\(\int \frac {c+d x^6}{x^{11} (a+b x^6)^{3/2}} \, dx\) [14]

Optimal result

Integrand size = 22, antiderivative size = 318 \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{3/2}} \, dx=-\frac {c}{10 a x^{10} \sqrt {a+b x^6}}-\frac {13 b c-10 a d}{30 a^2 x^4 \sqrt {a+b x^6}}+\frac {7 (13 b c-10 a d) \sqrt {a+b x^6}}{120 a^3 x^4}+\frac {7 \sqrt {2+\sqrt {3}} b^{2/3} (13 b c-10 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{120 \sqrt [4]{3} a^3 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/10*c/a/x^10/(b*x^6+a)^(1/2)-1/30*(-10*a*d+13*b*c)/a^2/x^4/(b*x^6+a)^(1/ 
2)+7/120*(-10*a*d+13*b*c)*(b*x^6+a)^(1/2)/a^3/x^4+7/360*(1/2*6^(1/2)+1/2*2 
^(1/2))*b^(2/3)*(-10*a*d+13*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)*Ellipti 
cF(((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^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 = 10.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.23 \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{3/2}} \, dx=\frac {-4 a c+(13 b c-10 a d) x^6 \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {3}{2},\frac {1}{3},-\frac {b x^6}{a}\right )}{40 a^2 x^{10} \sqrt {a+b x^6}} \] Input:

Integrate[(c + d*x^6)/(x^11*(a + b*x^6)^(3/2)),x]
 

Output:

(-4*a*c + (13*b*c - 10*a*d)*x^6*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[-2/3 
, 3/2, 1/3, -((b*x^6)/a)])/(40*a^2*x^10*Sqrt[a + b*x^6])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {955, 807, 819, 847, 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.

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

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(13 b c-10 a d) \int \frac {1}{x^5 \left (b x^6+a\right )^{3/2}}dx}{10 a}-\frac {c}{10 a x^{10} \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 807

\(\displaystyle -\frac {(13 b c-10 a d) \int \frac {1}{x^6 \left (b x^6+a\right )^{3/2}}dx^2}{20 a}-\frac {c}{10 a x^{10} \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(13 b c-10 a d) \left (\frac {7 \int \frac {1}{x^6 \sqrt {b x^6+a}}dx^2}{3 a}+\frac {2}{3 a x^4 \sqrt {a+b x^6}}\right )}{20 a}-\frac {c}{10 a x^{10} \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {(13 b c-10 a d) \left (\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {b x^6+a}}dx^2}{4 a}-\frac {\sqrt {a+b x^6}}{2 a x^4}\right )}{3 a}+\frac {2}{3 a x^4 \sqrt {a+b x^6}}\right )}{20 a}-\frac {c}{10 a x^{10} \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {(13 b c-10 a d) \left (\frac {7 \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {\sqrt {a+b x^6}}{2 a x^4}\right )}{3 a}+\frac {2}{3 a x^4 \sqrt {a+b x^6}}\right )}{20 a}-\frac {c}{10 a x^{10} \sqrt {a+b x^6}}\)

Input:

Int[(c + d*x^6)/(x^11*(a + b*x^6)^(3/2)),x]
 

Output:

-1/10*c/(a*x^10*Sqrt[a + b*x^6]) - ((13*b*c - 10*a*d)*(2/(3*a*x^4*Sqrt[a + 
 b*x^6]) + (7*(-1/2*Sqrt[a + b*x^6]/(a*x^4) - (Sqrt[2 + Sqrt[3]]*b^(2/3)*( 
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*Sqrt[ 
3]])/(2*3^(1/4)*a*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])))/(3*a)))/(20*a)
 

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 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]

\[\int \frac {d \,x^{6}+c}{x^{11} \left (b \,x^{6}+a \right )^{\frac {3}{2}}}d x\]

Input:

int((d*x^6+c)/x^11/(b*x^6+a)^(3/2),x)
 

Output:

int((d*x^6+c)/x^11/(b*x^6+a)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.38 \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{3/2}} \, dx=\frac {7 \, {\left ({\left (13 \, b^{2} c - 10 \, a b d\right )} x^{16} + {\left (13 \, a b c - 10 \, a^{2} d\right )} x^{10}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right ) + {\left (7 \, {\left (13 \, b^{2} c - 10 \, a b d\right )} x^{12} + 3 \, {\left (13 \, a b c - 10 \, a^{2} d\right )} x^{6} - 12 \, a^{2} c\right )} \sqrt {b x^{6} + a}}{120 \, {\left (a^{3} b x^{16} + a^{4} x^{10}\right )}} \] Input:

integrate((d*x^6+c)/x^11/(b*x^6+a)^(3/2),x, algorithm="fricas")
 

Output:

1/120*(7*((13*b^2*c - 10*a*b*d)*x^16 + (13*a*b*c - 10*a^2*d)*x^10)*sqrt(b) 
*weierstrassPInverse(0, -4*a/b, x^2) + (7*(13*b^2*c - 10*a*b*d)*x^12 + 3*( 
13*a*b*c - 10*a^2*d)*x^6 - 12*a^2*c)*sqrt(b*x^6 + a))/(a^3*b*x^16 + a^4*x^ 
10)
 

Sympy [A] (verification not implemented)

Time = 128.87 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.28 \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{3/2}} \, dx=\frac {c \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {3}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{10} \Gamma \left (- \frac {2}{3}\right )} + \frac {d \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{4} \Gamma \left (\frac {1}{3}\right )} \] Input:

integrate((d*x**6+c)/x**11/(b*x**6+a)**(3/2),x)
 

Output:

c*gamma(-5/3)*hyper((-5/3, 3/2), (-2/3,), b*x**6*exp_polar(I*pi)/a)/(6*a** 
(3/2)*x**10*gamma(-2/3)) + d*gamma(-2/3)*hyper((-2/3, 3/2), (1/3,), b*x**6 
*exp_polar(I*pi)/a)/(6*a**(3/2)*x**4*gamma(1/3))
 

Maxima [F]

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

integrate((d*x^6+c)/x^11/(b*x^6+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(3/2)*x^11), x)
 

Giac [F]

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

integrate((d*x^6+c)/x^11/(b*x^6+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(3/2)*x^11), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{3/2}} \, dx=\int \frac {d\,x^6+c}{x^{11}\,{\left (b\,x^6+a\right )}^{3/2}} \,d x \] Input:

int((c + d*x^6)/(x^11*(a + b*x^6)^(3/2)),x)
 

Output:

int((c + d*x^6)/(x^11*(a + b*x^6)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d -10 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{23}+2 a b \,x^{17}+a^{2} x^{11}}d x \right ) a^{2} d \,x^{10}+13 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{23}+2 a b \,x^{17}+a^{2} x^{11}}d x \right ) a b c \,x^{10}-10 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{23}+2 a b \,x^{17}+a^{2} x^{11}}d x \right ) a b d \,x^{16}+13 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{23}+2 a b \,x^{17}+a^{2} x^{11}}d x \right ) b^{2} c \,x^{16}}{13 b \,x^{10} \left (b \,x^{6}+a \right )} \] Input:

int((d*x^6+c)/x^11/(b*x^6+a)^(3/2),x)
 

Output:

( - sqrt(a + b*x**6)*d - 10*int(sqrt(a + b*x**6)/(a**2*x**11 + 2*a*b*x**17 
 + b**2*x**23),x)*a**2*d*x**10 + 13*int(sqrt(a + b*x**6)/(a**2*x**11 + 2*a 
*b*x**17 + b**2*x**23),x)*a*b*c*x**10 - 10*int(sqrt(a + b*x**6)/(a**2*x**1 
1 + 2*a*b*x**17 + b**2*x**23),x)*a*b*d*x**16 + 13*int(sqrt(a + b*x**6)/(a* 
*2*x**11 + 2*a*b*x**17 + b**2*x**23),x)*b**2*c*x**16)/(13*b*x**10*(a + b*x 
**6))