\(\int \frac {x^8}{(a+b x^3)^2 (c+d x^3)^{3/2}} \, dx\) [663]

Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=-\frac {2 c^2}{3 d (b c-a d)^2 \sqrt {c+d x^3}}-\frac {a^2 \sqrt {c+d x^3}}{3 b (b c-a d)^2 \left (a+b x^3\right )}+\frac {a (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{5/2}} \] Output:

-2/3*c^2/d/(-a*d+b*c)^2/(d*x^3+c)^(1/2)-1/3*a^2*(d*x^3+c)^(1/2)/b/(-a*d+b* 
c)^2/(b*x^3+a)+1/3*a*(-a*d+4*b*c)*arctanh(b^(1/2)*(d*x^3+c)^(1/2)/(-a*d+b* 
c)^(1/2))/b^(3/2)/(-a*d+b*c)^(5/2)
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-\frac {\sqrt {b} \left (2 a b c^2+2 b^2 c^2 x^3+a^2 d \left (c+d x^3\right )\right )}{d (b c-a d)^2 \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {a (-4 b c+a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}}{3 b^{3/2}} \] Input:

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

Output:

(-((Sqrt[b]*(2*a*b*c^2 + 2*b^2*c^2*x^3 + a^2*d*(c + d*x^3)))/(d*(b*c - a*d 
)^2*(a + b*x^3)*Sqrt[c + d*x^3])) + (a*(-4*b*c + a*d)*ArcTan[(Sqrt[b]*Sqrt 
[c + d*x^3])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(5/2))/(3*b^(3/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 100, 27, 87, 73, 221}

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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 \left (a^2 d^2+2 b^2 c^2\right )}{d \sqrt {c+d x^3} (b c-a d)}-\frac {2 a \sqrt {b} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}}{2 b^2 (b c-a d)}-\frac {a^2}{b^2 \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)}\right )\)

Input:

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

Output:

(-(a^2/(b^2*(b*c - a*d)*(a + b*x^3)*Sqrt[c + d*x^3])) - ((2*(2*b^2*c^2 + a 
^2*d^2))/(d*(b*c - a*d)*Sqrt[c + d*x^3]) - (2*a*Sqrt[b]*(4*b*c - a*d)*ArcT 
anh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/(2*b^2* 
(b*c - a*d)))/3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [A] (verified)

Time = 1.57 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(-\frac {-a d \sqrt {d \,x^{3}+c}\, \left (b \,x^{3}+a \right ) \left (a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (2 b^{2} c^{2} x^{3}+2 b \,c^{2} a +a^{2} d \left (d \,x^{3}+c \right )\right ) \sqrt {\left (a d -b c \right ) b}}{3 \sqrt {\left (a d -b c \right ) b}\, \sqrt {d \,x^{3}+c}\, b d \left (b \,x^{3}+a \right ) \left (a d -b c \right )^{2}}\)	\(146\)
default	\(-\frac {2}{3 b^{2} d \sqrt {d \,x^{3}+c}}+\frac {a^{2} d \left (-\frac {b \sqrt {d \,x^{3}+c}}{d \left (b \,x^{3}+a \right )}-\frac {3 b \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}-\frac {2}{\sqrt {d \,x^{3}+c}}\right )}{3 b^{2} \left (a d -b c \right )^{2}}-\frac {4 a \left (-\frac {b \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}-\frac {1}{\sqrt {d \,x^{3}+c}}\right )}{3 b^{2} \left (a d -b c \right )}\)	\(180\)
elliptic	\(-\frac {2 c^{2}}{3 d \left (a d -b c \right )^{2} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {a^{2} \sqrt {d \,x^{3}+c}}{3 \left (a d -b c \right )^{2} b \left (b \,x^{3}+a \right )}-\frac {i a \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (a d -4 b c \right ) \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {b \left (2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d \right )}{2 d \left (a d -b c \right )}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \left (a d -b c \right )^{3} \sqrt {d \,x^{3}+c}}\right )}{6 b \,d^{2}}\)	\(506\)

Input:

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

Output:

-1/3*(-a*d*(d*x^3+c)^(1/2)*(b*x^3+a)*(a*d-4*b*c)*arctan(b*(d*x^3+c)^(1/2)/ 
((a*d-b*c)*b)^(1/2))+(2*b^2*c^2*x^3+2*b*c^2*a+a^2*d*(d*x^3+c))*((a*d-b*c)* 
b)^(1/2))/((a*d-b*c)*b)^(1/2)/(d*x^3+c)^(1/2)/b/d/(b*x^3+a)/(a*d-b*c)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (113) = 226\).

Time = 0.26 (sec) , antiderivative size = 746, normalized size of antiderivative = 5.61 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} \sqrt {b^{2} c - a b d}}{b x^{3} + a}\right ) + 2 \, {\left (2 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + {\left (2 \, b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a b^{5} c^{4} d - 3 \, a^{2} b^{4} c^{3} d^{2} + 3 \, a^{3} b^{3} c^{2} d^{3} - a^{4} b^{2} c d^{4} + {\left (b^{6} c^{3} d^{2} - 3 \, a b^{5} c^{2} d^{3} + 3 \, a^{2} b^{4} c d^{4} - a^{3} b^{3} d^{5}\right )} x^{6} + {\left (b^{6} c^{4} d - 2 \, a b^{5} c^{3} d^{2} + 2 \, a^{3} b^{3} c d^{4} - a^{4} b^{2} d^{5}\right )} x^{3}\right )}}, -\frac {{\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-b^{2} c + a b d}}{b d x^{3} + b c}\right ) + {\left (2 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + {\left (2 \, b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a b^{5} c^{4} d - 3 \, a^{2} b^{4} c^{3} d^{2} + 3 \, a^{3} b^{3} c^{2} d^{3} - a^{4} b^{2} c d^{4} + {\left (b^{6} c^{3} d^{2} - 3 \, a b^{5} c^{2} d^{3} + 3 \, a^{2} b^{4} c d^{4} - a^{3} b^{3} d^{5}\right )} x^{6} + {\left (b^{6} c^{4} d - 2 \, a b^{5} c^{3} d^{2} + 2 \, a^{3} b^{3} c d^{4} - a^{4} b^{2} d^{5}\right )} x^{3}\right )}}\right ] \] Input:

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

Output:

[-1/6*(((4*a*b^2*c*d^2 - a^2*b*d^3)*x^6 + 4*a^2*b*c^2*d - a^3*c*d^2 + (4*a 
*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3)*sqrt(b^2*c - a*b*d)*log((b*d*x^ 
3 + 2*b*c - a*d - 2*sqrt(d*x^3 + c)*sqrt(b^2*c - a*b*d))/(b*x^3 + a)) + 2* 
(2*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 + (2*b^4*c^3 - 2*a*b^3*c^2*d + 
a^2*b^2*c*d^2 - a^3*b*d^3)*x^3)*sqrt(d*x^3 + c))/(a*b^5*c^4*d - 3*a^2*b^4* 
c^3*d^2 + 3*a^3*b^3*c^2*d^3 - a^4*b^2*c*d^4 + (b^6*c^3*d^2 - 3*a*b^5*c^2*d 
^3 + 3*a^2*b^4*c*d^4 - a^3*b^3*d^5)*x^6 + (b^6*c^4*d - 2*a*b^5*c^3*d^2 + 2 
*a^3*b^3*c*d^4 - a^4*b^2*d^5)*x^3), -1/3*(((4*a*b^2*c*d^2 - a^2*b*d^3)*x^6 
 + 4*a^2*b*c^2*d - a^3*c*d^2 + (4*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x 
^3)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d)/(b*d* 
x^3 + b*c)) + (2*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 + (2*b^4*c^3 - 2* 
a*b^3*c^2*d + a^2*b^2*c*d^2 - a^3*b*d^3)*x^3)*sqrt(d*x^3 + c))/(a*b^5*c^4* 
d - 3*a^2*b^4*c^3*d^2 + 3*a^3*b^3*c^2*d^3 - a^4*b^2*c*d^4 + (b^6*c^3*d^2 - 
 3*a*b^5*c^2*d^3 + 3*a^2*b^4*c*d^4 - a^3*b^3*d^5)*x^6 + (b^6*c^4*d - 2*a*b 
^5*c^3*d^2 + 2*a^3*b^3*c*d^4 - a^4*b^2*d^5)*x^3)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.53 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=-\frac {\frac {{\left (4 \, a b c d^{2} - a^{2} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (d x^{3} + c\right )} b^{2} c^{2} d - 2 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + {\left (d x^{3} + c\right )} a^{2} d^{3}}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} b - \sqrt {d x^{3} + c} b c + \sqrt {d x^{3} + c} a d\right )}}}{3 \, d^{2}} \] Input:

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

Output:

-1/3*((4*a*b*c*d^2 - a^2*d^3)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d 
))/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*sqrt(-b^2*c + a*b*d)) + (2*(d*x^3 
+ c)*b^2*c^2*d - 2*b^2*c^3*d + 2*a*b*c^2*d^2 + (d*x^3 + c)*a^2*d^3)/((b^3* 
c^2 - 2*a*b^2*c*d + a^2*b*d^2)*((d*x^3 + c)^(3/2)*b - sqrt(d*x^3 + c)*b*c 
+ sqrt(d*x^3 + c)*a*d)))/d^2
 

Mupad [B] (verification not implemented)

Time = 6.83 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.76 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {\sqrt {d\,x^3+c}\,\left (x^3\,\left (\frac {\left (\frac {3\,b\,d\,\left (a\,d+b\,c\right )-b\,d\,\left (a\,d+2\,b\,c\right )}{3\,\left (a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}-\frac {b\,d\,\left (a\,d+b\,c\right )}{a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d}\right )\,\left (a\,d+b\,c\right )}{b\,d}+\frac {a\,b\,c\,d}{a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d}\right )+\frac {a\,c\,\left (\frac {3\,b\,d\,\left (a\,d+b\,c\right )-b\,d\,\left (a\,d+2\,b\,c\right )}{3\,\left (a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}-\frac {b\,d\,\left (a\,d+b\,c\right )}{a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d}\right )}{b\,d}\right )}{b\,d\,x^6+\left (a\,d+b\,c\right )\,x^3+a\,c}+\frac {a\,\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\left (a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{5/2}} \] Input:

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

Output:

((c + d*x^3)^(1/2)*(x^3*((((3*b*d*(a*d + b*c) - b*d*(a*d + 2*b*c))/(3*(a^2 
*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) - (b*d*(a*d + b*c))/(a^2*b*d^3 + b^3* 
c^2*d - 2*a*b^2*c*d^2))*(a*d + b*c))/(b*d) + (a*b*c*d)/(a^2*b*d^3 + b^3*c^ 
2*d - 2*a*b^2*c*d^2)) + (a*c*((3*b*d*(a*d + b*c) - b*d*(a*d + 2*b*c))/(3*( 
a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) - (b*d*(a*d + b*c))/(a^2*b*d^3 + b 
^3*c^2*d - 2*a*b^2*c*d^2)))/(b*d)))/(a*c + x^3*(a*d + b*c) + b*d*x^6) + (a 
*log((2*b*c - a*d + b^(1/2)*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2)*2i + b*d*x 
^3)/(a + b*x^3))*(a*d - 4*b*c)*1i)/(6*b^(3/2)*(a*d - b*c)^(5/2))
 

Reduce [F]

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

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

Output:

( - 4*sqrt(c + d*x**3)*a*c - 2*sqrt(c + d*x**3)*a*d*x**3 - 4*sqrt(c + d*x* 
*3)*b*c*x**3 + 3*int((sqrt(c + d*x**3)*x**5)/(a**3*c**2*d + 2*a**3*c*d**2* 
x**3 + a**3*d**3*x**6 + 2*a**2*b*c**3 + 6*a**2*b*c**2*d*x**3 + 6*a**2*b*c* 
d**2*x**6 + 2*a**2*b*d**3*x**9 + 4*a*b**2*c**3*x**3 + 9*a*b**2*c**2*d*x**6 
 + 6*a*b**2*c*d**2*x**9 + a*b**2*d**3*x**12 + 2*b**3*c**3*x**6 + 4*b**3*c* 
*2*d*x**9 + 2*b**3*c*d**2*x**12),x)*a**4*c*d**3 + 3*int((sqrt(c + d*x**3)* 
x**5)/(a**3*c**2*d + 2*a**3*c*d**2*x**3 + a**3*d**3*x**6 + 2*a**2*b*c**3 + 
 6*a**2*b*c**2*d*x**3 + 6*a**2*b*c*d**2*x**6 + 2*a**2*b*d**3*x**9 + 4*a*b* 
*2*c**3*x**3 + 9*a*b**2*c**2*d*x**6 + 6*a*b**2*c*d**2*x**9 + a*b**2*d**3*x 
**12 + 2*b**3*c**3*x**6 + 4*b**3*c**2*d*x**9 + 2*b**3*c*d**2*x**12),x)*a** 
4*d**4*x**3 - 6*int((sqrt(c + d*x**3)*x**5)/(a**3*c**2*d + 2*a**3*c*d**2*x 
**3 + a**3*d**3*x**6 + 2*a**2*b*c**3 + 6*a**2*b*c**2*d*x**3 + 6*a**2*b*c*d 
**2*x**6 + 2*a**2*b*d**3*x**9 + 4*a*b**2*c**3*x**3 + 9*a*b**2*c**2*d*x**6 
+ 6*a*b**2*c*d**2*x**9 + a*b**2*d**3*x**12 + 2*b**3*c**3*x**6 + 4*b**3*c** 
2*d*x**9 + 2*b**3*c*d**2*x**12),x)*a**3*b*c**2*d**2 - 3*int((sqrt(c + d*x* 
*3)*x**5)/(a**3*c**2*d + 2*a**3*c*d**2*x**3 + a**3*d**3*x**6 + 2*a**2*b*c* 
*3 + 6*a**2*b*c**2*d*x**3 + 6*a**2*b*c*d**2*x**6 + 2*a**2*b*d**3*x**9 + 4* 
a*b**2*c**3*x**3 + 9*a*b**2*c**2*d*x**6 + 6*a*b**2*c*d**2*x**9 + a*b**2*d* 
*3*x**12 + 2*b**3*c**3*x**6 + 4*b**3*c**2*d*x**9 + 2*b**3*c*d**2*x**12),x) 
*a**3*b*c*d**3*x**3 + 3*int((sqrt(c + d*x**3)*x**5)/(a**3*c**2*d + 2*a*...