\(\int \frac {c+d x^6}{x^{12} (a+b x^6)^{3/2}} \, dx\) [18]

Optimal result

Integrand size = 22, antiderivative size = 307 \[ \int \frac {c+d x^6}{x^{12} \left (a+b x^6\right )^{3/2}} \, dx=-\frac {c}{11 a x^{11} \sqrt {a+b x^6}}-\frac {14 b c-11 a d}{33 a^2 x^5 \sqrt {a+b x^6}}+\frac {8 (14 b c-11 a d) \sqrt {a+b x^6}}{165 a^3 x^5}+\frac {8 b (14 b c-11 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{165 \sqrt [4]{3} a^{10/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/11*c/a/x^11/(b*x^6+a)^(1/2)-1/33*(-11*a*d+14*b*c)/a^2/x^5/(b*x^6+a)^(1/ 
2)+8/165*(-11*a*d+14*b*c)*(b*x^6+a)^(1/2)/a^3/x^5+8/495*b*(-11*a*d+14*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^(10/3)/(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 = 72, normalized size of antiderivative = 0.23 \[ \int \frac {c+d x^6}{x^{12} \left (a+b x^6\right )^{3/2}} \, dx=\frac {-5 a c+(14 b c-11 a d) x^6 \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {3}{2},\frac {1}{6},-\frac {b x^6}{a}\right )}{55 a^2 x^{11} \sqrt {a+b x^6}} \] Input:

Integrate[(c + d*x^6)/(x^12*(a + b*x^6)^(3/2)),x]
 

Output:

(-5*a*c + (14*b*c - 11*a*d)*x^6*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[-5/6 
, 3/2, 1/6, -((b*x^6)/a)])/(55*a^2*x^11*Sqrt[a + b*x^6])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {955, 819, 847, 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.

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

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(14 b c-11 a d) \int \frac {1}{x^6 \left (b x^6+a\right )^{3/2}}dx}{11 a}-\frac {c}{11 a x^{11} \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(14 b c-11 a d) \left (\frac {8 \int \frac {1}{x^6 \sqrt {b x^6+a}}dx}{3 a}+\frac {1}{3 a x^5 \sqrt {a+b x^6}}\right )}{11 a}-\frac {c}{11 a x^{11} \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {(14 b c-11 a d) \left (\frac {8 \left (-\frac {2 b \int \frac {1}{\sqrt {b x^6+a}}dx}{5 a}-\frac {\sqrt {a+b x^6}}{5 a x^5}\right )}{3 a}+\frac {1}{3 a x^5 \sqrt {a+b x^6}}\right )}{11 a}-\frac {c}{11 a x^{11} \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 766

\(\displaystyle -\frac {(14 b c-11 a d) \left (\frac {8 \left (-\frac {b x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt [4]{3} a^{4/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {\sqrt {a+b x^6}}{5 a x^5}\right )}{3 a}+\frac {1}{3 a x^5 \sqrt {a+b x^6}}\right )}{11 a}-\frac {c}{11 a x^{11} \sqrt {a+b x^6}}\)

Input:

Int[(c + d*x^6)/(x^12*(a + b*x^6)^(3/2)),x]
 

Output:

-1/11*c/(a*x^11*Sqrt[a + b*x^6]) - ((14*b*c - 11*a*d)*(1/(3*a*x^5*Sqrt[a + 
 b*x^6]) + (8*(-1/5*Sqrt[a + b*x^6]/(a*x^5) - (b*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])/(5*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])))/(3*a)))/(11*a)
 

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 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 847

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

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

Input:

int((d*x^6+c)/x^12/(b*x^6+a)^(3/2),x)
 

Output:

int((d*x^6+c)/x^12/(b*x^6+a)^(3/2),x)
 

Fricas [F]

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

integrate((d*x^6+c)/x^12/(b*x^6+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x^6 + a)*(d*x^6 + c)/(b^2*x^24 + 2*a*b*x^18 + a^2*x^12), x 
)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 143.67 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.29 \[ \int \frac {c+d x^6}{x^{12} \left (a+b x^6\right )^{3/2}} \, dx=\frac {c \Gamma \left (- \frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{6}, \frac {3}{2} \\ - \frac {5}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{11} \Gamma \left (- \frac {5}{6}\right )} + \frac {d \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {3}{2} \\ \frac {1}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{5} \Gamma \left (\frac {1}{6}\right )} \] Input:

integrate((d*x**6+c)/x**12/(b*x**6+a)**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate((d*x^6+c)/x^12/(b*x^6+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(3/2)*x^12), x)
 

Giac [F]

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

integrate((d*x^6+c)/x^12/(b*x^6+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(3/2)*x^12), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((c + d*x^6)/(x^12*(a + b*x^6)^(3/2)),x)
 

Output:

int((c + d*x^6)/(x^12*(a + b*x^6)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {c+d x^6}{x^{12} \left (a+b x^6\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d -11 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{24}+2 a b \,x^{18}+a^{2} x^{12}}d x \right ) a^{2} d \,x^{11}+14 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{24}+2 a b \,x^{18}+a^{2} x^{12}}d x \right ) a b c \,x^{11}-11 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{24}+2 a b \,x^{18}+a^{2} x^{12}}d x \right ) a b d \,x^{17}+14 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{24}+2 a b \,x^{18}+a^{2} x^{12}}d x \right ) b^{2} c \,x^{17}}{14 b \,x^{11} \left (b \,x^{6}+a \right )} \] Input:

int((d*x^6+c)/x^12/(b*x^6+a)^(3/2),x)
 

Output:

( - sqrt(a + b*x**6)*d - 11*int(sqrt(a + b*x**6)/(a**2*x**12 + 2*a*b*x**18 
 + b**2*x**24),x)*a**2*d*x**11 + 14*int(sqrt(a + b*x**6)/(a**2*x**12 + 2*a 
*b*x**18 + b**2*x**24),x)*a*b*c*x**11 - 11*int(sqrt(a + b*x**6)/(a**2*x**1 
2 + 2*a*b*x**18 + b**2*x**24),x)*a*b*d*x**17 + 14*int(sqrt(a + b*x**6)/(a* 
*2*x**12 + 2*a*b*x**18 + b**2*x**24),x)*b**2*c*x**17)/(14*b*x**11*(a + b*x 
**6))