\(\int \frac {c+d x^6}{(a+b x^6)^{3/2}} \, dx\) [16]

Optimal result

Integrand size = 19, antiderivative size = 256 \[ \int \frac {c+d x^6}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {(b c-a d) x}{3 a b \sqrt {a+b x^6}}+\frac {(2 b c+a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6 \sqrt [4]{3} a^{4/3} b \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

1/3*(-a*d+b*c)*x/a/b/(b*x^6+a)^(1/2)+1/18*(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^(4/3) 
/b/(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 = 10.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.28 \[ \int \frac {c+d x^6}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {x \left (b c-a d+(2 b c+a d) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {b x^6}{a}\right )\right )}{3 a b \sqrt {a+b x^6}} \] Input:

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

Output:

(x*(b*c - a*d + (2*b*c + a*d)*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[1/6, 1 
/2, 7/6, -((b*x^6)/a)]))/(3*a*b*Sqrt[a + b*x^6])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {910, 766}

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.

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

\(\Big \downarrow \) 910

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

\(\Big \downarrow \) 766

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

Input:

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

Output:

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

Defintions of rubi rules used

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 910

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - 
 b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1), x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ 
n + p, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x^6 + a)*(d*x^6 + c)/(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 = 7.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.30 \[ \int \frac {c+d x^6}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {c x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{2} \\ \frac {7}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {7}{6}\right )} + \frac {d x^{7} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{6}, \frac {3}{2} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {13}{6}\right )} \] Input:

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

Output:

c*x*gamma(1/6)*hyper((1/6, 3/2), (7/6,), b*x**6*exp_polar(I*pi)/a)/(6*a**( 
3/2)*gamma(7/6)) + d*x**7*gamma(7/6)*hyper((7/6, 3/2), (13/6,), b*x**6*exp 
_polar(I*pi)/a)/(6*a**(3/2)*gamma(13/6))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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