3.9.98 \(\int x^7 \sqrt [4]{-x+x^4} \, dx\) [898]

3.9.98.1 Optimal result

Integrand size = 15, antiderivative size = 68 \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\frac {1}{288} \sqrt [4]{-x+x^4} \left (-7 x^2-4 x^5+32 x^8\right )+\frac {7}{192} \arctan \left (\frac {x}{\sqrt [4]{-x+x^4}}\right )-\frac {7}{192} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \]

output

1/288*(x^4-x)^(1/4)*(32*x^8-4*x^5-7*x^2)+7/192*arctan(x/(x^4-x)^(1/4))-7/1 
92*arctanh(x/(x^4-x)^(1/4))
 

3.9.98.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\frac {x^2 \sqrt [4]{x \left (-1+x^3\right )} \left (\sqrt [4]{1-x^3} \left (-7-x^3+8 x^6\right )+7 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{4},\frac {7}{4},x^3\right )\right )}{72 \sqrt [4]{1-x^3}} \]

input

Integrate[x^7*(-x + x^4)^(1/4),x]
 

output

(x^2*(x*(-1 + x^3))^(1/4)*((1 - x^3)^(1/4)*(-7 - x^3 + 8*x^6) + 7*Hypergeo 
metric2F1[-1/4, 3/4, 7/4, x^3]))/(72*(1 - x^3)^(1/4))
 

3.9.98.3 Rubi [A] (warning: unable to verify)

Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1927, 1930, 1930, 1938, 851, 807, 854, 827, 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.

\(\displaystyle \int x^7 \sqrt [4]{x^4-x} \, dx\)

\(\Big \downarrow \) 1927

\(\displaystyle \frac {1}{9} x^8 \sqrt [4]{x^4-x}-\frac {1}{12} \int \frac {x^8}{\left (x^4-x\right )^{3/4}}dx\)

\(\Big \downarrow \) 1930

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \int \frac {x^5}{\left (x^4-x\right )^{3/4}}dx-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

\(\Big \downarrow \) 1930

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \left (\frac {3}{4} \int \frac {x^2}{\left (x^4-x\right )^{3/4}}dx+\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

\(\Big \downarrow \) 1938

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \left (\frac {3 \left (x^3-1\right )^{3/4} x^{3/4} \int \frac {x^{5/4}}{\left (x^3-1\right )^{3/4}}dx}{4 \left (x^4-x\right )^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \left (\frac {3 \left (x^3-1\right )^{3/4} x^{3/4} \int \frac {x^2}{\left (x^3-1\right )^{3/4}}d\sqrt [4]{x}}{\left (x^4-x\right )^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \left (\frac {\left (x^3-1\right )^{3/4} x^{3/4} \int \frac {\sqrt {x}}{(x-1)^{3/4}}dx^{3/4}}{\left (x^4-x\right )^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \left (\frac {\left (x^3-1\right )^{3/4} x^{3/4} \int \frac {\sqrt {x}}{1-x}d\frac {x^{3/4}}{\sqrt [4]{x-1}}}{\left (x^4-x\right )^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \left (\frac {\left (x^3-1\right )^{3/4} x^{3/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{x-1}}-\frac {1}{2} \int \frac {1}{\sqrt {x}+1}d\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )}{\left (x^4-x\right )^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \left (\frac {\left (x^3-1\right )^{3/4} x^{3/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{x-1}}-\frac {1}{2} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )\right )}{\left (x^4-x\right )^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{12} \left (-\frac {7}{8} \left (\frac {\left (x^3-1\right )^{3/4} x^{3/4} \left (\frac {1}{2} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )-\frac {1}{2} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )\right )}{\left (x^4-x\right )^{3/4}}+\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )-\frac {1}{6} \sqrt [4]{x^4-x} x^5\right )+\frac {1}{9} \sqrt [4]{x^4-x} x^8\)

input

Int[x^7*(-x + x^4)^(1/4),x]
 

output

(x^8*(-x + x^4)^(1/4))/9 + (-1/6*(x^5*(-x + x^4)^(1/4)) - (7*((x^2*(-x + x 
^4)^(1/4))/3 + (x^(3/4)*(-1 + x^3)^(3/4)*(-1/2*ArcTan[x^(3/4)/(-1 + x)^(1/ 
4)] + ArcTanh[x^(3/4)/(-1 + x)^(1/4)]/2))/(-x + x^4)^(3/4)))/8)/12
 

3.9.98.3.1 Defintions of rubi rules used

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 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 827

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

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1927

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 + n*p + 1))), x] + Simp[a* 
(n - j)*(p/(c^j*(m + n*p + 1)))   Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) 
, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (Int 
egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
 

rule 1930

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

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

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

Time = 4.48 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.49

method	result	size
meijerg	\(\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {33}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], x^{3}\right )}{33 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\)	\(33\)
pseudoelliptic	\(-\frac {x^{3} \left (\left (-128 x^{8}+16 x^{5}+28 x^{2}\right ) \left (x^{4}-x \right )^{\frac {1}{4}}+42 \arctan \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )-21 \ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}-x}{x}\right )+21 \ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}+x}{x}\right )\right )}{1152 {\left (-\left (x^{4}-x \right )^{\frac {1}{4}}+x \right )}^{3} \left (x^{2}+\sqrt {x^{4}-x}\right )^{3} {\left (\left (x^{4}-x \right )^{\frac {1}{4}}+x \right )}^{3}}\)	\(130\)
trager	\(\frac {x^{2} \left (32 x^{6}-4 x^{3}-7\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{288}+\frac {7 \ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+2 x^{3}-1\right )}{384}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{384}\)	\(145\)
risch	\(\frac {x^{2} \left (32 x^{6}-4 x^{3}-7\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}{288}+\frac {\left (-\frac {7 \ln \left (\frac {2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{384}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{384}\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\)	\(455\)

input

int(x^7*(x^4-x)^(1/4),x,method=_RETURNVERBOSE)
 

output

4/33*signum(x^3-1)^(1/4)/(-signum(x^3-1))^(1/4)*x^(33/4)*hypergeom([-1/4,1 
1/4],[15/4],x^3)
 

3.9.98.5 Fricas [A] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.53 \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\frac {1}{288} \, {\left (32 \, x^{8} - 4 \, x^{5} - 7 \, x^{2}\right )} {\left (x^{4} - x\right )}^{\frac {1}{4}} - \frac {7}{384} \, \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + \frac {7}{384} \, \log \left (2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1\right ) \]

input

integrate(x^7*(x^4-x)^(1/4),x, algorithm="fricas")
 

output

1/288*(32*x^8 - 4*x^5 - 7*x^2)*(x^4 - x)^(1/4) - 7/384*arctan(2*(x^4 - x)^ 
(1/4)*x^2 + 2*(x^4 - x)^(3/4)) + 7/384*log(2*x^3 - 2*(x^4 - x)^(1/4)*x^2 + 
 2*sqrt(x^4 - x)*x - 2*(x^4 - x)^(3/4) - 1)
 

3.9.98.6 Sympy [F]

\[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\int x^{7} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]

input

integrate(x**7*(x**4-x)**(1/4),x)
 

output

Integral(x**7*(x*(x - 1)*(x**2 + x + 1))**(1/4), x)
 

3.9.98.7 Maxima [F]

\[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\int { {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{7} \,d x } \]

input

integrate(x^7*(x^4-x)^(1/4),x, algorithm="maxima")
 

output

integrate((x^4 - x)^(1/4)*x^7, x)
 

3.9.98.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.29 \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=-\frac {1}{288} \, {\left (7 \, {\left (\frac {1}{x^{3}} - 1\right )}^{2} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 18 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} - 21 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )} x^{9} - \frac {7}{192} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {7}{384} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {7}{384} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]

input

integrate(x^7*(x^4-x)^(1/4),x, algorithm="giac")
 

output

-1/288*(7*(1/x^3 - 1)^2*(-1/x^3 + 1)^(1/4) - 18*(-1/x^3 + 1)^(5/4) - 21*(- 
1/x^3 + 1)^(1/4))*x^9 - 7/192*arctan((-1/x^3 + 1)^(1/4)) - 7/384*log((-1/x 
^3 + 1)^(1/4) + 1) + 7/384*log(abs((-1/x^3 + 1)^(1/4) - 1))
 

3.9.98.9 Mupad [F(-1)]

Timed out. \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\int x^7\,{\left (x^4-x\right )}^{1/4} \,d x \]

input

int(x^7*(x^4 - x)^(1/4),x)
 

output

int(x^7*(x^4 - x)^(1/4), x)