\(\int \frac {c+d x^6}{x^6 (a+b x^6)^{3/2}} \, dx\) [17]

Optimal result

Integrand size = 22, antiderivative size = 274 \[ \int \frac {c+d x^6}{x^6 \left (a+b x^6\right )^{3/2}} \, dx=-\frac {c}{5 a x^5 \sqrt {a+b x^6}}-\frac {(8 b c-5 a d) x}{15 a^2 \sqrt {a+b x^6}}-\frac {(8 b c-5 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{15 \sqrt [4]{3} a^{7/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/5*c/a/x^5/(b*x^6+a)^(1/2)-1/15*(-5*a*d+8*b*c)*x/a^2/(b*x^6+a)^(1/2)-1/4 
5*(-5*a*d+8*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(arcco 
s((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^(7/3)/(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 = 10.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.32 \[ \int \frac {c+d x^6}{x^6 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-3 a c-8 b c x^6+5 a d x^6+2 (-8 b c+5 a d) x^6 \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {b x^6}{a}\right )}{15 a^2 x^5 \sqrt {a+b x^6}} \] Input:

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

Output:

(-3*a*c - 8*b*c*x^6 + 5*a*d*x^6 + 2*(-8*b*c + 5*a*d)*x^6*Sqrt[1 + (b*x^6)/ 
a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^6)/a)])/(15*a^2*x^5*Sqrt[a + b* 
x^6])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {955, 749, 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.

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

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 749

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

\(\Big \downarrow \) 766

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

Input:

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

Output:

-1/5*c/(a*x^5*Sqrt[a + b*x^6]) - ((8*b*c - 5*a*d)*(x/(3*a*Sqrt[a + b*x^6]) 
 + (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])/(3*3^(1/4)*a^(4/3)*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])))/(5*a)
 

Defintions of rubi rules used

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

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 955

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x^6 + a)*(d*x^6 + c)/(b^2*x^18 + 2*a*b*x^12 + a^2*x^6), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 34.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.30 \[ \int \frac {c+d x^6}{x^6 \left (a+b x^6\right )^{3/2}} \, dx=\frac {c \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {3}{2} \\ \frac {1}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{5} \Gamma \left (\frac {1}{6}\right )} + \frac {d x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{2} \\ \frac {7}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {7}{6}\right )} \] Input:

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

Output:

c*gamma(-5/6)*hyper((-5/6, 3/2), (1/6,), b*x**6*exp_polar(I*pi)/a)/(6*a**( 
3/2)*x**5*gamma(1/6)) + d*x*gamma(1/6)*hyper((1/6, 3/2), (7/6,), b*x**6*ex 
p_polar(I*pi)/a)/(6*a**(3/2)*gamma(7/6))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {c+d x^6}{x^6 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d -5 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{18}+2 a b \,x^{12}+a^{2} x^{6}}d x \right ) a^{2} d \,x^{5}+8 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{18}+2 a b \,x^{12}+a^{2} x^{6}}d x \right ) a b c \,x^{5}-5 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{18}+2 a b \,x^{12}+a^{2} x^{6}}d x \right ) a b d \,x^{11}+8 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{18}+2 a b \,x^{12}+a^{2} x^{6}}d x \right ) b^{2} c \,x^{11}}{8 b \,x^{5} \left (b \,x^{6}+a \right )} \] Input:

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

Output:

( - sqrt(a + b*x**6)*d - 5*int(sqrt(a + b*x**6)/(a**2*x**6 + 2*a*b*x**12 + 
 b**2*x**18),x)*a**2*d*x**5 + 8*int(sqrt(a + b*x**6)/(a**2*x**6 + 2*a*b*x* 
*12 + b**2*x**18),x)*a*b*c*x**5 - 5*int(sqrt(a + b*x**6)/(a**2*x**6 + 2*a* 
b*x**12 + b**2*x**18),x)*a*b*d*x**11 + 8*int(sqrt(a + b*x**6)/(a**2*x**6 + 
 2*a*b*x**12 + b**2*x**18),x)*b**2*c*x**11)/(8*b*x**5*(a + b*x**6))