\(\int \frac {(a+b x^3)^{2/3} (A+B x^3)}{x^5} \, dx\) [323]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int \frac {\left (a+b x^3\right )^{2/3} \left (A+B x^3\right )}{x^5} \, dx=-\frac {A \left (a+b x^3\right )^{5/3}}{4 a x^4}-\frac {(A b+4 a B) \left (a+b x^3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )}{4 a x \left (1+\frac {b x^3}{a}\right )^{2/3}} \] Output:

-1/4*A*(b*x^3+a)^(5/3)/a/x^4-1/4*(A*b+4*B*a)*(b*x^3+a)^(2/3)*hypergeom([-2 
/3, -1/3],[2/3],-b*x^3/a)/a/x/(1+b*x^3/a)^(2/3)
 

Mathematica [A] (verified)

Time = 10.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^{2/3} \left (A+B x^3\right )}{x^5} \, dx=\frac {\left (a+b x^3\right )^{2/3} \left (-A \left (a+b x^3\right )-\frac {(A b+4 a B) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\right )}{4 a x^4} \] Input:

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

Output:

((a + b*x^3)^(2/3)*(-(A*(a + b*x^3)) - ((A*b + 4*a*B)*x^3*Hypergeometric2F 
1[-2/3, -1/3, 2/3, -((b*x^3)/a)])/(1 + (b*x^3)/a)^(2/3)))/(4*a*x^4)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {955, 889, 888}

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.

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

\(\Big \downarrow \) 955

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

\(\displaystyle -\frac {\left (a+b x^3\right )^{2/3} (4 a B+A b) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )}{4 a x \left (\frac {b x^3}{a}+1\right )^{2/3}}-\frac {A \left (a+b x^3\right )^{5/3}}{4 a x^4}\)

Input:

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

Output:

-1/4*(A*(a + b*x^3)^(5/3))/(a*x^4) - ((A*b + 4*a*B)*(a + b*x^3)^(2/3)*Hype 
rgeometric2F1[-2/3, -1/3, 2/3, -((b*x^3)/a)])/(4*a*x*(1 + (b*x^3)/a)^(2/3) 
)
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^3\right )^{2/3} \left (A+B x^3\right )}{x^5} \, dx=\frac {A a^{\frac {2}{3}} \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {2}{3} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {B a^{\frac {2}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3} \left (A+B x^3\right )}{x^5} \, dx=\frac {-3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a +2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b \,x^{3}-10 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b \,x^{8}+a \,x^{5}}d x \right ) a^{2} x^{4}}{2 x^{4}} \] Input:

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

Output:

( - 3*(a + b*x**3)**(2/3)*a + 2*(a + b*x**3)**(2/3)*b*x**3 - 10*int((a + b 
*x**3)**(2/3)/(a*x**5 + b*x**8),x)*a**2*x**4)/(2*x**4)