\(\int \frac {x^{10} (c+d x^6)}{(a+b x^6)^{5/2}} \, dx\) [48]

Optimal result

Integrand size = 22, antiderivative size = 592 \[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=-\frac {(b c-a d) x^5}{9 b^2 \left (a+b x^6\right )^{3/2}}+\frac {(5 b c-14 a d) x^5}{27 a b^2 \sqrt {a+b x^6}}-\frac {5 \left (1+\sqrt {3}\right ) (2 b c-11 a d) x \sqrt {a+b x^6}}{54 a b^{8/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )}+\frac {5 (2 b c-11 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{18\ 3^{3/4} a^{2/3} b^{8/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}+\frac {5 \left (1-\sqrt {3}\right ) (2 b c-11 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{108 \sqrt [4]{3} a^{2/3} b^{8/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/9*(-a*d+b*c)*x^5/b^2/(b*x^6+a)^(3/2)+1/27*(-14*a*d+5*b*c)*x^5/a/b^2/(b* 
x^6+a)^(1/2)-5/54*(1+3^(1/2))*(-11*a*d+2*b*c)*x*(b*x^6+a)^(1/2)/a/b^(8/3)/ 
(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)+5/54*(-11*a*d+2*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)/a^(2/3 
)/b^(8/3)/(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)+5/324*(1-3^(1/2))*(-11*a*d+2*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^(2/3)/b^(8/3)/(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 = 10.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.16 \[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {x^5 \left (a \left (-2 b c+11 a d+4 b d x^6\right )+(2 b c-11 a d) \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {5}{2},\frac {11}{6},-\frac {b x^6}{a}\right )\right )}{8 a b^2 \left (a+b x^6\right )^{3/2}} \] Input:

Integrate[(x^10*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {959, 817, 819, 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.

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {(2 b c-11 a d) \int \frac {x^{10}}{\left (b x^6+a\right )^{5/2}}dx}{2 b}+\frac {d x^{11}}{2 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 817

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

\(\Big \downarrow \) 819

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

\(\Big \downarrow \) 837

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 766

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

\(\Big \downarrow \) 2420

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

Input:

Int[(x^10*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(d*x^11)/(2*b*(a + b*x^6)^(3/2)) + ((2*b*c - 11*a*d)*(-1/9*x^5/(b*(a + b*x 
^6)^(3/2)) + (5*(x^5/(3*a*Sqrt[a + b*x^6]) - (2*((((1 + Sqrt[3])*x*Sqrt[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 - Sqr 
t[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*a)))/(9*b)))/(2*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 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 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]

\[\int \frac {x^{10} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

int(x^10*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

int(x^10*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Fricas [F]

\[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{10}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^10*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

integral((d*x^16 + c*x^10)*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. \[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**10*(d*x**6+c)/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{10}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^10*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)*x^10/(b*x^6 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{10}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^10*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)*x^10/(b*x^6 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int \frac {x^{10}\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((x^10*(c + d*x^6))/(a + b*x^6)^(5/2),x)
 

Output:

int((x^10*(c + d*x^6))/(a + b*x^6)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {11 \sqrt {b \,x^{6}+a}\, a d \,x^{5}-2 \sqrt {b \,x^{6}+a}\, b c \,x^{5}+4 \sqrt {b \,x^{6}+a}\, b d \,x^{11}-55 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} d +10 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b c -110 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b d \,x^{6}+20 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{2} c \,x^{6}-55 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{2} d \,x^{12}+10 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a \,b^{3} c \,x^{12}}{8 b^{2} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int(x^10*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

(11*sqrt(a + b*x**6)*a*d*x**5 - 2*sqrt(a + b*x**6)*b*c*x**5 + 4*sqrt(a + b 
*x**6)*b*d*x**11 - 55*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*d + 10*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*c - 110*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*d*x**6 + 20*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**2*c*x**6 - 55*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** 
2*d*x**12 + 10*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*b**3*c*x**12)/(8*b**2*(a**2 + 2*a*b*x**6 + b**2 
*x**12))