3.11.20 \(\int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx\) [1020]

3.11.20.1 Optimal result

Integrand size = 15, antiderivative size = 611 \[ \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx=-\frac {9 a x}{16 b \sqrt [6]{a+b x^2}}+\frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {9 a^2 x}{16 b \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )}-\frac {9 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{32 b^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}}+\frac {3\ 3^{3/4} a^2 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right ),-7+4 \sqrt {3}\right )}{8 \sqrt {2} b^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}} \]

output

-9/16*a*x/b/(b*x^2+a)^(1/6)+3/8*x*(b*x^2+a)^(5/6)/b-9/16*a^2*x/b/(a/(b*x^2 
+a))^(2/3)/(b*x^2+a)^(7/6)/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))+3/16*3^(3/4)*a^ 
2*(1-(a/(b*x^2+a))^(1/3))*EllipticF((1-(a/(b*x^2+a))^(1/3)+3^(1/2))/(1-(a/ 
(b*x^2+a))^(1/3)-3^(1/2)),2*I-I*3^(1/2))*((1+(a/(b*x^2+a))^(1/3)+(a/(b*x^2 
+a))^(2/3))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))^2)^(1/2)/b^2/x/(a/(b*x^2+a))^( 
2/3)/(b*x^2+a)^(1/6)*2^(1/2)/((-1+(a/(b*x^2+a))^(1/3))/(1-(a/(b*x^2+a))^(1 
/3)-3^(1/2))^2)^(1/2)-9/32*3^(1/4)*a^2*(1-(a/(b*x^2+a))^(1/3))*EllipticE(( 
1-(a/(b*x^2+a))^(1/3)+3^(1/2))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2)),2*I-I*3^(1/ 
2))*((1+(a/(b*x^2+a))^(1/3)+(a/(b*x^2+a))^(2/3))/(1-(a/(b*x^2+a))^(1/3)-3^ 
(1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))/b^2/x/(a/(b*x^2+a))^(2/3)/(b*x^2 
+a)^(1/6)/((-1+(a/(b*x^2+a))^(1/3))/(1-(a/(b*x^2+a))^(1/3)-3^(1/2))^2)^(1/ 
2)
 

3.11.20.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 6.98 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.10 \[ \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx=\frac {3 x \left (a+b x^2-a \sqrt [6]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{8 b \sqrt [6]{a+b x^2}} \]

input

Integrate[x^2/(a + b*x^2)^(1/6),x]
 

output

(3*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, 
 -((b*x^2)/a)]))/(8*b*(a + b*x^2)^(1/6))
 

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

Time = 0.42 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {262, 235, 214, 233, 833, 760, 2418}

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.

\(\displaystyle \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx\)

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 235

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

\(\Big \downarrow \) 214

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

\(\Big \downarrow \) 233

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

\(\Big \downarrow \) 833

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

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {3 a \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (-\int \frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{8 b}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 x \left (a+b x^2\right )^{5/6}}{8 b}-\frac {3 a \left (\frac {3 a \sqrt {-\frac {b x^2}{a+b x^2}} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1}\right )}{4 b x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}+\frac {3 x}{2 \sqrt [6]{a+b x^2}}\right )}{8 b}\)

input

Int[x^2/(a + b*x^2)^(1/6),x]
 

output

(3*x*(a + b*x^2)^(5/6))/(8*b) - (3*a*((3*x)/(2*(a + b*x^2)^(1/6)) + (3*a*S 
qrt[-((b*x^2)/(a + b*x^2))]*((-2*Sqrt[-1 + x^3/(a + b*x^2)^(3/2)])/(1 - Sq 
rt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - 
 (1 - (b*x^2)/(a + b*x^2))^(1/3))*Sqrt[(1 + x^2/(a + b*x^2) + (1 - (b*x^2) 
/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))^2]*El 
lipticE[ArcSin[(1 + Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3 
] - (1 - (b*x^2)/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[-1 + x^3/(a 
+ b*x^2)^(3/2)]*Sqrt[-((1 - (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] 
- (1 - (b*x^2)/(a + b*x^2))^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3 
])*(1 - (1 - (b*x^2)/(a + b*x^2))^(1/3))*Sqrt[(1 + x^2/(a + b*x^2) + (1 - 
(b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3) 
)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - 
 Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sq 
rt[-1 + x^3/(a + b*x^2)^(3/2)]*Sqrt[-((1 - (1 - (b*x^2)/(a + b*x^2))^(1/3) 
)/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))^2)])))/(4*b*x*(a/(a + b* 
x^2))^(2/3)*(a + b*x^2)^(1/6))))/(8*b)
 

3.11.20.3.1 Defintions of rubi rules used

rule 214

Int[((a_) + (b_.)*(x_)^2)^(-7/6), x_Symbol] :> Simp[1/((a + b*x^2)^(2/3)*(a 
/(a + b*x^2))^(2/3))   Subst[Int[1/(1 - b*x^2)^(1/3), x], x, x/Sqrt[a + b*x 
^2]], x] /; FreeQ[{a, b}, x]
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/6), x_Symbol] :> Simp[3*(x/(2*(a + b*x^2)^(1/ 
6))), x] - Simp[a/2   Int[1/(a + b*x^2)^(7/6), x], x] /; FreeQ[{a, b}, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

3.11.20.4 Maple [F]

\[\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{6}}}d x\]

input

int(x^2/(b*x^2+a)^(1/6),x)
 

output

int(x^2/(b*x^2+a)^(1/6),x)
 

3.11.20.5 Fricas [F]

\[ \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {1}{6}}} \,d x } \]

input

integrate(x^2/(b*x^2+a)^(1/6),x, algorithm="fricas")
 

output

integral(x^2/(b*x^2 + a)^(1/6), x)
 

3.11.20.6 Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.04 \[ \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx=\frac {x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 \sqrt [6]{a}} \]

input

integrate(x**2/(b*x**2+a)**(1/6),x)
 

output

x**3*hyper((1/6, 3/2), (5/2,), b*x**2*exp_polar(I*pi)/a)/(3*a**(1/6))
 

3.11.20.7 Maxima [F]

\[ \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {1}{6}}} \,d x } \]

input

integrate(x^2/(b*x^2+a)^(1/6),x, algorithm="maxima")
 

output

integrate(x^2/(b*x^2 + a)^(1/6), x)
 

3.11.20.8 Giac [F]

\[ \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {1}{6}}} \,d x } \]

input

integrate(x^2/(b*x^2+a)^(1/6),x, algorithm="giac")
 

output

integrate(x^2/(b*x^2 + a)^(1/6), x)
 

3.11.20.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx=\int \frac {x^2}{{\left (b\,x^2+a\right )}^{1/6}} \,d x \]

input

int(x^2/(a + b*x^2)^(1/6),x)
 

output

int(x^2/(a + b*x^2)^(1/6), x)