

\(\Big \downarrow \) 1938

\(\displaystyle \frac {(x-1)^{3/4} x^{9/4} \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x}}dx}{\left (x^4-x^3\right )^{3/4}}-\frac {4 \sqrt [4]{x^4-x^3}}{x}\)

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {4 (x-1)^{3/4} x^{9/4} \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )}{\left (x^4-x^3\right )^{3/4}}-\frac {4 \sqrt [4]{x^4-x^3}}{x}\)

Input:

Int[(-x^3 + x^4)^(1/4)/x^2,x]
 

Output:

(-4*(-x^3 + x^4)^(1/4))/x + (4*(-1 + x)^(3/4)*x^(9/4)*(ArcTan[(-1 + x)^(1/ 
4)/x^(1/4)]/2 + ArcTanh[(-1 + x)^(1/4)/x^(1/4)]/2))/(-x^3 + x^4)^(3/4)
 

Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 1926

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] 
 :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + j*p + 1))), x] - Simp[b*p 
*((n - j)/(c^n*(m + j*p + 1)))   Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), 
 x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (Integer 
sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [C] (warning: unable to verify)

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

Time = 0.65 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.49

method	result	size
meijerg	\(-\frac {4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x\right )}{\left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {1}{4}}}\)	\(27\)
pseudoelliptic	\(\frac {-\ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right ) x +\ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right ) x +2 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right ) x -4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\)	\(73\)
trager	\(-\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x}+\ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )\)	\(162\)
risch	\(-\frac {4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}+\frac {\left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}}{\left (-1+x \right )^{2}}\right )+\ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+2 x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+4 x -1}{\left (-1+x \right )^{2}}\right )\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\)	\(393\)

Input:

int((x^4-x^3)^(1/4)/x^2,x,method=_RETURNVERBOSE)
 

Output:

-4*signum(-1+x)^(1/4)/(-signum(-1+x))^(1/4)/x^(1/4)*hypergeom([-1/4,-1/4], 
[3/4],x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2} \, dx=-\frac {2 \, x \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {3}{4}}}{x^{3} - x^{2}}\right ) - x \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + x \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x} \] Input:

integrate((x^4-x^3)^(1/4)/x^2,x, algorithm="fricas")
 

Output:

-(2*x*arctan((x^4 - x^3)^(3/4)/(x^3 - x^2)) - x*log((x + (x^4 - x^3)^(1/4) 
)/x) + x*log(-(x - (x^4 - x^3)^(1/4))/x) + 4*(x^4 - x^3)^(1/4))/x
 

Sympy [F]

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

integrate((x**4-x**3)**(1/4)/x**2,x)
 

Output:

Integral((x**3*(x - 1))**(1/4)/x**2, x)
 

Maxima [F]

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

integrate((x^4-x^3)^(1/4)/x^2,x, algorithm="maxima")
 

Output:

integrate((x^4 - x^3)^(1/4)/x^2, x)
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2} \, dx=-4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 2 \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \] Input:

integrate((x^4-x^3)^(1/4)/x^2,x, algorithm="giac")
 

Output:

-4*(-1/x + 1)^(1/4) + 2*arctan((-1/x + 1)^(1/4)) + log((-1/x + 1)^(1/4) + 
1) - log(abs((-1/x + 1)^(1/4) - 1))
 

Mupad [B] (verification not implemented)

Time = 8.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x^2} \, dx=-\frac {4\,{\left (x^4-x^3\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ x\right )}{x\,{\left (1-x\right )}^{1/4}} \] Input:

int((x^4 - x^3)^(1/4)/x^2,x)
 

Output:

-(4*(x^4 - x^3)^(1/4)*hypergeom([-1/4, -1/4], 3/4, x))/(x*(1 - x)^(1/4))
 

Reduce [F]

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

int((x^4-x^3)^(1/4)/x^2,x)
 

Output:

int((x**(3/4)*(x - 1)**(1/4))/x**2,x)