\(\int \frac {1}{x (a+b x^3)^{4/3} (c+d x^3)} \, dx\) [767]

Optimal result

Integrand size = 24, antiderivative size = 271 \[ \int \frac {1}{x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {b}{a (b c-a d) \sqrt [3]{a+b x^3}}+\frac {\arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} c}-\frac {d^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c (b c-a d)^{4/3}}-\frac {\log (x)}{2 a^{4/3} c}+\frac {d^{4/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{4/3}}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{4/3} c}-\frac {d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{4/3}} \] Output:

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

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {1}{6} \left (\frac {6 b}{\left (a b c-a^2 d\right ) \sqrt [3]{a+b x^3}}+\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3} c}-\frac {2 \sqrt {3} d^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{c (b c-a d)^{4/3}}+\frac {2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )}{a^{4/3} c}-\frac {2 d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{c (b c-a d)^{4/3}}-\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{a^{4/3} c}+\frac {d^{4/3} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{c (b c-a d)^{4/3}}\right ) \] Input:

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

Output:

((6*b)/((a*b*c - a^2*d)*(a + b*x^3)^(1/3)) + (2*Sqrt[3]*ArcTan[(1 + (2*(a 
+ b*x^3)^(1/3))/a^(1/3))/Sqrt[3]])/(a^(4/3)*c) - (2*Sqrt[3]*d^(4/3)*ArcTan 
[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]])/(c*(b*c - 
 a*d)^(4/3)) + (2*Log[-a^(1/3) + (a + b*x^3)^(1/3)])/(a^(4/3)*c) - (2*d^(4 
/3)*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(c*(b*c - a*d)^(4/ 
3)) - Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]/(a^(4/3 
)*c) + (d^(4/3)*Log[(b*c - a*d)^(2/3) - d^(1/3)*(b*c - a*d)^(1/3)*(a + b*x 
^3)^(1/3) + d^(2/3)*(a + b*x^3)^(2/3)])/(c*(b*c - a*d)^(4/3)))/6
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {948, 96, 25, 174, 67, 16, 68, 16, 1082, 217}

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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 96

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 68

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

((3*b)/(a*(b*c - a*d)*(a + b*x^3)^(1/3)) + (((b*c - a*d)*((Sqrt[3]*ArcTan[ 
(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3) - Log[x^3]/(2*a^(1/3 
)) + (3*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(2*a^(1/3))))/c + (a*d^2*(-((Sqr 
t[3]*ArcTan[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]] 
)/(d^(2/3)*(b*c - a*d)^(1/3))) + Log[c + d*x^3]/(2*d^(2/3)*(b*c - a*d)^(1/ 
3)) - (3*Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*d^(2/3)*(b 
*c - a*d)^(1/3))))/c)/(a*(b*c - a*d)))/3
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

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

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 68

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
 

rule 96

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S 
imp[1/((b*e - a*f)*(d*e - c*f))   Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e 
 + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, 
 x] && LtQ[p, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 948

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {\left (-\left (a d -b c \right ) \left (-2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )-2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}-6 a^{\frac {1}{3}} b c \right ) \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}+d \left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}\right )-2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )\right ) a^{\frac {4}{3}}}{6 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{\frac {4}{3}} \left (a d -b c \right ) c}\)	\(304\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (216) = 432\).

Time = 0.13 (sec) , antiderivative size = 975, normalized size of antiderivative = 3.60 \[ \int \frac {1}{x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/6*(6*(b*x^3 + a)^(2/3)*a*b*c + 3*sqrt(1/3)*(a^2*b*c - a^3*d + (a*b^2*c 
- a^2*b*d)*x^3)*sqrt(-1/a^(2/3))*log((2*b*x^3 + 3*sqrt(1/3)*(2*(b*x^3 + a) 
^(2/3)*a^(2/3) - (b*x^3 + a)^(1/3)*a - a^(4/3))*sqrt(-1/a^(2/3)) - 3*(b*x^ 
3 + a)^(1/3)*a^(2/3) + 3*a)/x^3) + 2*sqrt(3)*(a^2*b*d*x^3 + a^3*d)*(d/(b*c 
 - a*d))^(1/3)*arctan(2/3*sqrt(3)*(b*x^3 + a)^(1/3)*(d/(b*c - a*d))^(1/3) 
- 1/3*sqrt(3)) - ((b^2*c - a*b*d)*x^3 + a*b*c - a^2*d)*a^(2/3)*log((b*x^3 
+ a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*((b^2*c - a*b*d)*x^3 
 + a*b*c - a^2*d)*a^(2/3)*log((b*x^3 + a)^(1/3) - a^(1/3)) + (a^2*b*d*x^3 
+ a^3*d)*(d/(b*c - a*d))^(1/3)*log(-(b*x^3 + a)^(1/3)*(b*c - a*d)*(d/(b*c 
- a*d))^(2/3) + (b*x^3 + a)^(2/3)*d + (b*c - a*d)*(d/(b*c - a*d))^(1/3)) - 
 2*(a^2*b*d*x^3 + a^3*d)*(d/(b*c - a*d))^(1/3)*log((b*c - a*d)*(d/(b*c - a 
*d))^(2/3) + (b*x^3 + a)^(1/3)*d))/(a^3*b*c^2 - a^4*c*d + (a^2*b^2*c^2 - a 
^3*b*c*d)*x^3), 1/6*(6*(b*x^3 + a)^(2/3)*a*b*c + 2*sqrt(3)*(a^2*b*d*x^3 + 
a^3*d)*(d/(b*c - a*d))^(1/3)*arctan(2/3*sqrt(3)*(b*x^3 + a)^(1/3)*(d/(b*c 
- a*d))^(1/3) - 1/3*sqrt(3)) - ((b^2*c - a*b*d)*x^3 + a*b*c - a^2*d)*a^(2/ 
3)*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3)) + 2*((b^2* 
c - a*b*d)*x^3 + a*b*c - a^2*d)*a^(2/3)*log((b*x^3 + a)^(1/3) - a^(1/3)) + 
 (a^2*b*d*x^3 + a^3*d)*(d/(b*c - a*d))^(1/3)*log(-(b*x^3 + a)^(1/3)*(b*c - 
 a*d)*(d/(b*c - a*d))^(2/3) + (b*x^3 + a)^(2/3)*d + (b*c - a*d)*(d/(b*c - 
a*d))^(1/3)) - 2*(a^2*b*d*x^3 + a^3*d)*(d/(b*c - a*d))^(1/3)*log((b*c -...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x/(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), x)
 

Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=-\frac {d^{2} \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c^{3} - 2 \, \sqrt {3} a b c^{2} d + \sqrt {3} a^{2} c d^{2}} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} + \frac {b}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a b c - a^{2} d\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {4}{3}} c} - \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {4}{3}} c} + \frac {\log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {4}{3}} c} \] Input:

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

Output:

-1/3*d^2*(-(b*c - a*d)/d)^(2/3)*log(abs((b*x^3 + a)^(1/3) - (-(b*c - a*d)/ 
d)^(1/3)))/(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2) - (-b*c*d^2 + a*d^3)^(2/3)* 
arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + (-(b*c - a*d)/d)^(1/3))/(-(b*c - 
 a*d)/d)^(1/3))/(sqrt(3)*b^2*c^3 - 2*sqrt(3)*a*b*c^2*d + sqrt(3)*a^2*c*d^2 
) + 1/6*(-b*c*d^2 + a*d^3)^(2/3)*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3) 
*(-(b*c - a*d)/d)^(1/3) + (-(b*c - a*d)/d)^(2/3))/(b^2*c^3 - 2*a*b*c^2*d + 
 a^2*c*d^2) + b/((b*x^3 + a)^(1/3)*(a*b*c - a^2*d)) + 1/3*sqrt(3)*arctan(1 
/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3))/(a^(4/3)*c) - 1/6*log( 
(b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/(a^(4/3)*c) + 1/3 
*log(abs((b*x^3 + a)^(1/3) - a^(1/3)))/(a^(4/3)*c)
 

Mupad [B] (verification not implemented)

Time = 4.04 (sec) , antiderivative size = 3804, normalized size of antiderivative = 14.04 \[ \int \frac {1}{x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

log(9*a^7*b^14*c^11*d^4 - ((a + b*x^3)^(1/3)*(27*a^7*b^15*c^13*d^3 - 297*a 
^8*b^14*c^12*d^4 + 1485*a^9*b^13*c^11*d^5 - 4455*a^10*b^12*c^10*d^6 + 8937 
*a^11*b^11*c^9*d^7 - 12663*a^12*b^10*c^8*d^8 + 13041*a^13*b^9*c^7*d^9 - 98 
55*a^14*b^8*c^6*d^10 + 5400*a^15*b^7*c^5*d^11 - 2052*a^16*b^6*c^4*d^12 + 4 
86*a^17*b^5*c^3*d^13 - 54*a^18*b^4*c^2*d^14) - (-d^4/(27*b^4*c^7 + 27*a^4* 
c^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(2/3 
)*(243*a^10*b^15*c^15*d^3 - 2916*a^11*b^14*c^14*d^4 + 15795*a^12*b^13*c^13 
*d^5 - 51030*a^13*b^12*c^12*d^6 + 109350*a^14*b^11*c^11*d^7 - 163296*a^15* 
b^10*c^10*d^8 + 173502*a^16*b^9*c^9*d^9 - 131220*a^17*b^8*c^8*d^10 + 69255 
*a^18*b^7*c^7*d^11 - 24300*a^19*b^6*c^6*d^12 + 5103*a^20*b^5*c^5*d^13 - 48 
6*a^21*b^4*c^4*d^14))*(-d^4/(27*b^4*c^7 + 27*a^4*c^3*d^4 - 108*a^3*b*c^4*d 
^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(1/3) - 90*a^8*b^13*c^10*d^5 
+ 405*a^9*b^12*c^9*d^6 - 1071*a^10*b^11*c^8*d^7 + 1827*a^11*b^10*c^7*d^8 - 
 2079*a^12*b^9*c^6*d^9 + 1575*a^13*b^8*c^5*d^10 - 765*a^14*b^7*c^4*d^11 + 
216*a^15*b^6*c^3*d^12 - 27*a^16*b^5*c^2*d^13)*(-d^4/(27*b^4*c^7 + 27*a^4*c 
^3*d^4 - 108*a^3*b*c^4*d^3 + 162*a^2*b^2*c^5*d^2 - 108*a*b^3*c^6*d))^(1/3) 
 + log(9*a^7*b^14*c^11*d^4 - ((a + b*x^3)^(1/3)*(27*a^7*b^15*c^13*d^3 - 29 
7*a^8*b^14*c^12*d^4 + 1485*a^9*b^13*c^11*d^5 - 4455*a^10*b^12*c^10*d^6 + 8 
937*a^11*b^11*c^9*d^7 - 12663*a^12*b^10*c^8*d^8 + 13041*a^13*b^9*c^7*d^9 - 
 9855*a^14*b^8*c^6*d^10 + 5400*a^15*b^7*c^5*d^11 - 2052*a^16*b^6*c^4*d^...
 

Reduce [F]

\[ \int \frac {1}{x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a c x +\left (b \,x^{3}+a \right )^{\frac {1}{3}} a d \,x^{4}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b c \,x^{4}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b d \,x^{7}}d x \] Input:

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

Output:

int(1/((a + b*x**3)**(1/3)*a*c*x + (a + b*x**3)**(1/3)*a*d*x**4 + (a + b*x 
**3)**(1/3)*b*c*x**4 + (a + b*x**3)**(1/3)*b*d*x**7),x)