\(\int \frac {c+d x^6}{x^9 (a+b x^6)^{3/2}} \, dx\) [26]

Optimal result

Integrand size = 22, antiderivative size = 610 \[ \int \frac {c+d x^6}{x^9 \left (a+b x^6\right )^{3/2}} \, dx=-\frac {c}{8 a x^8 \sqrt {a+b x^6}}-\frac {11 b c-8 a d}{24 a^2 x^2 \sqrt {a+b x^6}}+\frac {5 (11 b c-8 a d) \sqrt {a+b x^6}}{48 a^3 x^2}-\frac {5 \sqrt [3]{b} (11 b c-8 a d) \sqrt {a+b x^6}}{48 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )}+\frac {5 \sqrt {2-\sqrt {3}} \sqrt [3]{b} (11 b c-8 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right )|-7-4 \sqrt {3}\right )}{32\ 3^{3/4} a^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {5 \sqrt [3]{b} (11 b c-8 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{24 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/8*c/a/x^8/(b*x^6+a)^(1/2)-1/24*(-8*a*d+11*b*c)/a^2/x^2/(b*x^6+a)^(1/2)+ 
5/48*(-8*a*d+11*b*c)*(b*x^6+a)^(1/2)/a^3/x^2-5/48*b^(1/3)*(-8*a*d+11*b*c)* 
(b*x^6+a)^(1/2)/a^3/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^2)+5/96*(1/2*6^(1/2)-1/ 
2*2^(1/2))*b^(1/3)*(-8*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)*Ellip 
ticE(((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)/(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)-5/144*b^(1/3)*(-8*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)/(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 = 10.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.12 \[ \int \frac {c+d x^6}{x^9 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-2 a c+(11 b c-8 a d) x^6 \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{2},\frac {2}{3},-\frac {b x^6}{a}\right )}{16 a^2 x^8 \sqrt {a+b x^6}} \] Input:

Integrate[(c + d*x^6)/(x^9*(a + b*x^6)^(3/2)),x]
 

Output:

(-2*a*c + (11*b*c - 8*a*d)*x^6*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[-1/3, 
 3/2, 2/3, -((b*x^6)/a)])/(16*a^2*x^8*Sqrt[a + b*x^6])
 

Rubi [A] (warning: unable to verify)

Time = 1.01 (sec) , antiderivative size = 610, 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, 847, 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.

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

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 807

\(\displaystyle -\frac {(11 b c-8 a d) \int \frac {1}{x^4 \left (b x^6+a\right )^{3/2}}dx^2}{16 a}-\frac {c}{8 a x^8 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(11 b c-8 a d) \left (\frac {5 \int \frac {1}{x^4 \sqrt {b x^6+a}}dx^2}{3 a}+\frac {2}{3 a x^2 \sqrt {a+b x^6}}\right )}{16 a}-\frac {c}{8 a x^8 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {(11 b c-8 a d) \left (\frac {5 \left (\frac {b \int \frac {x^2}{\sqrt {b x^6+a}}dx^2}{2 a}-\frac {\sqrt {a+b x^6}}{a x^2}\right )}{3 a}+\frac {2}{3 a x^2 \sqrt {a+b x^6}}\right )}{16 a}-\frac {c}{8 a x^8 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 832

\(\displaystyle -\frac {(11 b c-8 a d) \left (\frac {5 \left (\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^6+a}}dx^2}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^6+a}}dx^2}{\sqrt [3]{b}}\right )}{2 a}-\frac {\sqrt {a+b x^6}}{a x^2}\right )}{3 a}+\frac {2}{3 a x^2 \sqrt {a+b x^6}}\right )}{16 a}-\frac {c}{8 a x^8 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {(11 b c-8 a d) \left (\frac {5 \left (\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^6+a}}dx^2}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{2 a}-\frac {\sqrt {a+b x^6}}{a x^2}\right )}{3 a}+\frac {2}{3 a x^2 \sqrt {a+b x^6}}\right )}{16 a}-\frac {c}{8 a x^8 \sqrt {a+b x^6}}\)

\(\Big \downarrow \) 2416

\(\displaystyle -\frac {(11 b c-8 a d) \left (\frac {5 \left (\frac {b \left (\frac {\frac {2 \sqrt {a+b x^6}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}\right )}{2 a}-\frac {\sqrt {a+b x^6}}{a x^2}\right )}{3 a}+\frac {2}{3 a x^2 \sqrt {a+b x^6}}\right )}{16 a}-\frac {c}{8 a x^8 \sqrt {a+b x^6}}\)

Input:

Int[(c + d*x^6)/(x^9*(a + b*x^6)^(3/2)),x]
 

Output:

-1/8*c/(a*x^8*Sqrt[a + b*x^6]) - ((11*b*c - 8*a*d)*(2/(3*a*x^2*Sqrt[a + b* 
x^6]) + (5*(-(Sqrt[a + b*x^6]/(a*x^2)) + (b*(((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*S 
qrt[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]*EllipticF[A 
rcSin[((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])))/(2*a) 
))/(3*a)))/(16*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 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 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]

\[\int \frac {d \,x^{6}+c}{x^{9} \left (b \,x^{6}+a \right )^{\frac {3}{2}}}d x\]

Input:

int((d*x^6+c)/x^9/(b*x^6+a)^(3/2),x)
 

Output:

int((d*x^6+c)/x^9/(b*x^6+a)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.21 \[ \int \frac {c+d x^6}{x^9 \left (a+b x^6\right )^{3/2}} \, dx=\frac {5 \, {\left ({\left (11 \, b^{2} c - 8 \, a b d\right )} x^{14} + {\left (11 \, a b c - 8 \, a^{2} d\right )} x^{8}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right )\right ) + {\left (5 \, {\left (11 \, b^{2} c - 8 \, a b d\right )} x^{12} + 3 \, {\left (11 \, a b c - 8 \, a^{2} d\right )} x^{6} - 6 \, a^{2} c\right )} \sqrt {b x^{6} + a}}{48 \, {\left (a^{3} b x^{14} + a^{4} x^{8}\right )}} \] Input:

integrate((d*x^6+c)/x^9/(b*x^6+a)^(3/2),x, algorithm="fricas")
 

Output:

1/48*(5*((11*b^2*c - 8*a*b*d)*x^14 + (11*a*b*c - 8*a^2*d)*x^8)*sqrt(b)*wei 
erstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x^2)) + (5*(11*b^2* 
c - 8*a*b*d)*x^12 + 3*(11*a*b*c - 8*a^2*d)*x^6 - 6*a^2*c)*sqrt(b*x^6 + a)) 
/(a^3*b*x^14 + a^4*x^8)
 

Sympy [A] (verification not implemented)

Time = 76.73 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.15 \[ \int \frac {c+d x^6}{x^9 \left (a+b x^6\right )^{3/2}} \, dx=\frac {c \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {3}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{8} \Gamma \left (- \frac {1}{3}\right )} + \frac {d \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{6} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{2}} x^{2} \Gamma \left (\frac {2}{3}\right )} \] Input:

integrate((d*x**6+c)/x**9/(b*x**6+a)**(3/2),x)
 

Output:

c*gamma(-4/3)*hyper((-4/3, 3/2), (-1/3,), b*x**6*exp_polar(I*pi)/a)/(6*a** 
(3/2)*x**8*gamma(-1/3)) + d*gamma(-1/3)*hyper((-1/3, 3/2), (2/3,), b*x**6* 
exp_polar(I*pi)/a)/(6*a**(3/2)*x**2*gamma(2/3))
 

Maxima [F]

\[ \int \frac {c+d x^6}{x^9 \left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} x^{9}} \,d x } \] Input:

integrate((d*x^6+c)/x^9/(b*x^6+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(3/2)*x^9), x)
 

Giac [F]

\[ \int \frac {c+d x^6}{x^9 \left (a+b x^6\right )^{3/2}} \, dx=\int { \frac {d x^{6} + c}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} x^{9}} \,d x } \] Input:

integrate((d*x^6+c)/x^9/(b*x^6+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x^6 + c)/((b*x^6 + a)^(3/2)*x^9), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^9 \left (a+b x^6\right )^{3/2}} \, dx=\int \frac {d\,x^6+c}{x^9\,{\left (b\,x^6+a\right )}^{3/2}} \,d x \] Input:

int((c + d*x^6)/(x^9*(a + b*x^6)^(3/2)),x)
 

Output:

int((c + d*x^6)/(x^9*(a + b*x^6)^(3/2)), x)
 

Reduce [F]

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

int((d*x^6+c)/x^9/(b*x^6+a)^(3/2),x)
 

Output:

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