3.11.10 \(\int x^6 \sqrt [6]{a+b x^2} \, dx\) [1010]

3.11.10.1 Optimal result

Integrand size = 15, antiderivative size = 345 \[ \int x^6 \sqrt [6]{a+b x^2} \, dx=\frac {81 a^3 x \sqrt [6]{a+b x^2}}{2816 b^3}-\frac {9 a^2 x^3 \sqrt [6]{a+b x^2}}{704 b^2}+\frac {3 a x^5 \sqrt [6]{a+b x^2}}{352 b}+\frac {3}{22} x^7 \sqrt [6]{a+b x^2}-\frac {81\ 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right ),-7+4 \sqrt {3}\right )}{2816 b^4 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}} \]

output

81/2816*a^3*x*(b*x^2+a)^(1/6)/b^3-9/704*a^2*x^3*(b*x^2+a)^(1/6)/b^2+3/352* 
a*x^5*(b*x^2+a)^(1/6)/b+3/22*x^7*(b*x^2+a)^(1/6)-81/2816*3^(3/4)*a^4*(b*x^ 
2+a)^(1/6)*(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))*(1/2*6^(1/2)-1/2*2^(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^4/x/(a/(b*x^2+a))^(1/3)/((-1+(a/(b*x^2+a))^(1/3))/(1-(a/(b 
*x^2+a))^(1/3)-3^(1/2))^2)^(1/2)
 

3.11.10.2 Mathematica [C] (verified)

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

Time = 7.62 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.30 \[ \int x^6 \sqrt [6]{a+b x^2} \, dx=\frac {3 x \sqrt [6]{a+b x^2} \left (\sqrt [6]{1+\frac {b x^2}{a}} \left (27 a^3-3 a^2 b x^2+2 a b^2 x^4+32 b^3 x^6\right )-27 a^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{704 b^3 \sqrt [6]{1+\frac {b x^2}{a}}} \]

input

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

output

(3*x*(a + b*x^2)^(1/6)*((1 + (b*x^2)/a)^(1/6)*(27*a^3 - 3*a^2*b*x^2 + 2*a* 
b^2*x^4 + 32*b^3*x^6) - 27*a^3*Hypergeometric2F1[-1/6, 1/2, 3/2, -((b*x^2) 
/a)]))/(704*b^3*(1 + (b*x^2)/a)^(1/6))
 

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

Time = 0.34 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.28, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {248, 262, 262, 262, 236, 234, 760}

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.

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

\(\Big \downarrow \) 248

\(\displaystyle \frac {1}{22} a \int \frac {x^6}{\left (b x^2+a\right )^{5/6}}dx+\frac {3}{22} x^7 \sqrt [6]{a+b x^2}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{22} a \left (\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {15 a \int \frac {x^4}{\left (b x^2+a\right )^{5/6}}dx}{16 b}\right )+\frac {3}{22} x^7 \sqrt [6]{a+b x^2}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{22} a \left (\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {15 a \left (\frac {3 x^3 \sqrt [6]{a+b x^2}}{10 b}-\frac {9 a \int \frac {x^2}{\left (b x^2+a\right )^{5/6}}dx}{10 b}\right )}{16 b}\right )+\frac {3}{22} x^7 \sqrt [6]{a+b x^2}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{22} a \left (\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {15 a \left (\frac {3 x^3 \sqrt [6]{a+b x^2}}{10 b}-\frac {9 a \left (\frac {3 x \sqrt [6]{a+b x^2}}{4 b}-\frac {3 a \int \frac {1}{\left (b x^2+a\right )^{5/6}}dx}{4 b}\right )}{10 b}\right )}{16 b}\right )+\frac {3}{22} x^7 \sqrt [6]{a+b x^2}\)

\(\Big \downarrow \) 236

\(\displaystyle \frac {1}{22} a \left (\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {15 a \left (\frac {3 x^3 \sqrt [6]{a+b x^2}}{10 b}-\frac {9 a \left (\frac {3 x \sqrt [6]{a+b x^2}}{4 b}-\frac {3 a \int \frac {1}{\left (1-\frac {b x^2}{b x^2+a}\right )^{2/3}}d\frac {x}{\sqrt {b x^2+a}}}{4 b \sqrt [3]{\frac {a}{a+b x^2}} \sqrt [3]{a+b x^2}}\right )}{10 b}\right )}{16 b}\right )+\frac {3}{22} x^7 \sqrt [6]{a+b x^2}\)

\(\Big \downarrow \) 234

\(\displaystyle \frac {1}{22} a \left (\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {15 a \left (\frac {3 x^3 \sqrt [6]{a+b x^2}}{10 b}-\frac {9 a \left (\frac {9 a \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{8 b^2 x \sqrt [3]{\frac {a}{a+b x^2}}}+\frac {3 x \sqrt [6]{a+b x^2}}{4 b}\right )}{10 b}\right )}{16 b}\right )+\frac {3}{22} x^7 \sqrt [6]{a+b x^2}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {1}{22} a \left (\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {15 a \left (\frac {3 x^3 \sqrt [6]{a+b x^2}}{10 b}-\frac {9 a \left (\frac {3 x \sqrt [6]{a+b x^2}}{4 b}-\frac {3\ 3^{3/4} \sqrt {2-\sqrt {3}} a \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{4 b^2 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}\right )}{10 b}\right )}{16 b}\right )+\frac {3}{22} x^7 \sqrt [6]{a+b x^2}\)

input

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

output

(3*x^7*(a + b*x^2)^(1/6))/22 + (a*((3*x^5*(a + b*x^2)^(1/6))/(16*b) - (15* 
a*((3*x^3*(a + b*x^2)^(1/6))/(10*b) - (9*a*((3*x*(a + b*x^2)^(1/6))/(4*b) 
- (3*3^(3/4)*Sqrt[2 - Sqrt[3]]*a*Sqrt[-((b*x^2)/(a + b*x^2))]*(a + b*x^2)^ 
(1/6)*(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]])/(4*b^2*x 
*(a/(a + b*x^2))^(1/3)*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)])))/(10*b)))/(16*b)))/22
 

3.11.10.3.1 Defintions of rubi rules used

rule 234

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

rule 236

Int[((a_) + (b_.)*(x_)^2)^(-5/6), x_Symbol] :> Simp[1/((a/(a + b*x^2))^(1/3 
)*(a + b*x^2)^(1/3))   Subst[Int[1/(1 - b*x^2)^(2/3), x], x, x/Sqrt[a + b*x 
^2]], x] /; FreeQ[{a, b}, x]
 

rule 248

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

3.11.10.4 Maple [F]

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

input

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

output

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

3.11.10.5 Fricas [F]

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

input

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

output

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

3.11.10.6 Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.08 \[ \int x^6 \sqrt [6]{a+b x^2} \, dx=\frac {\sqrt [6]{a} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7} \]

input

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

output

a**(1/6)*x**7*hyper((-1/6, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/a)/7
 

3.11.10.7 Maxima [F]

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

input

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

output

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

3.11.10.8 Giac [F]

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

input

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

output

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

3.11.10.9 Mupad [F(-1)]

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

input

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

output

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