\(\int \frac {x^{13} (c+d x^6)}{(a+b x^6)^{5/2}} \, dx\) [37]

Optimal result

Integrand size = 22, antiderivative size = 315 \[ \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {a (b c-a d) x^2}{9 b^3 \left (a+b x^6\right )^{3/2}}-\frac {(11 b c-20 a d) x^2}{27 b^3 \sqrt {a+b x^6}}+\frac {d x^2 \sqrt {a+b x^6}}{5 b^3}+\frac {16 \sqrt {2+\sqrt {3}} (5 b c-14 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{135 \sqrt [4]{3} b^{10/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

1/9*a*(-a*d+b*c)*x^2/b^3/(b*x^6+a)^(3/2)-1/27*(-20*a*d+11*b*c)*x^2/b^3/(b* 
x^6+a)^(1/2)+1/5*d*x^2*(b*x^6+a)^(1/2)/b^3+16/405*(1/2*6^(1/2)+1/2*2^(1/2) 
)*(-14*a*d+5*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)/b^(10/3)/(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 = 10.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.35 \[ \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {x^2 \left (112 a^2 d+b^2 x^6 \left (-55 c+27 d x^6\right )+a b \left (-40 c+154 d x^6\right )+8 (5 b c-14 a d) \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^6}{a}\right )\right )}{135 b^3 \left (a+b x^6\right )^{3/2}} \] Input:

Integrate[(x^13*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(x^2*(112*a^2*d + b^2*x^6*(-55*c + 27*d*x^6) + a*b*(-40*c + 154*d*x^6) + 8 
*(5*b*c - 14*a*d)*(a + b*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[1/3, 1 
/2, 4/3, -((b*x^6)/a)]))/(135*b^3*(a + b*x^6)^(3/2))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 313, 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 = {959, 807, 817, 817, 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.

\(\displaystyle \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {(5 b c-14 a d) \int \frac {x^{13}}{\left (b x^6+a\right )^{5/2}}dx}{5 b}+\frac {d x^{14}}{5 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {(5 b c-14 a d) \int \frac {x^{12}}{\left (b x^6+a\right )^{5/2}}dx^2}{10 b}+\frac {d x^{14}}{5 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {(5 b c-14 a d) \left (\frac {8 \int \frac {x^6}{\left (b x^6+a\right )^{3/2}}dx^2}{9 b}-\frac {2 x^8}{9 b \left (a+b x^6\right )^{3/2}}\right )}{10 b}+\frac {d x^{14}}{5 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {(5 b c-14 a d) \left (\frac {8 \left (\frac {2 \int \frac {1}{\sqrt {b x^6+a}}dx^2}{3 b}-\frac {2 x^2}{3 b \sqrt {a+b x^6}}\right )}{9 b}-\frac {2 x^8}{9 b \left (a+b x^6\right )^{3/2}}\right )}{10 b}+\frac {d x^{14}}{5 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {(5 b c-14 a d) \left (\frac {8 \left (\frac {4 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {2 x^2}{3 b \sqrt {a+b x^6}}\right )}{9 b}-\frac {2 x^8}{9 b \left (a+b x^6\right )^{3/2}}\right )}{10 b}+\frac {d x^{14}}{5 b \left (a+b x^6\right )^{3/2}}\)

Input:

Int[(x^13*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(d*x^14)/(5*b*(a + b*x^6)^(3/2)) + ((5*b*c - 14*a*d)*((-2*x^8)/(9*b*(a + b 
*x^6)^(3/2)) + (8*((-2*x^2)/(3*b*Sqrt[a + b*x^6]) + (4*Sqrt[2 + Sqrt[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]])/(3*3^(1/4)*b^(4/3)*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])))/(9*b)))/(10*b)
 

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 817

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

rule 959

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

Maple [F]

\[\int \frac {x^{13} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

int(x^13*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

int(x^13*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.50 \[ \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {16 \, {\left ({\left (5 \, b^{3} c - 14 \, a b^{2} d\right )} x^{12} + 2 \, {\left (5 \, a b^{2} c - 14 \, a^{2} b d\right )} x^{6} + 5 \, a^{2} b c - 14 \, a^{3} d\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right ) + {\left (27 \, b^{3} d x^{14} - 11 \, {\left (5 \, b^{3} c - 14 \, a b^{2} d\right )} x^{8} - 8 \, {\left (5 \, a b^{2} c - 14 \, a^{2} b d\right )} x^{2}\right )} \sqrt {b x^{6} + a}}{135 \, {\left (b^{6} x^{12} + 2 \, a b^{5} x^{6} + a^{2} b^{4}\right )}} \] Input:

integrate(x^13*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

1/135*(16*((5*b^3*c - 14*a*b^2*d)*x^12 + 2*(5*a*b^2*c - 14*a^2*b*d)*x^6 + 
5*a^2*b*c - 14*a^3*d)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x^2) + (27*b^ 
3*d*x^14 - 11*(5*b^3*c - 14*a*b^2*d)*x^8 - 8*(5*a*b^2*c - 14*a^2*b*d)*x^2) 
*sqrt(b*x^6 + a))/(b^6*x^12 + 2*a*b^5*x^6 + a^2*b^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**13*(d*x**6+c)/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{13}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^13*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)*x^13/(b*x^6 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{13}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^13*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)*x^13/(b*x^6 + a)^(5/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int \frac {x^{13}\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((x^13*(c + d*x^6))/(a + b*x^6)^(5/2),x)
 

Output:

int((x^13*(c + d*x^6))/(a + b*x^6)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^{13} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {112 \sqrt {b \,x^{6}+a}\, a^{2} d \,x^{2}-40 \sqrt {b \,x^{6}+a}\, a b c \,x^{2}+98 \sqrt {b \,x^{6}+a}\, a b d \,x^{8}-35 \sqrt {b \,x^{6}+a}\, b^{2} c \,x^{8}+7 \sqrt {b \,x^{6}+a}\, b^{2} d \,x^{14}-224 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{5} d +80 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} b c -448 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} b d \,x^{6}+160 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b^{2} c \,x^{6}-224 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b^{2} d \,x^{12}+80 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{3} c \,x^{12}}{35 b^{3} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int(x^13*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

(112*sqrt(a + b*x**6)*a**2*d*x**2 - 40*sqrt(a + b*x**6)*a*b*c*x**2 + 98*sq 
rt(a + b*x**6)*a*b*d*x**8 - 35*sqrt(a + b*x**6)*b**2*c*x**8 + 7*sqrt(a + b 
*x**6)*b**2*d*x**14 - 224*int((sqrt(a + b*x**6)*x)/(a**3 + 3*a**2*b*x**6 + 
 3*a*b**2*x**12 + b**3*x**18),x)*a**5*d + 80*int((sqrt(a + b*x**6)*x)/(a** 
3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**4*b*c - 448*int((sq 
rt(a + b*x**6)*x)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)* 
a**4*b*d*x**6 + 160*int((sqrt(a + b*x**6)*x)/(a**3 + 3*a**2*b*x**6 + 3*a*b 
**2*x**12 + b**3*x**18),x)*a**3*b**2*c*x**6 - 224*int((sqrt(a + b*x**6)*x) 
/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**3*b**2*d*x**12 
 + 80*int((sqrt(a + b*x**6)*x)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b* 
*3*x**18),x)*a**2*b**3*c*x**12)/(35*b**3*(a**2 + 2*a*b*x**6 + b**2*x**12))