\(\int \frac {A+B x^3}{x^5 (a+b x^3)^{3/2}} \, dx\) [221]

Optimal result

Integrand size = 22, antiderivative size = 580 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{3/2}} \, dx=-\frac {A}{4 a x^4 \sqrt {a+b x^3}}-\frac {11 A b-8 a B}{12 a^2 x \sqrt {a+b x^3}}+\frac {5 (11 A b-8 a B) \sqrt {a+b x^3}}{24 a^3 x}-\frac {5 \sqrt [3]{b} (11 A b-8 a B) \sqrt {a+b x^3}}{24 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {5 \sqrt {2-\sqrt {3}} \sqrt [3]{b} (11 A b-8 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{16\ 3^{3/4} a^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {5 \sqrt [3]{b} (11 A b-8 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{12 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

-1/4*A/a/x^4/(b*x^3+a)^(1/2)-1/12*(11*A*b-8*B*a)/a^2/x/(b*x^3+a)^(1/2)+5/2 
4*(11*A*b-8*B*a)*(b*x^3+a)^(1/2)/a^3/x-5/24*b^(1/3)*(11*A*b-8*B*a)*(b*x^3+ 
a)^(1/2)/a^3/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)+5/48*(1/2*6^(1/2)-1/2*2^(1/2) 
)*b^(1/3)*(11*A*b-8*B*a)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b 
^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2) 
)*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)*3^(1/4 
)/a^(8/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^ 
(1/2)/(b*x^3+a)^(1/2)-5/72*b^(1/3)*(11*A*b-8*B*a)*(a^(1/3)+b^(1/3)*x)*((a^ 
(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1 
/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3) 
*x),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)/(( 
1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b*x^3+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 = 72, normalized size of antiderivative = 0.12 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{3/2}} \, dx=\frac {-2 a A+(11 A b-8 a B) x^3 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{2},\frac {2}{3},-\frac {b x^3}{a}\right )}{8 a^2 x^4 \sqrt {a+b x^3}} \] Input:

Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^(3/2)),x]
 

Output:

(-2*a*A + (11*A*b - 8*a*B)*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/3, 
 3/2, 2/3, -((b*x^3)/a)])/(8*a^2*x^4*Sqrt[a + b*x^3])
 

Rubi [A] (warning: unable to verify)

Time = 1.01 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {955, 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.

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

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 819

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

\(\Big \downarrow \) 847

\(\displaystyle -\frac {(11 A b-8 a B) \left (\frac {5 \left (\frac {b \int \frac {x}{\sqrt {b x^3+a}}dx}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{3 a}+\frac {2}{3 a x \sqrt {a+b x^3}}\right )}{8 a}-\frac {A}{4 a x^4 \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 832

\(\displaystyle -\frac {(11 A b-8 a B) \left (\frac {5 \left (\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{3 a}+\frac {2}{3 a x \sqrt {a+b x^3}}\right )}{8 a}-\frac {A}{4 a x^4 \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2416

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

Input:

Int[(A + B*x^3)/(x^5*(a + b*x^3)^(3/2)),x]
 

Output:

-1/4*A/(a*x^4*Sqrt[a + b*x^3]) - ((11*A*b - 8*a*B)*(2/(3*a*x*Sqrt[a + b*x^ 
3]) + (5*(-(Sqrt[a + b*x^3]/(a*x)) + (b*(((2*Sqrt[a + b*x^3])/(b^(1/3)*((1 
 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(a^ 
(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + 
Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + 
 b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(b^(1/3 
)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x) 
^2]*Sqrt[a + b*x^3]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3) 
*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(( 
1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/ 
3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^ 
(1/4)*b^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) 
+ b^(1/3)*x)^2]*Sqrt[a + b*x^3])))/(2*a)))/(3*a)))/(8*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 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 [A] (verified)

Time = 1.99 (sec) , antiderivative size = 540, normalized size of antiderivative = 0.93

method	result	size
elliptic	\(-\frac {A \sqrt {b \,x^{3}+a}}{4 a^{2} x^{4}}+\frac {\left (13 A b -8 B a \right ) \sqrt {b \,x^{3}+a}}{8 a^{3} x}+\frac {2 b \,x^{2} \left (A b -B a \right )}{3 a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 i \left (-\frac {b \left (13 A b -8 B a \right )}{16 a^{3}}-\frac {\left (A b -B a \right ) b}{3 a^{3}}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{3 b \sqrt {b \,x^{3}+a}}\)	\(540\)
default	\(\text {Expression too large to display}\)	\(975\)
risch	\(\text {Expression too large to display}\)	\(983\)

Input:

int((B*x^3+A)/x^5/(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*A/a^2*(b*x^3+a)^(1/2)/x^4+1/8/a^3*(13*A*b-8*B*a)*(b*x^3+a)^(1/2)/x+2/ 
3*b*x^2/a^3*(A*b-B*a)/((x^3+a/b)*b)^(1/2)-2/3*I*(-1/16*b*(13*A*b-8*B*a)/a^ 
3-1/3*(A*b-B*a)/a^3*b)*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3) 
-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*( 
-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/ 
2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/( 
-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2) 
/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I* 
3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a 
*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)) 
+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I 
*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(- 
a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2) 
))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.22 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{3/2}} \, dx=-\frac {5 \, {\left ({\left (8 \, B a b - 11 \, A b^{2}\right )} x^{7} + {\left (8 \, B a^{2} - 11 \, A a b\right )} x^{4}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (5 \, {\left (8 \, B a b - 11 \, A b^{2}\right )} x^{6} + 3 \, {\left (8 \, B a^{2} - 11 \, A a b\right )} x^{3} + 6 \, A a^{2}\right )} \sqrt {b x^{3} + a}}{24 \, {\left (a^{3} b x^{7} + a^{4} x^{4}\right )}} \] Input:

integrate((B*x^3+A)/x^5/(b*x^3+a)^(3/2),x, algorithm="fricas")
 

Output:

-1/24*(5*((8*B*a*b - 11*A*b^2)*x^7 + (8*B*a^2 - 11*A*a*b)*x^4)*sqrt(b)*wei 
erstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (5*(8*B*a*b - 
 11*A*b^2)*x^6 + 3*(8*B*a^2 - 11*A*a*b)*x^3 + 6*A*a^2)*sqrt(b*x^3 + a))/(a 
^3*b*x^7 + a^4*x^4)
 

Sympy [A] (verification not implemented)

Time = 17.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.15 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {3}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {B \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x \Gamma \left (\frac {2}{3}\right )} \] Input:

integrate((B*x**3+A)/x**5/(b*x**3+a)**(3/2),x)
 

Output:

A*gamma(-4/3)*hyper((-4/3, 3/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*a** 
(3/2)*x**4*gamma(-1/3)) + B*gamma(-1/3)*hyper((-1/3, 3/2), (2/3,), b*x**3* 
exp_polar(I*pi)/a)/(3*a**(3/2)*x*gamma(2/3))
 

Maxima [F]

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

integrate((B*x^3+A)/x^5/(b*x^3+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((B*x^3 + A)/((b*x^3 + a)^(3/2)*x^5), x)
 

Giac [F]

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

integrate((B*x^3+A)/x^5/(b*x^3+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((B*x^3 + A)/((b*x^3 + a)^(3/2)*x^5), x)
 

Mupad [F(-1)]

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

int((A + B*x^3)/(x^5*(a + b*x^3)^(3/2)),x)
 

Output:

int((A + B*x^3)/(x^5*(a + b*x^3)^(3/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{3/2}} \, dx=\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{8}+a \,x^{5}}d x \] Input:

int((B*x^3+A)/x^5/(b*x^3+a)^(3/2),x)
 

Output:

int(sqrt(a + b*x**3)/(a*x**5 + b*x**8),x)