\(\int \frac {(a+b x^3)^{2/3}}{x^{12} (a d-b d x^3)} \, dx\) [813]

Optimal result

Integrand size = 28, antiderivative size = 236 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}-\frac {13 b \left (a+b x^3\right )^{2/3}}{88 a^2 d x^8}-\frac {49 b^2 \left (a+b x^3\right )^{2/3}}{220 a^3 d x^5}-\frac {293 b^3 \left (a+b x^3\right )^{2/3}}{440 a^4 d x^2}+\frac {2^{2/3} b^{11/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^4 d}+\frac {b^{11/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a^4 d}-\frac {b^{11/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^4 d} \] Output:

-1/11*(b*x^3+a)^(2/3)/a/d/x^11-13/88*b*(b*x^3+a)^(2/3)/a^2/d/x^8-49/220*b^ 
2*(b*x^3+a)^(2/3)/a^3/d/x^5-293/440*b^3*(b*x^3+a)^(2/3)/a^4/d/x^2+1/3*2^(2 
/3)*b^(11/3)*arctan(1/3*(1+2*2^(1/3)*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))*3 
^(1/2)/a^4/d+1/6*b^(11/3)*ln(-b*d*x^3+a*d)*2^(2/3)/a^4/d-1/2*b^(11/3)*ln(2 
^(1/3)*b^(1/3)*x-(b*x^3+a)^(1/3))*2^(2/3)/a^4/d
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx=\frac {-\frac {3 \left (a+b x^3\right )^{2/3} \left (40 a^3+65 a^2 b x^3+98 a b^2 x^6+293 b^3 x^9\right )}{x^{11}}+440\ 2^{2/3} \sqrt {3} b^{11/3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}}\right )-440\ 2^{2/3} b^{11/3} \log \left (-2 \sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}\right )+220\ 2^{2/3} b^{11/3} \log \left (2 b^{2/3} x^2+2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{1320 a^4 d} \] Input:

Integrate[(a + b*x^3)^(2/3)/(x^12*(a*d - b*d*x^3)),x]
 

Output:

((-3*(a + b*x^3)^(2/3)*(40*a^3 + 65*a^2*b*x^3 + 98*a*b^2*x^6 + 293*b^3*x^9 
))/x^11 + 440*2^(2/3)*Sqrt[3]*b^(11/3)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3) 
*x + 2^(2/3)*(a + b*x^3)^(1/3))] - 440*2^(2/3)*b^(11/3)*Log[-2*b^(1/3)*x + 
 2^(2/3)*(a + b*x^3)^(1/3)] + 220*2^(2/3)*b^(11/3)*Log[2*b^(2/3)*x^2 + 2^( 
2/3)*b^(1/3)*x*(a + b*x^3)^(1/3) + 2^(1/3)*(a + b*x^3)^(2/3)])/(1320*a^4*d 
)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {975, 27, 1053, 27, 1053, 25, 27, 1053, 27, 901}

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.

\(\displaystyle \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx\)

\(\Big \downarrow \) 975

\(\displaystyle \frac {\int \frac {b \left (9 b x^3+13 a\right )}{x^9 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \int \frac {9 b x^3+13 a}{x^9 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {b \left (-\frac {\int -\frac {2 a b \left (39 b x^3+49 a\right )}{x^6 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{8 a^2}-\frac {13 \left (a+b x^3\right )^{2/3}}{8 a x^8}\right )}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \int \frac {39 b x^3+49 a}{x^6 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{4 a}-\frac {13 \left (a+b x^3\right )^{2/3}}{8 a x^8}\right )}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {b \left (\frac {b \left (-\frac {\int -\frac {a b \left (147 b x^3+293 a\right )}{x^3 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{5 a^2}-\frac {49 \left (a+b x^3\right )^{2/3}}{5 a x^5}\right )}{4 a}-\frac {13 \left (a+b x^3\right )^{2/3}}{8 a x^8}\right )}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {a b \left (147 b x^3+293 a\right )}{x^3 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{5 a^2}-\frac {49 \left (a+b x^3\right )^{2/3}}{5 a x^5}\right )}{4 a}-\frac {13 \left (a+b x^3\right )^{2/3}}{8 a x^8}\right )}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \int \frac {147 b x^3+293 a}{x^3 \left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{5 a}-\frac {49 \left (a+b x^3\right )^{2/3}}{5 a x^5}\right )}{4 a}-\frac {13 \left (a+b x^3\right )^{2/3}}{8 a x^8}\right )}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (-\frac {\int -\frac {880 a^2 b}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{2 a^2}-\frac {293 \left (a+b x^3\right )^{2/3}}{2 a x^2}\right )}{5 a}-\frac {49 \left (a+b x^3\right )^{2/3}}{5 a x^5}\right )}{4 a}-\frac {13 \left (a+b x^3\right )^{2/3}}{8 a x^8}\right )}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (440 b \int \frac {1}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx-\frac {293 \left (a+b x^3\right )^{2/3}}{2 a x^2}\right )}{5 a}-\frac {49 \left (a+b x^3\right )^{2/3}}{5 a x^5}\right )}{4 a}-\frac {13 \left (a+b x^3\right )^{2/3}}{8 a x^8}\right )}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

\(\Big \downarrow \) 901

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (440 b \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a \sqrt [3]{b}}+\frac {\log \left (a-b x^3\right )}{6 \sqrt [3]{2} a \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{2} a \sqrt [3]{b}}\right )-\frac {293 \left (a+b x^3\right )^{2/3}}{2 a x^2}\right )}{5 a}-\frac {49 \left (a+b x^3\right )^{2/3}}{5 a x^5}\right )}{4 a}-\frac {13 \left (a+b x^3\right )^{2/3}}{8 a x^8}\right )}{11 a d}-\frac {\left (a+b x^3\right )^{2/3}}{11 a d x^{11}}\)

Input:

Int[(a + b*x^3)^(2/3)/(x^12*(a*d - b*d*x^3)),x]
 

Output:

-1/11*(a + b*x^3)^(2/3)/(a*d*x^11) + (b*((-13*(a + b*x^3)^(2/3))/(8*a*x^8) 
 + (b*((-49*(a + b*x^3)^(2/3))/(5*a*x^5) + (b*((-293*(a + b*x^3)^(2/3))/(2 
*a*x^2) + 440*b*(ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt 
[3]]/(2^(1/3)*Sqrt[3]*a*b^(1/3)) + Log[a - b*x^3]/(6*2^(1/3)*a*b^(1/3)) - 
Log[2^(1/3)*b^(1/3)*x - (a + b*x^3)^(1/3)]/(2*2^(1/3)*a*b^(1/3)))))/(5*a)) 
)/(4*a)))/(11*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 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

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]
 

Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.73

method	result	size
pseudoelliptic	\(\frac {220 x^{11} 2^{\frac {2}{3}} \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{3 b^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {b^{\frac {2}{3}} 2^{\frac {2}{3}} x^{2}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{\frac {1}{3}} 2^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right ) b^{\frac {11}{3}}-3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (293 b^{3} x^{9}+98 a \,b^{2} x^{6}+65 a^{2} b \,x^{3}+40 a^{3}\right )}{1320 x^{11} a^{4} d}\)	\(172\)

Input:

int((b*x^3+a)^(2/3)/x^12/(-b*d*x^3+a*d),x,method=_RETURNVERBOSE)
 

Output:

1/1320*(220*x^11*2^(2/3)*(-2*arctan(1/3*3^(1/2)*(2^(2/3)*(b*x^3+a)^(1/3)+b 
^(1/3)*x)/b^(1/3)/x)*3^(1/2)+ln((b^(2/3)*2^(2/3)*x^2+(b*x^3+a)^(1/3)*b^(1/ 
3)*2^(1/3)*x+(b*x^3+a)^(2/3))/x^2)-2*ln((-2^(1/3)*b^(1/3)*x+(b*x^3+a)^(1/3 
))/x))*b^(11/3)-3*(b*x^3+a)^(2/3)*(293*b^3*x^9+98*a*b^2*x^6+65*a^2*b*x^3+4 
0*a^3))/x^11/a^4/d
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^3+a)^(2/3)/x^12/(-b*d*x^3+a*d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x**3+a)**(2/3)/x**12/(-b*d*x**3+a*d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{12}} \,d x } \] Input:

integrate((b*x^3+a)^(2/3)/x^12/(-b*d*x^3+a*d),x, algorithm="maxima")
 

Output:

-integrate((b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^12), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{12}} \,d x } \] Input:

integrate((b*x^3+a)^(2/3)/x^12/(-b*d*x^3+a*d),x, algorithm="giac")
 

Output:

integrate(-(b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^12), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^{12}\,\left (a\,d-b\,d\,x^3\right )} \,d x \] Input:

int((a + b*x^3)^(2/3)/(x^12*(a*d - b*d*x^3)),x)
 

Output:

int((a + b*x^3)^(2/3)/(x^12*(a*d - b*d*x^3)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx=\frac {-40 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{3}-65 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} b \,x^{3}-98 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,b^{2} x^{6}+147 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{3} x^{9}+880 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b^{2} x^{9}+a^{2} x^{3}}d x \right ) a^{2} b^{3} x^{11}}{440 a^{4} d \,x^{11}} \] Input:

int((b*x^3+a)^(2/3)/x^12/(-b*d*x^3+a*d),x)
 

Output:

( - 40*(a + b*x**3)**(2/3)*a**3 - 65*(a + b*x**3)**(2/3)*a**2*b*x**3 - 98* 
(a + b*x**3)**(2/3)*a*b**2*x**6 + 147*(a + b*x**3)**(2/3)*b**3*x**9 + 880* 
int((a + b*x**3)**(2/3)/(a**2*x**3 - b**2*x**9),x)*a**2*b**3*x**11)/(440*a 
**4*d*x**11)