\(\int \frac {x^{12} (c+d x^6)}{(a+b x^6)^{5/2}} \, dx\) [42]

Optimal result

Integrand size = 22, antiderivative size = 303 \[ \int \frac {x^{12} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {a (b c-a d) x}{9 b^3 \left (a+b x^6\right )^{3/2}}-\frac {(10 b c-19 a d) x}{27 b^3 \sqrt {a+b x^6}}+\frac {d x \sqrt {a+b x^6}}{4 b^3}+\frac {7 (4 b c-13 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{216 \sqrt [4]{3} \sqrt [3]{a} b^3 \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

1/9*a*(-a*d+b*c)*x/b^3/(b*x^6+a)^(3/2)-1/27*(-19*a*d+10*b*c)*x/b^3/(b*x^6+ 
a)^(1/2)+1/4*d*x*(b*x^6+a)^(1/2)/b^3+7/648*(-13*a*d+4*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^ 
(1/3)/b^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 = 10.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.35 \[ \int \frac {x^{12} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {x \left (91 a^2 d+b^2 x^6 \left (-40 c+27 d x^6\right )+a b \left (-28 c+130 d x^6\right )+7 (4 b c-13 a d) \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {b x^6}{a}\right )\right )}{108 b^3 \left (a+b x^6\right )^{3/2}} \] Input:

Integrate[(x^12*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(x*(91*a^2*d + b^2*x^6*(-40*c + 27*d*x^6) + a*b*(-28*c + 130*d*x^6) + 7*(4 
*b*c - 13*a*d)*(a + b*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[1/6, 1/2, 
 7/6, -((b*x^6)/a)]))/(108*b^3*(a + b*x^6)^(3/2))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {959, 817, 817, 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.

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {(4 b c-13 a d) \int \frac {x^{12}}{\left (b x^6+a\right )^{5/2}}dx}{4 b}+\frac {d x^{13}}{4 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 817

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

\(\Big \downarrow \) 817

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

\(\Big \downarrow \) 766

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

Input:

Int[(x^12*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(d*x^13)/(4*b*(a + b*x^6)^(3/2)) + ((4*b*c - 13*a*d)*(-1/9*x^7/(b*(a + b*x 
^6)^(3/2)) + (7*(-1/3*x/(b*Sqrt[a + b*x^6]) + (x*(a^(1/3) + b^(1/3)*x^2)*S 
qrt[(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])/(6*3^(1/4)*a^(1/3) 
*b*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])))/(9*b)))/(4*b)
 

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 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]

\[\int \frac {x^{12} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

int(x^12*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

int(x^12*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Fricas [F]

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

integrate(x^12*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

integral((d*x^18 + c*x^12)*sqrt(b*x^6 + a)/(b^3*x^18 + 3*a*b^2*x^12 + 3*a^ 
2*b*x^6 + a^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{12} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**12*(d*x**6+c)/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^12*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)*x^12/(b*x^6 + a)^(5/2), x)
 

Giac [F]

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

integrate(x^12*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)*x^12/(b*x^6 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{12} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int \frac {x^{12}\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((x^12*(c + d*x^6))/(a + b*x^6)^(5/2),x)
 

Output:

int((x^12*(c + d*x^6))/(a + b*x^6)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^{12} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {91 \sqrt {b \,x^{6}+a}\, a^{2} d x -28 \sqrt {b \,x^{6}+a}\, a b c x +104 \sqrt {b \,x^{6}+a}\, a b d \,x^{7}-32 \sqrt {b \,x^{6}+a}\, b^{2} c \,x^{7}+16 \sqrt {b \,x^{6}+a}\, b^{2} d \,x^{13}-91 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{5} d +28 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} b c -182 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} b d \,x^{6}+56 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b^{2} c \,x^{6}-91 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b^{2} d \,x^{12}+28 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{3} c \,x^{12}}{64 b^{3} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int(x^12*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

(91*sqrt(a + b*x**6)*a**2*d*x - 28*sqrt(a + b*x**6)*a*b*c*x + 104*sqrt(a + 
 b*x**6)*a*b*d*x**7 - 32*sqrt(a + b*x**6)*b**2*c*x**7 + 16*sqrt(a + b*x**6 
)*b**2*d*x**13 - 91*int(sqrt(a + b*x**6)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2* 
x**12 + b**3*x**18),x)*a**5*d + 28*int(sqrt(a + b*x**6)/(a**3 + 3*a**2*b*x 
**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**4*b*c - 182*int(sqrt(a + b*x**6)/ 
(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 + 56 
*int(sqrt(a + b*x**6)/(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 - 91*int(sqrt(a + b*x**6)/(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 + 28*int(sqrt(a + b*x**6)/( 
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)/ 
(64*b**3*(a**2 + 2*a*b*x**6 + b**2*x**12))