\(\int x^2 \sqrt [4]{a-b x^4} \, dx\) [651]

Optimal result

Integrand size = 16, antiderivative size = 170 \[ \int x^2 \sqrt [4]{a-b x^4} \, dx=\frac {1}{4} x^3 \sqrt [4]{a-b x^4}-\frac {a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{3/4}}+\frac {a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt {2} b^{3/4}}-\frac {a \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4} \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}\right )}{8 \sqrt {2} b^{3/4}} \] Output:

1/4*x^3*(-b*x^4+a)^(1/4)+1/16*a*arctan(-1+2^(1/2)*b^(1/4)*x/(-b*x^4+a)^(1/ 
4))*2^(1/2)/b^(3/4)+1/16*a*arctan(1+2^(1/2)*b^(1/4)*x/(-b*x^4+a)^(1/4))*2^ 
(1/2)/b^(3/4)-1/16*a*arctanh(2^(1/2)*b^(1/4)*x/(-b*x^4+a)^(1/4)/(1+b^(1/2) 
*x^2/(-b*x^4+a)^(1/2)))*2^(1/2)/b^(3/4)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.86 \[ \int x^2 \sqrt [4]{a-b x^4} \, dx=\frac {4 b^{3/4} x^3 \sqrt [4]{a-b x^4}+\sqrt {2} a \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}{-\sqrt {b} x^2+\sqrt {a-b x^4}}\right )-\sqrt {2} a \text {arctanh}\left (\frac {\sqrt {b} x^2+\sqrt {a-b x^4}}{\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}\right )}{16 b^{3/4}} \] Input:

Integrate[x^2*(a - b*x^4)^(1/4),x]
 

Output:

(4*b^(3/4)*x^3*(a - b*x^4)^(1/4) + Sqrt[2]*a*ArcTan[(Sqrt[2]*b^(1/4)*x*(a 
- b*x^4)^(1/4))/(-(Sqrt[b]*x^2) + Sqrt[a - b*x^4])] - Sqrt[2]*a*ArcTanh[(S 
qrt[b]*x^2 + Sqrt[a - b*x^4])/(Sqrt[2]*b^(1/4)*x*(a - b*x^4)^(1/4))])/(16* 
b^(3/4))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.46, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {811, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 811

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

\(\Big \downarrow \) 854

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{4} a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}\right )+\frac {1}{4} x^3 \sqrt [4]{a-b x^4}\)

Input:

Int[x^2*(a - b*x^4)^(1/4),x]
 

Output:

(x^3*(a - b*x^4)^(1/4))/4 + (a*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x 
^4)^(1/4)]/(Sqrt[2]*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4) 
^(1/4)]/(Sqrt[2]*b^(1/4)))/(2*Sqrt[b]) - (-1/2*Log[1 + (Sqrt[b]*x^2)/Sqrt[ 
a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(Sqrt[2]*b^(1/4)) + Lo 
g[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4 
)]/(2*Sqrt[2]*b^(1/4)))/(2*Sqrt[b])))/4
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 811

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 
))   Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I 
GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m 
, p, x]
 

rule 826

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

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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(-\frac {a \left (\ln \left (\frac {b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {b}\, x^{2}+\sqrt {-b \,x^{4}+a}}{-b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {b}\, x^{2}+\sqrt {-b \,x^{4}+a}}\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}+1\right )\right ) \sqrt {2}-8 \left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{3} b^{\frac {3}{4}}}{32 b^{\frac {3}{4}}}\)	\(159\)

Input:

int(x^2*(-b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

-1/32*(a*(ln((b^(1/4)*(-b*x^4+a)^(1/4)*2^(1/2)*x+b^(1/2)*x^2+(-b*x^4+a)^(1 
/2))/(-b^(1/4)*(-b*x^4+a)^(1/4)*2^(1/2)*x+b^(1/2)*x^2+(-b*x^4+a)^(1/2)))-2 
*arctan(-2^(1/2)/b^(1/4)*(-b*x^4+a)^(1/4)/x+1)+2*arctan(2^(1/2)/b^(1/4)*(- 
b*x^4+a)^(1/4)/x+1))*2^(1/2)-8*(-b*x^4+a)^(1/4)*x^3*b^(3/4))/b^(3/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.13 \[ \int x^2 \sqrt [4]{a-b x^4} \, dx=\frac {1}{4} \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{3} - \frac {1}{16} \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {\left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b x + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a}{x}\right ) + \frac {1}{16} \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b x - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a}{x}\right ) - \frac {1}{16} i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b x + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a}{x}\right ) + \frac {1}{16} i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (-\frac {a^{4}}{b^{3}}\right )^{\frac {1}{4}} b x + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a}{x}\right ) \] Input:

integrate(x^2*(-b*x^4+a)^(1/4),x, algorithm="fricas")
 

Output:

1/4*(-b*x^4 + a)^(1/4)*x^3 - 1/16*(-a^4/b^3)^(1/4)*log(((-a^4/b^3)^(1/4)*b 
*x + (-b*x^4 + a)^(1/4)*a)/x) + 1/16*(-a^4/b^3)^(1/4)*log(-((-a^4/b^3)^(1/ 
4)*b*x - (-b*x^4 + a)^(1/4)*a)/x) - 1/16*I*(-a^4/b^3)^(1/4)*log((I*(-a^4/b 
^3)^(1/4)*b*x + (-b*x^4 + a)^(1/4)*a)/x) + 1/16*I*(-a^4/b^3)^(1/4)*log((-I 
*(-a^4/b^3)^(1/4)*b*x + (-b*x^4 + a)^(1/4)*a)/x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.65 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.24 \[ \int x^2 \sqrt [4]{a-b x^4} \, dx=\frac {\sqrt [4]{a} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:

integrate(x**2*(-b*x**4+a)**(1/4),x)
 

Output:

a**(1/4)*x**3*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), b*x**4*exp_polar(2*I*p 
i)/a)/(4*gamma(7/4))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.26 \[ \int x^2 \sqrt [4]{a-b x^4} \, dx=-\frac {\sqrt {2} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{16 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{16 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} a \log \left (\sqrt {b} + \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{32 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} a \log \left (\sqrt {b} - \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{32 \, b^{\frac {3}{4}}} + \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a}{4 \, {\left (b - \frac {b x^{4} - a}{x^{4}}\right )} x} \] Input:

integrate(x^2*(-b*x^4+a)^(1/4),x, algorithm="maxima")
 

Output:

-1/16*sqrt(2)*a*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(-b*x^4 + a)^(1/4) 
/x)/b^(1/4))/b^(3/4) - 1/16*sqrt(2)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) 
 - 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4))/b^(3/4) - 1/32*sqrt(2)*a*log(sqrt(b) + 
 sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(3/4) + 1/ 
32*sqrt(2)*a*log(sqrt(b) - sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b* 
x^4 + a)/x^2)/b^(3/4) + 1/4*(-b*x^4 + a)^(1/4)*a/((b - (b*x^4 - a)/x^4)*x)
 

Giac [F]

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

integrate(x^2*(-b*x^4+a)^(1/4),x, algorithm="giac")
 

Output:

integrate((-b*x^4 + a)^(1/4)*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt [4]{a-b x^4} \, dx=\int x^2\,{\left (a-b\,x^4\right )}^{1/4} \,d x \] Input:

int(x^2*(a - b*x^4)^(1/4),x)
 

Output:

int(x^2*(a - b*x^4)^(1/4), x)
 

Reduce [F]

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

int(x^2*(-b*x^4+a)^(1/4),x)
 

Output:

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