\(\int \frac {1}{x^6 (a+b x^3)^{4/3} (c+d x^3)} \, dx\) [774]

Optimal result

Integrand size = 24, antiderivative size = 287 \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {b}{a (b c-a d) x^5 \sqrt [3]{a+b x^3}}-\frac {(6 b c-a d) \left (a+b x^3\right )^{2/3}}{5 a^2 c (b c-a d) x^5}+\frac {\left (18 b^2 c^2-3 a b c d-5 a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{10 a^3 c^2 (b c-a d) x^2}-\frac {d^3 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{8/3} (b c-a d)^{4/3}}-\frac {d^3 \log \left (c+d x^3\right )}{6 c^{8/3} (b c-a d)^{4/3}}+\frac {d^3 \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{8/3} (b c-a d)^{4/3}} \] Output:

b/a/(-a*d+b*c)/x^5/(b*x^3+a)^(1/3)-1/5*(-a*d+6*b*c)*(b*x^3+a)^(2/3)/a^2/c/ 
(-a*d+b*c)/x^5+1/10*(-5*a^2*d^2-3*a*b*c*d+18*b^2*c^2)*(b*x^3+a)^(2/3)/a^3/ 
c^2/(-a*d+b*c)/x^2-1/3*d^3*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x 
^3+a)^(1/3))*3^(1/2))*3^(1/2)/c^(8/3)/(-a*d+b*c)^(4/3)-1/6*d^3*ln(d*x^3+c) 
/c^(8/3)/(-a*d+b*c)^(4/3)+1/2*d^3*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^ 
(1/3))/c^(8/3)/(-a*d+b*c)^(4/3)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.38 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {\frac {6 c^{2/3} \left (-18 b^3 c^2 x^6+3 a b^2 c x^3 \left (-2 c+d x^3\right )+a^3 d \left (-2 c+5 d x^3\right )+a^2 b \left (2 c^2+c d x^3+5 d^2 x^6\right )\right )}{a^3 (-b c+a d) x^5 \sqrt [3]{a+b x^3}}+\frac {10 \sqrt {-6+6 i \sqrt {3}} d^3 \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{(b c-a d)^{4/3}}-\frac {10 i \left (-i+\sqrt {3}\right ) d^3 \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{4/3}}+\frac {5 \left (1+i \sqrt {3}\right ) d^3 \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{4/3}}}{60 c^{8/3}} \] Input:

Integrate[1/(x^6*(a + b*x^3)^(4/3)*(c + d*x^3)),x]
 

Output:

((6*c^(2/3)*(-18*b^3*c^2*x^6 + 3*a*b^2*c*x^3*(-2*c + d*x^3) + a^3*d*(-2*c 
+ 5*d*x^3) + a^2*b*(2*c^2 + c*d*x^3 + 5*d^2*x^6)))/(a^3*(-(b*c) + a*d)*x^5 
*(a + b*x^3)^(1/3)) + (10*Sqrt[-6 + (6*I)*Sqrt[3]]*d^3*ArcTan[(3*(b*c - a* 
d)^(1/3)*x)/(Sqrt[3]*(b*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b* 
x^3)^(1/3))])/(b*c - a*d)^(4/3) - ((10*I)*(-I + Sqrt[3])*d^3*Log[2*(b*c - 
a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)])/(b*c - a*d)^(4/ 
3) + (5*(1 + I*Sqrt[3])*d^3*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3]) 
*c^(1/3)*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*( 
a + b*x^3)^(2/3)])/(b*c - a*d)^(4/3))/(60*c^(8/3))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {972, 25, 1053, 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.

\(\displaystyle \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx\)

\(\Big \downarrow \) 972

\(\displaystyle \frac {b}{a x^5 \sqrt [3]{a+b x^3} (b c-a d)}-\frac {\int -\frac {6 b d x^3+6 b c-a d}{x^6 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{a (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {6 b d x^3+6 b c-a d}{x^6 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{a (b c-a d)}+\frac {b}{a x^5 \sqrt [3]{a+b x^3} (b c-a d)}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {\int \frac {3 b d (6 b c-a d) x^3+18 b^2 c^2-5 a^2 d^2-3 a b c d}{x^3 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{5 a c}-\frac {\left (a+b x^3\right )^{2/3} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b}{a x^5 \sqrt [3]{a+b x^3} (b c-a d)}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {-\frac {\int -\frac {10 a^3 d^3}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{2 a c}-\frac {\left (a+b x^3\right )^{2/3} \left (\frac {18 b^2 c}{a}-\frac {5 a d^2}{c}-3 b d\right )}{2 x^2}}{5 a c}-\frac {\left (a+b x^3\right )^{2/3} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b}{a x^5 \sqrt [3]{a+b x^3} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {5 a^2 d^3 \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{c}-\frac {\left (a+b x^3\right )^{2/3} \left (\frac {18 b^2 c}{a}-\frac {5 a d^2}{c}-3 b d\right )}{2 x^2}}{5 a c}-\frac {\left (a+b x^3\right )^{2/3} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b}{a x^5 \sqrt [3]{a+b x^3} (b c-a d)}\)

\(\Big \downarrow \) 901

\(\displaystyle \frac {-\frac {\frac {5 a^2 d^3 \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{c}-\frac {\left (a+b x^3\right )^{2/3} \left (\frac {18 b^2 c}{a}-\frac {5 a d^2}{c}-3 b d\right )}{2 x^2}}{5 a c}-\frac {\left (a+b x^3\right )^{2/3} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b}{a x^5 \sqrt [3]{a+b x^3} (b c-a d)}\)

Input:

Int[1/(x^6*(a + b*x^3)^(4/3)*(c + d*x^3)),x]
 

Output:

b/(a*(b*c - a*d)*x^5*(a + b*x^3)^(1/3)) + (-1/5*((6*b*c - a*d)*(a + b*x^3) 
^(2/3))/(a*c*x^5) - (-1/2*(((18*b^2*c)/a - 3*b*d - (5*a*d^2)/c)*(a + b*x^3 
)^(2/3))/x^2 + (5*a^2*d^3*(ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a 
 + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^(2/3)*(b*c - a*d)^(1/3)) + Log[c + d 
*x^3]/(6*c^(2/3)*(b*c - a*d)^(1/3)) - Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - 
(a + b*x^3)^(1/3)]/(2*c^(2/3)*(b*c - a*d)^(1/3))))/c)/(5*a*c))/(a*(b*c - 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 972

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x 
^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 
 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( 
b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ 
a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & 
& IntBinomialQ[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 = 1.64 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a^{3} d^{3} x^{5} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{2}+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a^{3} d^{3} x^{5} \left (b \,x^{3}+a \right )^{\frac {1}{3}}-\frac {3 c \left (\left (-\frac {5}{2} a^{2} b \,d^{2}-\frac {3}{2} a \,b^{2} c d +9 c^{2} b^{3}\right ) x^{6}+\left (-\frac {5}{2} a^{3} d^{2}-\frac {1}{2} a^{2} b c d +3 a \,b^{2} c^{2}\right ) x^{3}+\left (a d -b c \right ) a^{2} c \right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{5}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right ) a^{3} d^{3} x^{5} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} c^{3} x^{5} \left (a d -b c \right ) a^{3}}\)	\(324\)

Input:

int(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

1/3/((a*d-b*c)/c)^(1/3)*(-1/2*ln((((a*d-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1 
/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)*a^3*d^3*x^5*(b*x^3+a)^(1/3)+ln 
((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)*a^3*d^3*x^5*(b*x^3+a)^(1/3)-3/ 
5*c*((-5/2*a^2*b*d^2-3/2*a*b^2*c*d+9*c^2*b^3)*x^6+(-5/2*a^3*d^2-1/2*a^2*b* 
c*d+3*a*b^2*c^2)*x^3+(a*d-b*c)*a^2*c)*((a*d-b*c)/c)^(1/3)+3^(1/2)*arctan(1 
/3*3^(1/2)*(-2/((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)+x)/x)*a^3*d^3*x^5*(b*x^ 
3+a)^(1/3))/(b*x^3+a)^(1/3)/c^3/x^5/(a*d-b*c)/a^3
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{x^{6} \left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \] Input:

integrate(1/x**6/(b*x**3+a)**(4/3)/(d*x**3+c),x)
 

Output:

Integral(1/(x**6*(a + b*x**3)**(4/3)*(c + d*x**3)), x)
 

Maxima [F]

\[ \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)*x^6), x)
 

Giac [F]

\[ \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")
 

Output:

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)*x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{x^6\,{\left (b\,x^3+a\right )}^{4/3}\,\left (d\,x^3+c\right )} \,d x \] Input:

int(1/(x^6*(a + b*x^3)^(4/3)*(c + d*x^3)),x)
 

Output:

int(1/(x^6*(a + b*x^3)^(4/3)*(c + d*x^3)), x)
 

Reduce [F]

\[ \int \frac {1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a c \,x^{6}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} a d \,x^{9}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b c \,x^{9}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b d \,x^{12}}d x \] Input:

int(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x)
 

Output:

int(1/((a + b*x**3)**(1/3)*a*c*x**6 + (a + b*x**3)**(1/3)*a*d*x**9 + (a + 
b*x**3)**(1/3)*b*c*x**9 + (a + b*x**3)**(1/3)*b*d*x**12),x)