\(\int \frac {(a+b x^3)^{2/3}}{x^5 (a d-b d x^3)} \, dx\) [818]

Optimal result

Integrand size = 28, antiderivative size = 512 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}-\frac {3 b \left (a+b x^3\right )^{2/3}}{2 a^2 d x}+\frac {2^{2/3} b^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^{5/3} d}+\frac {b^{4/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a^{5/3} d}+\frac {3 b^2 x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{4 a^2 d \sqrt [3]{a+b x^3}}+\frac {b^{4/3} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} a^{5/3} d}+\frac {b^{4/3} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{2} a^{5/3} d}-\frac {2^{2/3} b^{4/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 a^{5/3} d}-\frac {b^{4/3} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} a^{5/3} d} \] Output:

-1/4*(b*x^3+a)^(2/3)/a/d/x^4-3/2*b*(b*x^3+a)^(2/3)/a^2/d/x+1/3*2^(2/3)*b^( 
4/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2)) 
*3^(1/2)/a^(5/3)/d+1/6*b^(4/3)*arctan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/( 
b*x^3+a)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)/a^(5/3)/d+3/4*b^2*x^2*(1+b*x^3/a) 
^(1/3)*hypergeom([1/3, 2/3],[5/3],-b*x^3/a)/a^2/d/(b*x^3+a)^(1/3)+1/12*b^( 
4/3)*ln((a^(1/3)-b^(1/3)*x)^2*(a^(1/3)+b^(1/3)*x)/a)*2^(2/3)/a^(5/3)/d+1/6 
*b^(4/3)*ln(1+2^(2/3)*(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^(2/3)-2^(1/3)*(a^(1/ 
3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(2/3)/a^(5/3)/d-1/3*2^(2/3)*b^(4/3)*ln(1+ 
2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))/a^(5/3)/d-1/4*b^(4/3)*ln(b^(1 
/3)*(a^(1/3)+b^(1/3)*x)/a^(1/3)-2^(2/3)*b^(1/3)*(b*x^3+a)^(1/3)/a^(1/3))*2 
^(2/3)/a^(5/3)/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 11.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.29 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^5 \left (a d-b d x^3\right )} \, dx=\frac {-5 a \left (a^2+7 a b x^3+6 b^2 x^6\right )+35 a b^2 x^6 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )-6 b^3 x^9 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{20 a^3 d x^4 \sqrt [3]{a+b x^3}} \] Input:

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

Output:

(-5*a*(a^2 + 7*a*b*x^3 + 6*b^2*x^6) + 35*a*b^2*x^6*(1 + (b*x^3)/a)^(1/3)*A 
ppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a] - 6*b^3*x^9*(1 + (b*x^3 
)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a])/(20*a^3*d* 
x^4*(a + b*x^3)^(1/3))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {975, 27, 1053, 25, 27, 1054, 2009}

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.

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

\(\Big \downarrow \) 975

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1053

\(\displaystyle \frac {b \left (-\frac {\int -\frac {a b x \left (7 a-3 b x^3\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{a^2}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {\int \frac {a b x \left (7 a-3 b x^3\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{a^2}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \int \frac {x \left (7 a-3 b x^3\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{a}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\)

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {b \left (\frac {2\ 2^{2/3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {2^{2/3} \sqrt [3]{a} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {2^{2/3} \sqrt [3]{a} \log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {2\ 2^{2/3} \sqrt [3]{a} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {\sqrt [3]{a} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{\sqrt [3]{2} b^{2/3}}+\frac {\sqrt [3]{a} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{3 \sqrt [3]{2} b^{2/3}}+\frac {3 x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}}\right )}{a}-\frac {3 \left (a+b x^3\right )^{2/3}}{a x}\right )}{2 a d}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4}\)

Input:

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

Output:

-1/4*(a + b*x^3)^(2/3)/(a*d*x^4) + (b*((-3*(a + b*x^3)^(2/3))/(a*x) + (b*( 
(2*2^(2/3)*a^(1/3)*ArcTan[(1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^ 
3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)) + (2^(2/3)*a^(1/3)*ArcTan[(1 + (2^(1 
/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)) 
+ (3*x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/ 
a)])/(2*(a + b*x^3)^(1/3)) + (a^(1/3)*Log[((a^(1/3) - b^(1/3)*x)^2*(a^(1/3 
) + b^(1/3)*x))/a])/(3*2^(1/3)*b^(2/3)) + (2^(2/3)*a^(1/3)*Log[1 + (2^(2/3 
)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3) 
*x))/(a + b*x^3)^(1/3)])/(3*b^(2/3)) - (2*2^(2/3)*a^(1/3)*Log[1 + (2^(1/3) 
*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)])/(3*b^(2/3)) - (a^(1/3)*Log[(b^ 
(1/3)*(a^(1/3) + b^(1/3)*x))/a^(1/3) - (2^(2/3)*b^(1/3)*(a + b*x^3)^(1/3)) 
/a^(1/3)])/(2^(1/3)*b^(2/3))))/a))/(2*a*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 975

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

rule 1053

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ 
))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b 
*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( 
m + 1))   Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) 
- e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 
) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 
0] && LtQ[m, -1]
 

rule 1054

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, p}, x] && IGtQ[n, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

-Integral((a + b*x**3)**(2/3)/(-a*x**5 + b*x**8), x)/d
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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