\(\int \frac {c+d x^6}{x^8 (a+b x^6)^{3/2}} \, dx\) [22]

Optimal result

Integrand size = 22, antiderivative size = 612 \[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{3/2}} \, dx=-\frac {c}{7 a x^7 \sqrt {a+b x^6}}-\frac {10 b c-7 a d}{21 a^2 x \sqrt {a+b x^6}}+\frac {4 (10 b c-7 a d) \sqrt {a+b x^6}}{21 a^3 x}-\frac {4 \left (1+\sqrt {3}\right ) \sqrt [3]{b} (10 b c-7 a d) x \sqrt {a+b x^6}}{21 a^3 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )}+\frac {4 \sqrt [3]{b} (10 b c-7 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7\ 3^{3/4} a^{8/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}+\frac {2 \left (1-\sqrt {3}\right ) \sqrt [3]{b} (10 b c-7 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{21 \sqrt [4]{3} a^{8/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/7*c/a/x^7/(b*x^6+a)^(1/2)-1/21*(-7*a*d+10*b*c)/a^2/x/(b*x^6+a)^(1/2)+4/ 
21*(-7*a*d+10*b*c)*(b*x^6+a)^(1/2)/a^3/x-4/21*(1+3^(1/2))*b^(1/3)*(-7*a*d+ 
10*b*c)*x*(b*x^6+a)^(1/2)/a^3/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)+4/21*b^(1/ 
3)*(-7*a*d+10*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^(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)+2/63*(1-3^(1/2 
))*b^(1/3)*(-7*a*d+10*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)*InverseJaco 
biAM(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^(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 = 10.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.12 \[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-a c+(10 b c-7 a d) x^6 \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {3}{2},\frac {5}{6},-\frac {b x^6}{a}\right )}{7 a^2 x^7 \sqrt {a+b x^6}} \] Input:

Integrate[(c + d*x^6)/(x^8*(a + b*x^6)^(3/2)),x]
 

Output:

(-(a*c) + (10*b*c - 7*a*d)*x^6*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[-1/6, 
 3/2, 5/6, -((b*x^6)/a)])/(7*a^2*x^7*Sqrt[a + b*x^6])
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {955, 819, 847, 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.

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

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(10 b c-7 a d) \int \frac {1}{x^2 \left (b x^6+a\right )^{3/2}}dx}{7 a}-\frac {c}{7 a x^7 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(10 b c-7 a d) \left (\frac {4 \int \frac {1}{x^2 \sqrt {b x^6+a}}dx}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{7 a}-\frac {c}{7 a x^7 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {(10 b c-7 a d) \left (\frac {4 \left (\frac {2 b \int \frac {x^4}{\sqrt {b x^6+a}}dx}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{7 a}-\frac {c}{7 a x^7 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 837

\(\displaystyle -\frac {(10 b c-7 a d) \left (\frac {4 \left (\frac {2 b \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}\right )}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{7 a}-\frac {c}{7 a x^7 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(10 b c-7 a d) \left (\frac {4 \left (\frac {2 b \left (\frac {\int \frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}\right )}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{7 a}-\frac {c}{7 a x^7 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 766

\(\displaystyle -\frac {(10 b c-7 a d) \left (\frac {4 \left (\frac {2 b \left (\frac {\int \frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{7 a}-\frac {c}{7 a x^7 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 2420

\(\displaystyle -\frac {(10 b c-7 a d) \left (\frac {4 \left (\frac {2 b \left (\frac {\frac {\left (1+\sqrt {3}\right ) x \sqrt {a+b x^6}}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}-\frac {\sqrt [4]{3} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{7 a}-\frac {c}{7 a x^7 \sqrt {a+b x^6}}\)

Input:

Int[(c + d*x^6)/(x^8*(a + b*x^6)^(3/2)),x]
 

Output:

-1/7*c/(a*x^7*Sqrt[a + b*x^6]) - ((10*b*c - 7*a*d)*(1/(3*a*x*Sqrt[a + b*x^ 
6]) + (4*(-(Sqrt[a + b*x^6]/(a*x)) + (2*b*((((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])/(Sqr 
t[(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])))/a))/(3*a)))/(7*a)
 

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 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 847

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

rule 955

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

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]

\[\int \frac {d \,x^{6}+c}{x^{8} \left (b \,x^{6}+a \right )^{\frac {3}{2}}}d x\]

Input:

int((d*x^6+c)/x^8/(b*x^6+a)^(3/2),x)
 

Output:

int((d*x^6+c)/x^8/(b*x^6+a)^(3/2),x)
 

Fricas [F]

\[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} x^{8}} \,d x } \] Input:

integrate((d*x^6+c)/x^8/(b*x^6+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x^6 + a)*(d*x^6 + c)/(b^2*x^20 + 2*a*b*x^14 + a^2*x^8), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 67.80 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.14 \[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{3/2}} \, dx=\frac {c \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, \frac {3}{2} \\ - \frac {1}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{7} \Gamma \left (- \frac {1}{6}\right )} + \frac {d \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{2} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x \Gamma \left (\frac {5}{6}\right )} \] Input:

integrate((d*x**6+c)/x**8/(b*x**6+a)**(3/2),x)
 

Output:

c*gamma(-7/6)*hyper((-7/6, 3/2), (-1/6,), b*x**6*exp_polar(I*pi)/a)/(6*a** 
(3/2)*x**7*gamma(-1/6)) + d*gamma(-1/6)*hyper((-1/6, 3/2), (5/6,), b*x**6* 
exp_polar(I*pi)/a)/(6*a**(3/2)*x*gamma(5/6))
 

Maxima [F]

\[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} x^{8}} \,d x } \] Input:

integrate((d*x^6+c)/x^8/(b*x^6+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(3/2)*x^8), x)
 

Giac [F]

\[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} x^{8}} \,d x } \] Input:

integrate((d*x^6+c)/x^8/(b*x^6+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(3/2)*x^8), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{3/2}} \, dx=\int \frac {d\,x^6+c}{x^8\,{\left (b\,x^6+a\right )}^{3/2}} \,d x \] Input:

int((c + d*x^6)/(x^8*(a + b*x^6)^(3/2)),x)
 

Output:

int((c + d*x^6)/(x^8*(a + b*x^6)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d -7 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{20}+2 a b \,x^{14}+a^{2} x^{8}}d x \right ) a^{2} d \,x^{7}+10 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{20}+2 a b \,x^{14}+a^{2} x^{8}}d x \right ) a b c \,x^{7}-7 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{20}+2 a b \,x^{14}+a^{2} x^{8}}d x \right ) a b d \,x^{13}+10 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{2} x^{20}+2 a b \,x^{14}+a^{2} x^{8}}d x \right ) b^{2} c \,x^{13}}{10 b \,x^{7} \left (b \,x^{6}+a \right )} \] Input:

int((d*x^6+c)/x^8/(b*x^6+a)^(3/2),x)
 

Output:

( - sqrt(a + b*x**6)*d - 7*int(sqrt(a + b*x**6)/(a**2*x**8 + 2*a*b*x**14 + 
 b**2*x**20),x)*a**2*d*x**7 + 10*int(sqrt(a + b*x**6)/(a**2*x**8 + 2*a*b*x 
**14 + b**2*x**20),x)*a*b*c*x**7 - 7*int(sqrt(a + b*x**6)/(a**2*x**8 + 2*a 
*b*x**14 + b**2*x**20),x)*a*b*d*x**13 + 10*int(sqrt(a + b*x**6)/(a**2*x**8 
 + 2*a*b*x**14 + b**2*x**20),x)*b**2*c*x**13)/(10*b*x**7*(a + b*x**6))