\(\int \frac {x^{10} (c+d x^6)}{(a+b x^6)^{3/2}} \, dx\) [19]

Optimal result

Integrand size = 22, antiderivative size = 578 \[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=-\frac {(b c-a d) x^5}{3 b^2 \sqrt {a+b x^6}}+\frac {d x^5 \sqrt {a+b x^6}}{8 b^2}+\frac {5 \left (1+\sqrt {3}\right ) (8 b c-11 a d) x \sqrt {a+b x^6}}{48 b^{8/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )}-\frac {5 \sqrt [3]{a} (8 b c-11 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{16\ 3^{3/4} b^{8/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (8 b c-11 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{96 \sqrt [4]{3} b^{8/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/3*(-a*d+b*c)*x^5/b^2/(b*x^6+a)^(1/2)+1/8*d*x^5*(b*x^6+a)^(1/2)/b^2+5/48 
*(1+3^(1/2))*(-11*a*d+8*b*c)*x*(b*x^6+a)^(1/2)/b^(8/3)/(a^(1/3)+(1+3^(1/2) 
)*b^(1/3)*x^2)-5/48*a^(1/3)*(-11*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^(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)-5/288*(1-3^(1/2))*a^(1/3)*(-11*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)*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)/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 = 80, normalized size of antiderivative = 0.14 \[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {x^5 \left (8 b c-11 a d+2 b d x^6+(-8 b c+11 a d) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {3}{2},\frac {11}{6},-\frac {b x^6}{a}\right )\right )}{16 b^2 \sqrt {a+b x^6}} \] Input:

Integrate[(x^10*(c + d*x^6))/(a + b*x^6)^(3/2),x]
 

Output:

(x^5*(8*b*c - 11*a*d + 2*b*d*x^6 + (-8*b*c + 11*a*d)*Sqrt[1 + (b*x^6)/a]*H 
ypergeometric2F1[5/6, 3/2, 11/6, -((b*x^6)/a)]))/(16*b^2*Sqrt[a + b*x^6])
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 563, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {959, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 817

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

\(\Big \downarrow \) 837

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 766

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

\(\Big \downarrow \) 2420

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

Input:

Int[(x^10*(c + d*x^6))/(a + b*x^6)^(3/2),x]
 

Output:

(d*x^11)/(8*b*Sqrt[a + b*x^6]) + ((8*b*c - 11*a*d)*(-1/3*x^5/(b*Sqrt[a + b 
*x^6]) + (5*((((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 + Sq 
rt[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 + S 
qrt[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)))/(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]

\[\int \frac {x^{10} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {3}{2}}}d x\]

Input:

int(x^10*(d*x^6+c)/(b*x^6+a)^(3/2),x)
 

Output:

int(x^10*(d*x^6+c)/(b*x^6+a)^(3/2),x)
 

Fricas [F]

\[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{10}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^10*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="fricas")
 

Output:

integral((d*x^16 + c*x^10)*sqrt(b*x^6 + a)/(b^2*x^12 + 2*a*b*x^6 + a^2), x 
)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 79.36 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.14 \[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {c x^{11} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {17}{6}\right )} + \frac {d x^{17} \Gamma \left (\frac {17}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {17}{6} \\ \frac {23}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} \Gamma \left (\frac {23}{6}\right )} \] Input:

integrate(x**10*(d*x**6+c)/(b*x**6+a)**(3/2),x)
 

Output:

c*x**11*gamma(11/6)*hyper((3/2, 11/6), (17/6,), b*x**6*exp_polar(I*pi)/a)/ 
(6*a**(3/2)*gamma(17/6)) + d*x**17*gamma(17/6)*hyper((3/2, 17/6), (23/6,), 
 b*x**6*exp_polar(I*pi)/a)/(6*a**(3/2)*gamma(23/6))
 

Maxima [F]

\[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{10}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^10*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)*x^10/(b*x^6 + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{6} + c\right )} x^{10}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^10*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)*x^10/(b*x^6 + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int \frac {x^{10}\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{3/2}} \,d x \] Input:

int((x^10*(c + d*x^6))/(a + b*x^6)^(3/2),x)
 

Output:

int((x^10*(c + d*x^6))/(a + b*x^6)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^{10} \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {-11 \sqrt {b \,x^{6}+a}\, a d \,x^{5}+8 \sqrt {b \,x^{6}+a}\, b c \,x^{5}+2 \sqrt {b \,x^{6}+a}\, b d \,x^{11}+55 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{2} x^{12}+2 a b \,x^{6}+a^{2}}d x \right ) a^{3} d -40 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{2} x^{12}+2 a b \,x^{6}+a^{2}}d x \right ) a^{2} b c +55 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{2} x^{12}+2 a b \,x^{6}+a^{2}}d x \right ) a^{2} b d \,x^{6}-40 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x^{4}}{b^{2} x^{12}+2 a b \,x^{6}+a^{2}}d x \right ) a \,b^{2} c \,x^{6}}{16 b^{2} \left (b \,x^{6}+a \right )} \] Input:

int(x^10*(d*x^6+c)/(b*x^6+a)^(3/2),x)
 

Output:

( - 11*sqrt(a + b*x**6)*a*d*x**5 + 8*sqrt(a + b*x**6)*b*c*x**5 + 2*sqrt(a 
+ b*x**6)*b*d*x**11 + 55*int((sqrt(a + b*x**6)*x**4)/(a**2 + 2*a*b*x**6 + 
b**2*x**12),x)*a**3*d - 40*int((sqrt(a + b*x**6)*x**4)/(a**2 + 2*a*b*x**6 
+ b**2*x**12),x)*a**2*b*c + 55*int((sqrt(a + b*x**6)*x**4)/(a**2 + 2*a*b*x 
**6 + b**2*x**12),x)*a**2*b*d*x**6 - 40*int((sqrt(a + b*x**6)*x**4)/(a**2 
+ 2*a*b*x**6 + b**2*x**12),x)*a*b**2*c*x**6)/(16*b**2*(a + b*x**6))