\(\int \frac {c+d x^6}{x^3 (a+b x^6)^{5/2}} \, dx\) [55]

Optimal result

Integrand size = 22, antiderivative size = 610 \[ \int \frac {c+d x^6}{x^3 \left (a+b x^6\right )^{5/2}} \, dx=-\frac {c}{2 a x^2 \left (a+b x^6\right )^{3/2}}-\frac {(11 b c-2 a d) x^4}{18 a^2 \left (a+b x^6\right )^{3/2}}-\frac {5 (11 b c-2 a d) x^4}{54 a^3 \sqrt {a+b x^6}}+\frac {5 (11 b c-2 a d) \sqrt {a+b x^6}}{54 a^3 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )}-\frac {5 \sqrt {2-\sqrt {3}} (11 b c-2 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right )|-7-4 \sqrt {3}\right )}{36\ 3^{3/4} a^{8/3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}+\frac {5 (11 b c-2 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{27 \sqrt {2} \sqrt [4]{3} a^{8/3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/2*c/a/x^2/(b*x^6+a)^(3/2)-1/18*(-2*a*d+11*b*c)*x^4/a^2/(b*x^6+a)^(3/2)- 
5/54*(-2*a*d+11*b*c)*x^4/a^3/(b*x^6+a)^(1/2)+5/54*(-2*a*d+11*b*c)*(b*x^6+a 
)^(1/2)/a^3/b^(2/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^2)-5/108*(1/2*6^(1/2)-1 
/2*2^(1/2))*(-2*a*d+11*b*c)*(a^(1/3)+b^(1/3)*x^2)*((a^(2/3)-a^(1/3)*b^(1/3 
)*x^2+b^(2/3)*x^4)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^2)^2)^(1/2)*EllipticE((( 
1-3^(1/2))*a^(1/3)+b^(1/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^2),I*3^(1/2 
)+2*I)*3^(1/4)/a^(8/3)/b^(2/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x^2)/((1+3^(1/2)) 
*a^(1/3)+b^(1/3)*x^2)^2)^(1/2)/(b*x^6+a)^(1/2)+5/162*(-2*a*d+11*b*c)*(a^(1 
/3)+b^(1/3)*x^2)*((a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/((1+3^(1/2))*a 
^(1/3)+b^(1/3)*x^2)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x^2)/( 
(1+3^(1/2))*a^(1/3)+b^(1/3)*x^2),I*3^(1/2)+2*I)*2^(1/2)*3^(3/4)/a^(8/3)/b^ 
(2/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x^2)/((1+3^(1/2))*a^(1/3)+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 = 10.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.13 \[ \int \frac {c+d x^6}{x^3 \left (a+b x^6\right )^{5/2}} \, dx=\frac {-4 a^2 c+(-11 b c+2 a d) x^6 \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{2},\frac {5}{3},-\frac {b x^6}{a}\right )}{8 a^3 x^2 \left (a+b x^6\right )^{3/2}} \] Input:

Integrate[(c + d*x^6)/(x^3*(a + b*x^6)^(5/2)),x]
 

Output:

(-4*a^2*c + (-11*b*c + 2*a*d)*x^6*(a + b*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeo 
metric2F1[2/3, 5/2, 5/3, -((b*x^6)/a)])/(8*a^3*x^2*(a + b*x^6)^(3/2))
 

Rubi [A] (warning: unable to verify)

Time = 1.00 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {955, 807, 819, 819, 832, 759, 2416}

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.

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

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 819

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

\(\Big \downarrow \) 819

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

\(\Big \downarrow \) 832

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

\(\Big \downarrow \) 759

\(\displaystyle -\frac {(11 b c-2 a d) \left (\frac {5 \left (\frac {2 x^4}{3 a \sqrt {a+b x^6}}-\frac {\frac {\int \frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^6+a}}dx^2}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{3 a}\right )}{9 a}+\frac {2 x^4}{9 a \left (a+b x^6\right )^{3/2}}\right )}{4 a}-\frac {c}{2 a x^2 \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 2416

\(\displaystyle -\frac {(11 b c-2 a d) \left (\frac {5 \left (\frac {2 x^4}{3 a \sqrt {a+b x^6}}-\frac {\frac {\frac {2 \sqrt {a+b x^6}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{3 a}\right )}{9 a}+\frac {2 x^4}{9 a \left (a+b x^6\right )^{3/2}}\right )}{4 a}-\frac {c}{2 a x^2 \left (a+b x^6\right )^{3/2}}\)

Input:

Int[(c + d*x^6)/(x^3*(a + b*x^6)^(5/2)),x]
 

Output:

-1/2*c/(a*x^2*(a + b*x^6)^(3/2)) - ((11*b*c - 2*a*d)*((2*x^4)/(9*a*(a + b* 
x^6)^(3/2)) + (5*((2*x^4)/(3*a*Sqrt[a + b*x^6]) - (((2*Sqrt[a + b*x^6])/(b 
^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)) - (3^(1/4)*Sqrt[2 - Sqrt[3]] 
*a^(1/3)*(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)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)^2]*EllipticE[ArcSin[((1 - 
Sqrt[3])*a^(1/3) + b^(1/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)], -7 
 - 4*Sqrt[3]])/(b^(1/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x^2))/((1 + Sqrt[ 
3])*a^(1/3) + b^(1/3)*x^2)^2]*Sqrt[a + b*x^6]))/b^(1/3) - (2*(1 - Sqrt[3]) 
*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)^2]*Ellip 
ticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + 
 b^(1/3)*x^2)], -7 - 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) + 
 b^(1/3)*x^2))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)^2]*Sqrt[a + b*x^6]))/ 
(3*a)))/(9*a)))/(4*a)
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

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 832

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

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 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {d \,x^{6}+c}{x^{3} \left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

int((d*x^6+c)/x^3/(b*x^6+a)^(5/2),x)
 

Output:

int((d*x^6+c)/x^3/(b*x^6+a)^(5/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.28 \[ \int \frac {c+d x^6}{x^3 \left (a+b x^6\right )^{5/2}} \, dx=-\frac {5 \, {\left ({\left (11 \, b^{3} c - 2 \, a b^{2} d\right )} x^{14} + 2 \, {\left (11 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{8} + {\left (11 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right )\right ) + {\left (5 \, {\left (11 \, b^{3} c - 2 \, a b^{2} d\right )} x^{12} + 8 \, {\left (11 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{6} + 27 \, a^{2} b c\right )} \sqrt {b x^{6} + a}}{54 \, {\left (a^{3} b^{3} x^{14} + 2 \, a^{4} b^{2} x^{8} + a^{5} b x^{2}\right )}} \] Input:

integrate((d*x^6+c)/x^3/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

-1/54*(5*((11*b^3*c - 2*a*b^2*d)*x^14 + 2*(11*a*b^2*c - 2*a^2*b*d)*x^8 + ( 
11*a^2*b*c - 2*a^3*d)*x^2)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassP 
Inverse(0, -4*a/b, x^2)) + (5*(11*b^3*c - 2*a*b^2*d)*x^12 + 8*(11*a*b^2*c 
- 2*a^2*b*d)*x^6 + 27*a^2*b*c)*sqrt(b*x^6 + a))/(a^3*b^3*x^14 + 2*a^4*b^2* 
x^8 + a^5*b*x^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^3 \left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x**6+c)/x**3/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {c+d x^6}{x^3 \left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:

integrate((d*x^6+c)/x^3/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(5/2)*x^3), x)
 

Giac [F]

\[ \int \frac {c+d x^6}{x^3 \left (a+b x^6\right )^{5/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {5}{2}} x^{3}} \,d x } \] Input:

integrate((d*x^6+c)/x^3/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(5/2)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^3 \left (a+b x^6\right )^{5/2}} \, dx=\int \frac {d\,x^6+c}{x^3\,{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((c + d*x^6)/(x^3*(a + b*x^6)^(5/2)),x)
 

Output:

int((c + d*x^6)/(x^3*(a + b*x^6)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {c+d x^6}{x^3 \left (a+b x^6\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{6}+a}\, d -2 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{21}+3 a \,b^{2} x^{15}+3 a^{2} b \,x^{9}+a^{3} x^{3}}d x \right ) a^{3} d \,x^{2}+11 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{21}+3 a \,b^{2} x^{15}+3 a^{2} b \,x^{9}+a^{3} x^{3}}d x \right ) a^{2} b c \,x^{2}-4 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{21}+3 a \,b^{2} x^{15}+3 a^{2} b \,x^{9}+a^{3} x^{3}}d x \right ) a^{2} b d \,x^{8}+22 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{21}+3 a \,b^{2} x^{15}+3 a^{2} b \,x^{9}+a^{3} x^{3}}d x \right ) a \,b^{2} c \,x^{8}-2 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{21}+3 a \,b^{2} x^{15}+3 a^{2} b \,x^{9}+a^{3} x^{3}}d x \right ) a \,b^{2} d \,x^{14}+11 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{21}+3 a \,b^{2} x^{15}+3 a^{2} b \,x^{9}+a^{3} x^{3}}d x \right ) b^{3} c \,x^{14}}{11 b \,x^{2} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int((d*x^6+c)/x^3/(b*x^6+a)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

( - sqrt(a + b*x**6)*d - 2*int(sqrt(a + b*x**6)/(a**3*x**3 + 3*a**2*b*x**9 
 + 3*a*b**2*x**15 + b**3*x**21),x)*a**3*d*x**2 + 11*int(sqrt(a + b*x**6)/( 
a**3*x**3 + 3*a**2*b*x**9 + 3*a*b**2*x**15 + b**3*x**21),x)*a**2*b*c*x**2 
- 4*int(sqrt(a + b*x**6)/(a**3*x**3 + 3*a**2*b*x**9 + 3*a*b**2*x**15 + b** 
3*x**21),x)*a**2*b*d*x**8 + 22*int(sqrt(a + b*x**6)/(a**3*x**3 + 3*a**2*b* 
x**9 + 3*a*b**2*x**15 + b**3*x**21),x)*a*b**2*c*x**8 - 2*int(sqrt(a + b*x* 
*6)/(a**3*x**3 + 3*a**2*b*x**9 + 3*a*b**2*x**15 + b**3*x**21),x)*a*b**2*d* 
x**14 + 11*int(sqrt(a + b*x**6)/(a**3*x**3 + 3*a**2*b*x**9 + 3*a*b**2*x**1 
5 + b**3*x**21),x)*b**3*c*x**14)/(11*b*x**2*(a**2 + 2*a*b*x**6 + b**2*x**1 
2))