\(\int \frac {x^{16} (c+d x^6)}{(a+b x^6)^{5/2}} \, dx\) [47]

Optimal result

Integrand size = 22, antiderivative size = 608 \[ \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=-\frac {(b c-a d) x^{11}}{9 b^2 \left (a+b x^6\right )^{3/2}}-\frac {(11 b c-20 a d) x^5}{27 b^3 \sqrt {a+b x^6}}+\frac {d x^5 \sqrt {a+b x^6}}{8 b^3}+\frac {55 \left (1+\sqrt {3}\right ) (8 b c-17 a d) x \sqrt {a+b x^6}}{432 b^{11/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )}-\frac {55 \sqrt [3]{a} (8 b c-17 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{144\ 3^{3/4} b^{11/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (8 b c-17 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{864 \sqrt [4]{3} b^{11/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/9*(-a*d+b*c)*x^11/b^2/(b*x^6+a)^(3/2)-1/27*(-20*a*d+11*b*c)*x^5/b^3/(b* 
x^6+a)^(1/2)+1/8*d*x^5*(b*x^6+a)^(1/2)/b^3+55/432*(1+3^(1/2))*(-17*a*d+8*b 
*c)*x*(b*x^6+a)^(1/2)/b^(11/3)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)-55/432*a^ 
(1/3)*(-17*a*d+8*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)*EllipticE((1-(a^ 
(1/3)+(1-3^(1/2))*b^(1/3)*x^2)^2/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)^2)^(1/2 
),1/4*6^(1/2)+1/4*2^(1/2))*3^(1/4)/b^(11/3)/(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)-55/2592*(1 
-3^(1/2))*a^(1/3)*(-17*a*d+8*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)*Inve 
rseJacobiAM(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)/b^(11/3)/(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 = 10.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.18 \[ \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {x^5 \left (-187 a^2 d+a b \left (88 c-68 d x^6\right )+8 b^2 x^6 \left (4 c+d x^6\right )+11 (-8 b c+17 a d) \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {5}{2},\frac {11}{6},-\frac {b x^6}{a}\right )\right )}{64 b^3 \left (a+b x^6\right )^{3/2}} \] Input:

Integrate[(x^16*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(x^5*(-187*a^2*d + a*b*(88*c - 68*d*x^6) + 8*b^2*x^6*(4*c + d*x^6) + 11*(- 
8*b*c + 17*a*d)*(a + b*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[5/6, 5/2 
, 11/6, -((b*x^6)/a)]))/(64*b^3*(a + b*x^6)^(3/2))
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 592, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {959, 817, 817, 837, 25, 766, 2420}

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.

\(\displaystyle \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {(8 b c-17 a d) \int \frac {x^{16}}{\left (b x^6+a\right )^{5/2}}dx}{8 b}+\frac {d x^{17}}{8 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {(8 b c-17 a d) \left (\frac {11 \int \frac {x^{10}}{\left (b x^6+a\right )^{3/2}}dx}{9 b}-\frac {x^{11}}{9 b \left (a+b x^6\right )^{3/2}}\right )}{8 b}+\frac {d x^{17}}{8 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {(8 b c-17 a d) \left (\frac {11 \left (\frac {5 \int \frac {x^4}{\sqrt {b x^6+a}}dx}{3 b}-\frac {x^5}{3 b \sqrt {a+b x^6}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^6\right )^{3/2}}\right )}{8 b}+\frac {d x^{17}}{8 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 837

\(\displaystyle \frac {(8 b c-17 a d) \left (\frac {11 \left (\frac {5 \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}\right )}{3 b}-\frac {x^5}{3 b \sqrt {a+b x^6}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^6\right )^{3/2}}\right )}{8 b}+\frac {d x^{17}}{8 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(8 b c-17 a d) \left (\frac {11 \left (\frac {5 \left (\frac {\int \frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}\right )}{3 b}-\frac {x^5}{3 b \sqrt {a+b x^6}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^6\right )^{3/2}}\right )}{8 b}+\frac {d x^{17}}{8 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {(8 b c-17 a d) \left (\frac {11 \left (\frac {5 \left (\frac {\int \frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{3 b}-\frac {x^5}{3 b \sqrt {a+b x^6}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^6\right )^{3/2}}\right )}{8 b}+\frac {d x^{17}}{8 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 2420

\(\displaystyle \frac {(8 b c-17 a d) \left (\frac {11 \left (\frac {5 \left (\frac {\frac {\left (1+\sqrt {3}\right ) x \sqrt {a+b x^6}}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}-\frac {\sqrt [4]{3} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{3 b}-\frac {x^5}{3 b \sqrt {a+b x^6}}\right )}{9 b}-\frac {x^{11}}{9 b \left (a+b x^6\right )^{3/2}}\right )}{8 b}+\frac {d x^{17}}{8 b \left (a+b x^6\right )^{3/2}}\)

Input:

Int[(x^16*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(d*x^17)/(8*b*(a + b*x^6)^(3/2)) + ((8*b*c - 17*a*d)*(-1/9*x^11/(b*(a + b* 
x^6)^(3/2)) + (11*(-1/3*x^5/(b*Sqrt[a + b*x^6]) + (5*((((1 + Sqrt[3])*x*Sq 
rt[a + b*x^6])/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2) - (3^(1/4)*a^(1/3)*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]*EllipticE[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])/(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]))/(2*b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*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])/(4*3^(1/4)*b^(2/3)*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*b)))/(9*b)))/(8 
*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 837

Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 
3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2))   Int[1/Sqrt[ 
a + b*x^6], x], x] - Simp[1/(2*r^2)   Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S 
qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, 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]
 

rule 2420

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = 
 Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr 
t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*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 
*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) 
)*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 
 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 
- Sqrt[3])*d, 0]
 

Maple [F]

\[\int \frac {x^{16} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

int(x^16*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

int(x^16*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Fricas [F]

\[ \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{16}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^16*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

integral((d*x^22 + c*x^16)*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. \[ \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**16*(d*x**6+c)/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{16}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^16*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)*x^16/(b*x^6 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{16}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^16*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)*x^16/(b*x^6 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int \frac {x^{16}\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((x^16*(c + d*x^6))/(a + b*x^6)^(5/2),x)
 

Output:

int((x^16*(c + d*x^6))/(a + b*x^6)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^{16} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-187 \sqrt {b \,x^{6}+a}\, a^{2} d \,x^{5}+88 \sqrt {b \,x^{6}+a}\, a b c \,x^{5}-68 \sqrt {b \,x^{6}+a}\, a b d \,x^{11}+32 \sqrt {b \,x^{6}+a}\, b^{2} c \,x^{11}+8 \sqrt {b \,x^{6}+a}\, b^{2} d \,x^{17}+935 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{5} d -440 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} b c +1870 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} b d \,x^{6}-880 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b^{2} c \,x^{6}+935 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b^{2} d \,x^{12}-440 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{3} c \,x^{12}}{64 b^{3} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int(x^16*(d*x^6+c)/(b*x^6+a)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

( - 187*sqrt(a + b*x**6)*a**2*d*x**5 + 88*sqrt(a + b*x**6)*a*b*c*x**5 - 68 
*sqrt(a + b*x**6)*a*b*d*x**11 + 32*sqrt(a + b*x**6)*b**2*c*x**11 + 8*sqrt( 
a + b*x**6)*b**2*d*x**17 + 935*int((sqrt(a + b*x**6)*x**4)/(a**3 + 3*a**2* 
b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**5*d - 440*int((sqrt(a + b*x**6 
)*x**4)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**4*b*c + 
 1870*int((sqrt(a + b*x**6)*x**4)/(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 - 880*int((sqrt(a + b*x**6)*x**4)/(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 + 935*int(( 
sqrt(a + b*x**6)*x**4)/(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 - 440*int((sqrt(a + b*x**6)*x**4)/(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))