\(\int \frac {c+d x^6}{x^8 (a+b x^6)^{5/2}} \, dx\) [51]

Optimal result

Integrand size = 22, antiderivative size = 642 \[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{5/2}} \, dx=-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}-\frac {16 b c-7 a d}{63 a^2 x \left (a+b x^6\right )^{3/2}}-\frac {10 (16 b c-7 a d)}{189 a^3 x \sqrt {a+b x^6}}+\frac {40 (16 b c-7 a d) \sqrt {a+b x^6}}{189 a^4 x}-\frac {40 \left (1+\sqrt {3}\right ) \sqrt [3]{b} (16 b c-7 a d) x \sqrt {a+b x^6}}{189 a^4 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )}+\frac {40 \sqrt [3]{b} (16 b c-7 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{63\ 3^{3/4} a^{11/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}+\frac {20 \left (1-\sqrt {3}\right ) \sqrt [3]{b} (16 b c-7 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{189 \sqrt [4]{3} a^{11/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/7*c/a/x^7/(b*x^6+a)^(3/2)-1/63*(-7*a*d+16*b*c)/a^2/x/(b*x^6+a)^(3/2)-10 
/189*(-7*a*d+16*b*c)/a^3/x/(b*x^6+a)^(1/2)+40/189*(-7*a*d+16*b*c)*(b*x^6+a 
)^(1/2)/a^4/x-40/189*(1+3^(1/2))*b^(1/3)*(-7*a*d+16*b*c)*x*(b*x^6+a)^(1/2) 
/a^4/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)+40/189*b^(1/3)*(-7*a*d+16*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^(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)+20/567*(1-3^(1/2))*b^(1/3)*(-7*a*d+1 
6*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^(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 = 10.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.13 \[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{5/2}} \, dx=\frac {-a^2 c+(16 b c-7 a d) x^6 \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{2},\frac {5}{6},-\frac {b x^6}{a}\right )}{7 a^3 x^7 \left (a+b x^6\right )^{3/2}} \] Input:

Integrate[(c + d*x^6)/(x^8*(a + b*x^6)^(5/2)),x]
 

Output:

(-(a^2*c) + (16*b*c - 7*a*d)*x^6*(a + b*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeom 
etric2F1[-1/6, 5/2, 5/6, -((b*x^6)/a)])/(7*a^3*x^7*(a + b*x^6)^(3/2))
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 618, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {955, 819, 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.

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

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(16 b c-7 a d) \int \frac {1}{x^2 \left (b x^6+a\right )^{5/2}}dx}{7 a}-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(16 b c-7 a d) \left (\frac {10 \int \frac {1}{x^2 \left (b x^6+a\right )^{3/2}}dx}{9 a}+\frac {1}{9 a x \left (a+b x^6\right )^{3/2}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(16 b c-7 a d) \left (\frac {10 \left (\frac {4 \int \frac {1}{x^2 \sqrt {b x^6+a}}dx}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^6\right )^{3/2}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {(16 b c-7 a d) \left (\frac {10 \left (\frac {4 \left (\frac {2 b \int \frac {x^4}{\sqrt {b x^6+a}}dx}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^6\right )^{3/2}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 837

\(\displaystyle -\frac {(16 b c-7 a d) \left (\frac {10 \left (\frac {4 \left (\frac {2 b \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}\right )}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^6\right )^{3/2}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(16 b c-7 a d) \left (\frac {10 \left (\frac {4 \left (\frac {2 b \left (\frac {\int \frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}\right )}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^6\right )^{3/2}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 766

\(\displaystyle -\frac {(16 b c-7 a d) \left (\frac {10 \left (\frac {4 \left (\frac {2 b \left (\frac {\int \frac {2 b^{2/3} x^4+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^6+a}}dx}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^6\right )^{3/2}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 2420

\(\displaystyle -\frac {(16 b c-7 a d) \left (\frac {10 \left (\frac {4 \left (\frac {2 b \left (\frac {\frac {\left (1+\sqrt {3}\right ) x \sqrt {a+b x^6}}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}-\frac {\sqrt [4]{3} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{a}-\frac {\sqrt {a+b x^6}}{a x}\right )}{3 a}+\frac {1}{3 a x \sqrt {a+b x^6}}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^6\right )^{3/2}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^6\right )^{3/2}}\)

Input:

Int[(c + d*x^6)/(x^8*(a + b*x^6)^(5/2)),x]
 

Output:

-1/7*c/(a*x^7*(a + b*x^6)^(3/2)) - ((16*b*c - 7*a*d)*(1/(9*a*x*(a + b*x^6) 
^(3/2)) + (10*(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]*Ellipt 
icE[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]*Ellip 
ticF[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)))/(9*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]

\[\int \frac {d \,x^{6}+c}{x^{8} \left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

int((d*x^6+c)/x^8/(b*x^6+a)^(5/2),x)
 

Output:

int((d*x^6+c)/x^8/(b*x^6+a)^(5/2),x)
 

Fricas [F]

\[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {5}{2}} x^{8}} \,d x } \] Input:

integrate((d*x^6+c)/x^8/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x^6 + a)*(d*x^6 + c)/(b^3*x^26 + 3*a*b^2*x^20 + 3*a^2*b*x^ 
14 + a^3*x^8), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x**6+c)/x**8/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {5}{2}} x^{8}} \,d x } \] Input:

integrate((d*x^6+c)/x^8/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(5/2)*x^8), x)
 

Giac [F]

\[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {5}{2}} x^{8}} \,d x } \] Input:

integrate((d*x^6+c)/x^8/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(5/2)*x^8), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{5/2}} \, dx=\int \frac {d\,x^6+c}{x^8\,{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((c + d*x^6)/(x^8*(a + b*x^6)^(5/2)),x)
 

Output:

int((c + d*x^6)/(x^8*(a + b*x^6)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {c+d x^6}{x^8 \left (a+b x^6\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d -7 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{26}+3 a \,b^{2} x^{20}+3 a^{2} b \,x^{14}+a^{3} x^{8}}d x \right ) a^{3} d \,x^{7}+16 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{26}+3 a \,b^{2} x^{20}+3 a^{2} b \,x^{14}+a^{3} x^{8}}d x \right ) a^{2} b c \,x^{7}-14 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{26}+3 a \,b^{2} x^{20}+3 a^{2} b \,x^{14}+a^{3} x^{8}}d x \right ) a^{2} b d \,x^{13}+32 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{26}+3 a \,b^{2} x^{20}+3 a^{2} b \,x^{14}+a^{3} x^{8}}d x \right ) a \,b^{2} c \,x^{13}-7 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{26}+3 a \,b^{2} x^{20}+3 a^{2} b \,x^{14}+a^{3} x^{8}}d x \right ) a \,b^{2} d \,x^{19}+16 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{26}+3 a \,b^{2} x^{20}+3 a^{2} b \,x^{14}+a^{3} x^{8}}d x \right ) b^{3} c \,x^{19}}{16 b \,x^{7} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int((d*x^6+c)/x^8/(b*x^6+a)^(5/2),x)
 

Output:

( - sqrt(a + b*x**6)*d - 7*int(sqrt(a + b*x**6)/(a**3*x**8 + 3*a**2*b*x**1 
4 + 3*a*b**2*x**20 + b**3*x**26),x)*a**3*d*x**7 + 16*int(sqrt(a + b*x**6)/ 
(a**3*x**8 + 3*a**2*b*x**14 + 3*a*b**2*x**20 + b**3*x**26),x)*a**2*b*c*x** 
7 - 14*int(sqrt(a + b*x**6)/(a**3*x**8 + 3*a**2*b*x**14 + 3*a*b**2*x**20 + 
 b**3*x**26),x)*a**2*b*d*x**13 + 32*int(sqrt(a + b*x**6)/(a**3*x**8 + 3*a* 
*2*b*x**14 + 3*a*b**2*x**20 + b**3*x**26),x)*a*b**2*c*x**13 - 7*int(sqrt(a 
 + b*x**6)/(a**3*x**8 + 3*a**2*b*x**14 + 3*a*b**2*x**20 + b**3*x**26),x)*a 
*b**2*d*x**19 + 16*int(sqrt(a + b*x**6)/(a**3*x**8 + 3*a**2*b*x**14 + 3*a* 
b**2*x**20 + b**3*x**26),x)*b**3*c*x**19)/(16*b*x**7*(a**2 + 2*a*b*x**6 + 
b**2*x**12))