\(\int \frac {x^4 (c+d x^6)}{(a+b x^6)^{3/2}} \, dx\) [20]

Optimal result

Integrand size = 22, antiderivative size = 562 \[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {(b c-a d) x^5}{3 a b \sqrt {a+b x^6}}-\frac {\left (1+\sqrt {3}\right ) (2 b c-5 a d) x \sqrt {a+b x^6}}{6 a b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )}+\frac {(2 b c-5 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 3^{3/4} a^{2/3} b^{5/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}+\frac {\left (1-\sqrt {3}\right ) (2 b c-5 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{12 \sqrt [4]{3} a^{2/3} b^{5/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

1/3*(-a*d+b*c)*x^5/a/b/(b*x^6+a)^(1/2)-1/6*(1+3^(1/2))*(-5*a*d+2*b*c)*x*(b 
*x^6+a)^(1/2)/a/b^(5/3)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)+1/6*(-5*a*d+2*b* 
c)*x*(a^(1/3)+b^(1/3)*x^2)*((a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(a^( 
1/3)+(1+3^(1/2))*b^(1/3)*x^2)^2)^(1/2)*EllipticE((1-(a^(1/3)+(1-3^(1/2))*b 
^(1/3)*x^2)^2/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)^2)^(1/2),1/4*6^(1/2)+1/4*2 
^(1/2))*3^(1/4)/a^(2/3)/b^(5/3)/(b^(1/3)*x^2*(a^(1/3)+b^(1/3)*x^2)/(a^(1/3 
)+(1+3^(1/2))*b^(1/3)*x^2)^2)^(1/2)/(b*x^6+a)^(1/2)+1/36*(1-3^(1/2))*(-5*a 
*d+2*b*c)*x*(a^(1/3)+b^(1/3)*x^2)*((a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^ 
4)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)^2)^(1/2)*InverseJacobiAM(arccos((a^(1 
/3)+(1-3^(1/2))*b^(1/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)),1/4*6^(1/2 
)+1/4*2^(1/2))*3^(3/4)/a^(2/3)/b^(5/3)/(b^(1/3)*x^2*(a^(1/3)+b^(1/3)*x^2)/ 
(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)^2)^(1/2)/(b*x^6+a)^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.13 \[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {x^5 \left (5 a d+(2 b c-5 a d) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {3}{2},\frac {11}{6},-\frac {b x^6}{a}\right )\right )}{10 a b \sqrt {a+b x^6}} \] Input:

Integrate[(x^4*(c + d*x^6))/(a + b*x^6)^(3/2),x]
 

Output:

(x^5*(5*a*d + (2*b*c - 5*a*d)*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[5/6, 3 
/2, 11/6, -((b*x^6)/a)]))/(10*a*b*Sqrt[a + b*x^6])
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {959, 819, 837, 25, 766, 2420}

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.

\(\displaystyle \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 819

\(\displaystyle \frac {(2 b c-5 a d) \left (\frac {x^5}{3 a \sqrt {a+b x^6}}-\frac {2 \int \frac {x^4}{\sqrt {b x^6+a}}dx}{3 a}\right )}{2 b}+\frac {d x^5}{2 b \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 837

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 766

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

\(\Big \downarrow \) 2420

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

Input:

Int[(x^4*(c + d*x^6))/(a + b*x^6)^(3/2),x]
 

Output:

(d*x^5)/(2*b*Sqrt[a + b*x^6]) + ((2*b*c - 5*a*d)*(x^5/(3*a*Sqrt[a + b*x^6] 
) - (2*((((1 + Sqrt[3])*x*Sqrt[a + b*x^6])/(a^(1/3) + (1 + Sqrt[3])*b^(1/3 
)*x^2) - (3^(1/4)*a^(1/3)*x*(a^(1/3) + b^(1/3)*x^2)*Sqrt[(a^(2/3) - a^(1/3 
)*b^(1/3)*x^2 + b^(2/3)*x^4)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)^2]*Elli 
pticE[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x^2)/(a^(1/3) + (1 + Sqrt[3] 
)*b^(1/3)*x^2)], (2 + Sqrt[3])/4])/(Sqrt[(b^(1/3)*x^2*(a^(1/3) + b^(1/3)*x 
^2))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)^2]*Sqrt[a + b*x^6]))/(2*b^(2/3) 
) - ((1 - Sqrt[3])*a^(1/3)*x*(a^(1/3) + b^(1/3)*x^2)*Sqrt[(a^(2/3) - a^(1/ 
3)*b^(1/3)*x^2 + b^(2/3)*x^4)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)^2]*Ell 
ipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x^2)/(a^(1/3) + (1 + Sqrt[3 
])*b^(1/3)*x^2)], (2 + Sqrt[3])/4])/(4*3^(1/4)*b^(2/3)*Sqrt[(b^(1/3)*x^2*( 
a^(1/3) + b^(1/3)*x^2))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)^2]*Sqrt[a + 
b*x^6])))/(3*a)))/(2*b)
 

Defintions of rubi rules used

rule 25

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

rule 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

rule 819

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

rule 837

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 
3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2))   Int[1/Sqrt[ 
a + b*x^6], x], x] - Simp[1/(2*r^2)   Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S 
qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
 

rule 959

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

rule 2420

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = 
 Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr 
t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* 
(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 
*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) 
)*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 
 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 
- Sqrt[3])*d, 0]
 

Maple [F]

\[\int \frac {x^{4} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {3}{2}}}d x\]

Input:

int(x^4*(d*x^6+c)/(b*x^6+a)^(3/2),x)
 

Output:

int(x^4*(d*x^6+c)/(b*x^6+a)^(3/2),x)
 

Fricas [F]

\[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{4}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="fricas")
 

Output:

integral((d*x^10 + c*x^4)*sqrt(b*x^6 + a)/(b^2*x^12 + 2*a*b*x^6 + a^2), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 15.00 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.14 \[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {c x^{5} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{6}, \frac {3}{2} \\ \frac {11}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {11}{6}\right )} + \frac {d x^{11} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {17}{6}\right )} \] Input:

integrate(x**4*(d*x**6+c)/(b*x**6+a)**(3/2),x)
 

Output:

c*x**5*gamma(5/6)*hyper((5/6, 3/2), (11/6,), b*x**6*exp_polar(I*pi)/a)/(6* 
a**(3/2)*gamma(11/6)) + d*x**11*gamma(11/6)*hyper((3/2, 11/6), (17/6,), b* 
x**6*exp_polar(I*pi)/a)/(6*a**(3/2)*gamma(17/6))
 

Maxima [F]

\[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{4}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)*x^4/(b*x^6 + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{4}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)*x^4/(b*x^6 + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int \frac {x^4\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{3/2}} \,d x \] Input:

int((x^4*(c + d*x^6))/(a + b*x^6)^(3/2),x)
 

Output:

int((x^4*(c + d*x^6))/(a + b*x^6)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {\sqrt {b \,x^{6}+a}\, d \,x^{5}-5 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{2} x^{12}+2 a b \,x^{6}+a^{2}}d x \right ) a^{2} d +2 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{2} x^{12}+2 a b \,x^{6}+a^{2}}d x \right ) a b c -5 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{2} x^{12}+2 a b \,x^{6}+a^{2}}d x \right ) a b d \,x^{6}+2 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{2} x^{12}+2 a b \,x^{6}+a^{2}}d x \right ) b^{2} c \,x^{6}}{2 b \left (b \,x^{6}+a \right )} \] Input:

int(x^4*(d*x^6+c)/(b*x^6+a)^(3/2),x)
 

Output:

(sqrt(a + b*x**6)*d*x**5 - 5*int((sqrt(a + b*x**6)*x**4)/(a**2 + 2*a*b*x** 
6 + b**2*x**12),x)*a**2*d + 2*int((sqrt(a + b*x**6)*x**4)/(a**2 + 2*a*b*x* 
*6 + b**2*x**12),x)*a*b*c - 5*int((sqrt(a + b*x**6)*x**4)/(a**2 + 2*a*b*x* 
*6 + b**2*x**12),x)*a*b*d*x**6 + 2*int((sqrt(a + b*x**6)*x**4)/(a**2 + 2*a 
*b*x**6 + b**2*x**12),x)*b**2*c*x**6)/(2*b*(a + b*x**6))