\(\int \frac {x (c+d x^6)}{(a+b x^6)^{3/2}} \, dx\) [12]

Optimal result

Integrand size = 20, antiderivative size = 267 \[ \int \frac {x \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {(b c-a d) x^2}{3 a b \sqrt {a+b x^6}}+\frac {\sqrt {2+\sqrt {3}} (b c+2 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

1/3*(-a*d+b*c)*x^2/a/b/(b*x^6+a)^(1/2)+1/9*(1/2*6^(1/2)+1/2*2^(1/2))*(2*a* 
d+b*c)*(a^(1/3)+b^(1/3)*x^2)*((a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(( 
1+3^(1/2))*a^(1/3)+b^(1/3)*x^2)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^ 
(1/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^2),I*3^(1/2)+2*I)*3^(3/4)/a/b^(4 
/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x^2)/((1+3^(1/2))*a^(1/3)+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 = 75, normalized size of antiderivative = 0.28 \[ \int \frac {x \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {x^2 \left (2 b c-2 a d+(b c+2 a d) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^6}{a}\right )\right )}{6 a b \sqrt {a+b x^6}} \] Input:

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

Output:

(x^2*(2*b*c - 2*a*d + (b*c + 2*a*d)*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[ 
1/3, 1/2, 4/3, -((b*x^6)/a)]))/(6*a*b*Sqrt[a + b*x^6])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {957, 807, 759}

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.

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

\(\Big \downarrow \) 957

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 759

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

Input:

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

Output:

((b*c - a*d)*x^2)/(3*a*b*Sqrt[a + b*x^6]) + (Sqrt[2 + Sqrt[3]]*(b*c + 2*a* 
d)*(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)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)^2]*EllipticF[ArcSin[((1 - Sqrt[3 
])*a^(1/3) + b^(1/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)], -7 - 4*S 
qrt[3]])/(3*3^(1/4)*a*b^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x^2))/((1 + 
 Sqrt[3])*a^(1/3) + b^(1/3)*x^2)^2]*Sqrt[a + b*x^6])
 

Defintions of rubi rules used

rule 759

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] & 
& PosQ[a]
 

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 957

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a 
*b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* 
(p + 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.32 \[ \int \frac {x \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {\sqrt {b x^{6} + a} {\left (b^{2} c - a b d\right )} x^{2} + {\left ({\left (b^{2} c + 2 \, a b d\right )} x^{6} + a b c + 2 \, a^{2} d\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right )}{3 \, {\left (a b^{3} x^{6} + a^{2} b^{2}\right )}} \] Input:

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

Output:

1/3*(sqrt(b*x^6 + a)*(b^2*c - a*b*d)*x^2 + ((b^2*c + 2*a*b*d)*x^6 + a*b*c 
+ 2*a^2*d)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x^2))/(a*b^3*x^6 + a^2*b 
^2)
 

Sympy [A] (verification not implemented)

Time = 7.90 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.30 \[ \int \frac {x \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {c x^{2} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{8} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {3}{2} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {7}{3}\right )} \] Input:

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

Output:

c*x**2*gamma(1/3)*hyper((1/3, 3/2), (4/3,), b*x**6*exp_polar(I*pi)/a)/(6*a 
**(3/2)*gamma(4/3)) + d*x**8*gamma(4/3)*hyper((4/3, 3/2), (7/3,), b*x**6*e 
xp_polar(I*pi)/a)/(6*a**(3/2)*gamma(7/3))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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