\(\int \frac {c+d x^6}{x^{11} (a+b x^6)^{5/2}} \, dx\) [41]

Optimal result

Integrand size = 22, antiderivative size = 348 \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{5/2}} \, dx=-\frac {c}{10 a x^{10} \left (a+b x^6\right )^{3/2}}-\frac {19 b c-10 a d}{90 a^2 x^4 \left (a+b x^6\right )^{3/2}}-\frac {13 (19 b c-10 a d)}{270 a^3 x^4 \sqrt {a+b x^6}}+\frac {91 (19 b c-10 a d) \sqrt {a+b x^6}}{1080 a^4 x^4}+\frac {91 \sqrt {2+\sqrt {3}} b^{2/3} (19 b c-10 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{1080 \sqrt [4]{3} a^4 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/10*c/a/x^10/(b*x^6+a)^(3/2)-1/90*(-10*a*d+19*b*c)/a^2/x^4/(b*x^6+a)^(3/ 
2)-13/270*(-10*a*d+19*b*c)/a^3/x^4/(b*x^6+a)^(1/2)+91/1080*(-10*a*d+19*b*c 
)*(b*x^6+a)^(1/2)/a^4/x^4+91/3240*(1/2*6^(1/2)+1/2*2^(1/2))*b^(2/3)*(-10*a 
*d+19*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^ 
4/(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 = 10.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.23 \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{5/2}} \, dx=\frac {-4 a^2 c+(19 b c-10 a d) x^6 \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {5}{2},\frac {1}{3},-\frac {b x^6}{a}\right )}{40 a^3 x^{10} \left (a+b x^6\right )^{3/2}} \] Input:

Integrate[(c + d*x^6)/(x^11*(a + b*x^6)^(5/2)),x]
 

Output:

(-4*a^2*c + (19*b*c - 10*a*d)*x^6*(a + b*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeo 
metric2F1[-2/3, 5/2, 1/3, -((b*x^6)/a)])/(40*a^3*x^10*(a + b*x^6)^(3/2))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {955, 807, 819, 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.

\(\displaystyle \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(19 b c-10 a d) \int \frac {1}{x^5 \left (b x^6+a\right )^{5/2}}dx}{10 a}-\frac {c}{10 a x^{10} \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 807

\(\displaystyle -\frac {(19 b c-10 a d) \int \frac {1}{x^6 \left (b x^6+a\right )^{5/2}}dx^2}{20 a}-\frac {c}{10 a x^{10} \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(19 b c-10 a d) \left (\frac {13 \int \frac {1}{x^6 \left (b x^6+a\right )^{3/2}}dx^2}{9 a}+\frac {2}{9 a x^4 \left (a+b x^6\right )^{3/2}}\right )}{20 a}-\frac {c}{10 a x^{10} \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(19 b c-10 a d) \left (\frac {13 \left (\frac {7 \int \frac {1}{x^6 \sqrt {b x^6+a}}dx^2}{3 a}+\frac {2}{3 a x^4 \sqrt {a+b x^6}}\right )}{9 a}+\frac {2}{9 a x^4 \left (a+b x^6\right )^{3/2}}\right )}{20 a}-\frac {c}{10 a x^{10} \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {(19 b c-10 a d) \left (\frac {13 \left (\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {b x^6+a}}dx^2}{4 a}-\frac {\sqrt {a+b x^6}}{2 a x^4}\right )}{3 a}+\frac {2}{3 a x^4 \sqrt {a+b x^6}}\right )}{9 a}+\frac {2}{9 a x^4 \left (a+b x^6\right )^{3/2}}\right )}{20 a}-\frac {c}{10 a x^{10} \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {(19 b c-10 a d) \left (\frac {13 \left (\frac {7 \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {\sqrt {a+b x^6}}{2 a x^4}\right )}{3 a}+\frac {2}{3 a x^4 \sqrt {a+b x^6}}\right )}{9 a}+\frac {2}{9 a x^4 \left (a+b x^6\right )^{3/2}}\right )}{20 a}-\frac {c}{10 a x^{10} \left (a+b x^6\right )^{3/2}}\)

Input:

Int[(c + d*x^6)/(x^11*(a + b*x^6)^(5/2)),x]
 

Output:

-1/10*c/(a*x^10*(a + b*x^6)^(3/2)) - ((19*b*c - 10*a*d)*(2/(9*a*x^4*(a + b 
*x^6)^(3/2)) + (13*(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)))/(9*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]

\[\int \frac {d \,x^{6}+c}{x^{11} \left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

int((d*x^6+c)/x^11/(b*x^6+a)^(5/2),x)
 

Output:

int((d*x^6+c)/x^11/(b*x^6+a)^(5/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.52 \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{5/2}} \, dx=\frac {91 \, {\left ({\left (19 \, b^{3} c - 10 \, a b^{2} d\right )} x^{22} + 2 \, {\left (19 \, a b^{2} c - 10 \, a^{2} b d\right )} x^{16} + {\left (19 \, a^{2} b c - 10 \, a^{3} d\right )} x^{10}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right ) + {\left (91 \, {\left (19 \, b^{3} c - 10 \, a b^{2} d\right )} x^{18} + 130 \, {\left (19 \, a b^{2} c - 10 \, a^{2} b d\right )} x^{12} + 27 \, {\left (19 \, a^{2} b c - 10 \, a^{3} d\right )} x^{6} - 108 \, a^{3} c\right )} \sqrt {b x^{6} + a}}{1080 \, {\left (a^{4} b^{2} x^{22} + 2 \, a^{5} b x^{16} + a^{6} x^{10}\right )}} \] Input:

integrate((d*x^6+c)/x^11/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

1/1080*(91*((19*b^3*c - 10*a*b^2*d)*x^22 + 2*(19*a*b^2*c - 10*a^2*b*d)*x^1 
6 + (19*a^2*b*c - 10*a^3*d)*x^10)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x 
^2) + (91*(19*b^3*c - 10*a*b^2*d)*x^18 + 130*(19*a*b^2*c - 10*a^2*b*d)*x^1 
2 + 27*(19*a^2*b*c - 10*a^3*d)*x^6 - 108*a^3*c)*sqrt(b*x^6 + a))/(a^4*b^2* 
x^22 + 2*a^5*b*x^16 + a^6*x^10)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x**6+c)/x**11/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((d*x^6+c)/x^11/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(5/2)*x^11), x)
 

Giac [F]

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

integrate((d*x^6+c)/x^11/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(5/2)*x^11), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{5/2}} \, dx=\int \frac {d\,x^6+c}{x^{11}\,{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((c + d*x^6)/(x^11*(a + b*x^6)^(5/2)),x)
 

Output:

int((c + d*x^6)/(x^11*(a + b*x^6)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {c+d x^6}{x^{11} \left (a+b x^6\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d -10 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{29}+3 a \,b^{2} x^{23}+3 a^{2} b \,x^{17}+a^{3} x^{11}}d x \right ) a^{3} d \,x^{10}+19 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{29}+3 a \,b^{2} x^{23}+3 a^{2} b \,x^{17}+a^{3} x^{11}}d x \right ) a^{2} b c \,x^{10}-20 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{29}+3 a \,b^{2} x^{23}+3 a^{2} b \,x^{17}+a^{3} x^{11}}d x \right ) a^{2} b d \,x^{16}+38 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{29}+3 a \,b^{2} x^{23}+3 a^{2} b \,x^{17}+a^{3} x^{11}}d x \right ) a \,b^{2} c \,x^{16}-10 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{29}+3 a \,b^{2} x^{23}+3 a^{2} b \,x^{17}+a^{3} x^{11}}d x \right ) a \,b^{2} d \,x^{22}+19 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{29}+3 a \,b^{2} x^{23}+3 a^{2} b \,x^{17}+a^{3} x^{11}}d x \right ) b^{3} c \,x^{22}}{19 b \,x^{10} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int((d*x^6+c)/x^11/(b*x^6+a)^(5/2),x)
 

Output:

( - sqrt(a + b*x**6)*d - 10*int(sqrt(a + b*x**6)/(a**3*x**11 + 3*a**2*b*x* 
*17 + 3*a*b**2*x**23 + b**3*x**29),x)*a**3*d*x**10 + 19*int(sqrt(a + b*x** 
6)/(a**3*x**11 + 3*a**2*b*x**17 + 3*a*b**2*x**23 + b**3*x**29),x)*a**2*b*c 
*x**10 - 20*int(sqrt(a + b*x**6)/(a**3*x**11 + 3*a**2*b*x**17 + 3*a*b**2*x 
**23 + b**3*x**29),x)*a**2*b*d*x**16 + 38*int(sqrt(a + b*x**6)/(a**3*x**11 
 + 3*a**2*b*x**17 + 3*a*b**2*x**23 + b**3*x**29),x)*a*b**2*c*x**16 - 10*in 
t(sqrt(a + b*x**6)/(a**3*x**11 + 3*a**2*b*x**17 + 3*a*b**2*x**23 + b**3*x* 
*29),x)*a*b**2*d*x**22 + 19*int(sqrt(a + b*x**6)/(a**3*x**11 + 3*a**2*b*x* 
*17 + 3*a*b**2*x**23 + b**3*x**29),x)*b**3*c*x**22)/(19*b*x**10*(a**2 + 2* 
a*b*x**6 + b**2*x**12))