\(\int \frac {1}{(a+b x^3)^{4/3} (c+d x^3)^2} \, dx\) [107]

Optimal result

Integrand size = 21, antiderivative size = 261 \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx=\frac {b (3 b c+a d) x}{3 a c (b c-a d)^2 \sqrt [3]{a+b x^3}}-\frac {d x}{3 c (b c-a d) \sqrt [3]{a+b x^3} \left (c+d x^3\right )}-\frac {2 d (3 b c-a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{7/3}}-\frac {d (3 b c-a d) \log \left (c+d x^3\right )}{9 c^{5/3} (b c-a d)^{7/3}}+\frac {d (3 b c-a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} (b c-a d)^{7/3}} \] Output:

1/3*b*(a*d+3*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^3+a)^(1/3)-1/3*d*x/c/(-a*d+b*c)/ 
(b*x^3+a)^(1/3)/(d*x^3+c)-2/9*d*(-a*d+3*b*c)*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^(5/3)/(-a*d+b*c)^(7/3)-1 
/9*d*(-a*d+3*b*c)*ln(d*x^3+c)/c^(5/3)/(-a*d+b*c)^(7/3)+1/3*d*(-a*d+3*b*c)* 
ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(5/3)/(-a*d+b*c)^(7/3)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.40 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx=\frac {\frac {6 c^{2/3} x \left (a^2 d^2+a b d^2 x^3+3 b^2 c \left (c+d x^3\right )\right )}{a (b c-a d)^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {2 i \left (3 i+\sqrt {3}\right ) d (3 b c-a d) \text {arctanh}\left (\frac {i+\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d} x}}{\sqrt {3}}\right )}{(b c-a d)^{7/3}}+\frac {2 \left (1+i \sqrt {3}\right ) d (-3 b c+a d) \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{7/3}}+\frac {\left (1+i \sqrt {3}\right ) d (3 b c-a d) \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{7/3}}}{18 c^{5/3}} \] Input:

Integrate[1/((a + b*x^3)^(4/3)*(c + d*x^3)^2),x]
 

Output:

((6*c^(2/3)*x*(a^2*d^2 + a*b*d^2*x^3 + 3*b^2*c*(c + d*x^3)))/(a*(b*c - a*d 
)^2*(a + b*x^3)^(1/3)*(c + d*x^3)) + ((2*I)*(3*I + Sqrt[3])*d*(3*b*c - a*d 
)*ArcTanh[(I + ((-I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))/((b*c - a*d)^(1/ 
3)*x))/Sqrt[3]])/(b*c - a*d)^(7/3) + (2*(1 + I*Sqrt[3])*d*(-3*b*c + a*d)*L 
og[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)^(7/3) + ((1 + I*Sqrt[3])*d*(3*b*c - a*d)*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)^(7/3))/(18*c^(5/3))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {931, 1024, 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.

\(\displaystyle \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx\)

\(\Big \downarrow \) 931

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

\(\Big \downarrow \) 1024

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 901

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

Input:

Int[1/((a + b*x^3)^(4/3)*(c + d*x^3)^2),x]
 

Output:

-1/3*(d*x)/(c*(b*c - a*d)*(a + b*x^3)^(1/3)*(c + d*x^3)) + ((b*(3*b*c + a* 
d)*x)/(a*(b*c - a*d)*(a + b*x^3)^(1/3)) - (2*d*(3*b*c - a*d)*(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)) - L 
og[((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))))/(b*c - a*d))/(3*c*(b*c - a*d))
 

Defintions of rubi rules used

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 931

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d))   Int[(a + b*x^n)^(p + 1)*(c 
 + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, 
 x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, 
-1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, 
 c, d, n, p, q, x]
 

rule 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a d \left (d \,x^{3}+c \right ) \left (a d -3 b c \right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{9}+\frac {c \left (a \left (b \,x^{3}+a \right ) d^{2}+3 x^{3} b^{2} c d +3 b^{2} c^{2}\right ) x \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{3}+\frac {2 a d \left (d \,x^{3}+c \right ) \left (\arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right ) \sqrt {3}-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (a d -3 b c \right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{9}}{c^{2} \left (d \,x^{3}+c \right ) \left (a d -b c \right )^{2} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a}\)	\(288\)

Input:

int(1/(b*x^3+a)^(4/3)/(d*x^3+c)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^3+a)^(4/3)/(d*x^3+c)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx=\int \frac {1}{\left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )^{2}}\, dx \] Input:

integrate(1/(b*x**3+a)**(4/3)/(d*x**3+c)**2,x)
 

Output:

Integral(1/((a + b*x**3)**(4/3)*(c + d*x**3)**2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2}} \,d x } \] Input:

integrate(1/(b*x^3+a)^(4/3)/(d*x^3+c)^2,x, algorithm="maxima")
 

Output:

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)^2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{2}} \,d x } \] Input:

integrate(1/(b*x^3+a)^(4/3)/(d*x^3+c)^2,x, algorithm="giac")
 

Output:

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{4/3}\,{\left (d\,x^3+c\right )}^2} \,d x \] Input:

int(1/((a + b*x^3)^(4/3)*(c + d*x^3)^2),x)
 

Output:

int(1/((a + b*x^3)^(4/3)*(c + d*x^3)^2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,c^{2}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a c d \,x^{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,d^{2} x^{6}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b \,c^{2} x^{3}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b c d \,x^{6}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b \,d^{2} x^{9}}d x \] Input:

int(1/(b*x^3+a)^(4/3)/(d*x^3+c)^2,x)
 

Output:

int(1/((a + b*x**3)**(1/3)*a*c**2 + 2*(a + b*x**3)**(1/3)*a*c*d*x**3 + (a 
+ b*x**3)**(1/3)*a*d**2*x**6 + (a + b*x**3)**(1/3)*b*c**2*x**3 + 2*(a + b* 
x**3)**(1/3)*b*c*d*x**6 + (a + b*x**3)**(1/3)*b*d**2*x**9),x)