\(\int \frac {1}{x^4 (a+b x^3)^{4/3} (c+d x^3)} \, dx\) [768]

Optimal result

Integrand size = 24, antiderivative size = 339 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=-\frac {b (4 b c-a d)}{3 a^2 c (b c-a d) \sqrt [3]{a+b x^3}}-\frac {1}{3 a c x^3 \sqrt [3]{a+b x^3}}-\frac {(4 b c+3 a d) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} c^2}+\frac {d^{7/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^2 (b c-a d)^{4/3}}+\frac {(4 b c+3 a d) \log (x)}{6 a^{7/3} c^2}-\frac {d^{7/3} \log \left (c+d x^3\right )}{6 c^2 (b c-a d)^{4/3}}-\frac {(4 b c+3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{7/3} c^2}+\frac {d^{7/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 (b c-a d)^{4/3}} \] Output:

-1/3*b*(-a*d+4*b*c)/a^2/c/(-a*d+b*c)/(b*x^3+a)^(1/3)-1/3/a/c/x^3/(b*x^3+a) 
^(1/3)-1/9*(3*a*d+4*b*c)*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))*3^(1/2)/a^ 
(1/3))*3^(1/2)/a^(7/3)/c^2+1/3*d^(7/3)*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^2/(-a*d+b*c)^(4/3)+1/6*(3*a*d+4* 
b*c)*ln(x)/a^(7/3)/c^2-1/6*d^(7/3)*ln(d*x^3+c)/c^2/(-a*d+b*c)^(4/3)-1/6*(3 
*a*d+4*b*c)*ln(a^(1/3)-(b*x^3+a)^(1/3))/a^(7/3)/c^2+1/2*d^(7/3)*ln((-a*d+b 
*c)^(1/3)+d^(1/3)*(b*x^3+a)^(1/3))/c^2/(-a*d+b*c)^(4/3)
 

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {\frac {6 c \left (-a^2 d+4 b^2 c x^3+a b \left (c-d x^3\right )\right )}{a^2 (-b c+a d) x^3 \sqrt [3]{a+b x^3}}-\frac {2 \sqrt {3} (4 b c+3 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{7/3}}+\frac {6 \sqrt {3} d^{7/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{(b c-a d)^{4/3}}-\frac {2 (4 b c+3 a d) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )}{a^{7/3}}+\frac {6 d^{7/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{4/3}}+\frac {(4 b c+3 a d) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{a^{7/3}}-\frac {3 d^{7/3} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{4/3}}}{18 c^2} \] Input:

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

Output:

((6*c*(-(a^2*d) + 4*b^2*c*x^3 + a*b*(c - d*x^3)))/(a^2*(-(b*c) + a*d)*x^3* 
(a + b*x^3)^(1/3)) - (2*Sqrt[3]*(4*b*c + 3*a*d)*ArcTan[(1 + (2*(a + b*x^3) 
^(1/3))/a^(1/3))/Sqrt[3]])/a^(7/3) + (6*Sqrt[3]*d^(7/3)*ArcTan[(1 - (2*d^( 
1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]])/(b*c - a*d)^(4/3) - ( 
2*(4*b*c + 3*a*d)*Log[-a^(1/3) + (a + b*x^3)^(1/3)])/a^(7/3) + (6*d^(7/3)* 
Log[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(b*c - a*d)^(4/3) + (( 
4*b*c + 3*a*d)*Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3) 
])/a^(7/3) - (3*d^(7/3)*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)])/(b*c - a*d)^(4/3))/(18*c^2 
)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {948, 114, 27, 174, 61, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 61

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {(3 a d+4 b c) \left (\frac {\int \frac {1}{x^3 \sqrt [3]{b x^3+a}}dx^3}{a}+\frac {3}{a \sqrt [3]{a+b x^3}}\right )}{c}-\frac {3 a d^2 \left (-\frac {d \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx^3}{b c-a d}-\frac {3}{\sqrt [3]{a+b x^3} (b c-a d)}\right )}{c}}{3 a c}-\frac {1}{a c x^3 \sqrt [3]{a+b x^3}}\right )\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {(3 a d+4 b c) \left (\frac {\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^3}}\right )}{c}-\frac {3 a d^2 \left (-\frac {d \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx^3}{b c-a d}-\frac {3}{\sqrt [3]{a+b x^3} (b c-a d)}\right )}{c}}{3 a c}-\frac {1}{a c x^3 \sqrt [3]{a+b x^3}}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {(3 a d+4 b c) \left (\frac {\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^3}}\right )}{c}-\frac {3 a d^2 \left (-\frac {d \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx^3}{b c-a d}-\frac {3}{\sqrt [3]{a+b x^3} (b c-a d)}\right )}{c}}{3 a c}-\frac {1}{a c x^3 \sqrt [3]{a+b x^3}}\right )\)

\(\Big \downarrow \) 68

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {(3 a d+4 b c) \left (\frac {\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^3}}\right )}{c}-\frac {3 a d^2 \left (-\frac {d \left (-\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 d^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d}+\frac {\log \left (c+d x^3\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}\right )}{b c-a d}-\frac {3}{\sqrt [3]{a+b x^3} (b c-a d)}\right )}{c}}{3 a c}-\frac {1}{a c x^3 \sqrt [3]{a+b x^3}}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {(3 a d+4 b c) \left (\frac {\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^3}}\right )}{c}-\frac {3 a d^2 \left (-\frac {d \left (\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d}+\frac {\log \left (c+d x^3\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}-\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}\right )}{b c-a d}-\frac {3}{\sqrt [3]{a+b x^3} (b c-a d)}\right )}{c}}{3 a c}-\frac {1}{a c x^3 \sqrt [3]{a+b x^3}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} \left (-\frac {\frac {(3 a d+4 b c) \left (\frac {-\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^3}}\right )}{c}-\frac {3 a d^2 \left (-\frac {d \left (\frac {3 \int \frac {1}{-x^6-3}d\left (1-\frac {2 \sqrt [3]{d} \sqrt [3]{b x^3+a}}{\sqrt [3]{b c-a d}}\right )}{d^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}-\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}\right )}{b c-a d}-\frac {3}{\sqrt [3]{a+b x^3} (b c-a d)}\right )}{c}}{3 a c}-\frac {1}{a c x^3 \sqrt [3]{a+b x^3}}\right )\)

\(\Big \downarrow \) 217

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

Input:

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

Output:

(-(1/(a*c*x^3*(a + b*x^3)^(1/3))) - (((4*b*c + 3*a*d)*(3/(a*(a + b*x^3)^(1 
/3)) + ((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)))/a))/c - (3*a*d^2*(-3/((b*c - a*d)*(a + b*x^3)^(1/3)) - (d*(-((Sqrt[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))))/(b*c - a*d)))/c)/(3*a*c))/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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, 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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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 = 1.70 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(\frac {\left (\left (a d -b c \right ) \left (a d +\frac {4 b c}{3}\right ) \left (-2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )-2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )\right ) x^{3} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+2 c \left (-a^{\frac {7}{3}} d +\left (\left (-d \,x^{3}+c \right ) a^{\frac {4}{3}}+4 b c \,x^{3} a^{\frac {1}{3}}\right ) b \right )\right ) \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{\frac {7}{3}} d^{2} \left (\arctan \left (\frac {\sqrt {3}\, \left (2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}\right )}{2}\right ) x^{3}}{6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{\frac {7}{3}} \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} c^{2} x^{3} \left (a d -b c \right )}\)	\(350\)

Input:

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

Output:

1/6/(b*x^3+a)^(1/3)/a^(7/3)/((a*d-b*c)/d)^(1/3)*(((a*d-b*c)*(a*d+4/3*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) 
))*x^3*(b*x^3+a)^(1/3)+2*c*(-a^(7/3)*d+((-d*x^3+c)*a^(4/3)+4*b*c*x^3*a^(1/ 
3))*b))*((a*d-b*c)/d)^(1/3)+2*(b*x^3+a)^(1/3)*a^(7/3)*d^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)+l 
n((b*x^3+a)^(1/3)-((a*d-b*c)/d)^(1/3))-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)))*x^3)/c^2/x^3/(a*d-b*c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (276) = 552\).

Time = 0.78 (sec) , antiderivative size = 1386, normalized size of antiderivative = 4.09 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/18*(3*sqrt(1/3)*((4*a*b^3*c^2 - a^2*b^2*c*d - 3*a^3*b*d^2)*x^6 + (4*a^2 
*b^2*c^2 - a^3*b*c*d - 3*a^4*d^2)*x^3)*sqrt((-a)^(1/3)/a)*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)^(1 
/3)*a)*sqrt((-a)^(1/3)/a) - 3*(b*x^3 + a)^(1/3)*(-a)^(2/3) + 3*a)/x^3) - 6 
*sqrt(3)*(a^3*b*d^2*x^6 + a^4*d^2*x^3)*(-d/(b*c - a*d))^(1/3)*arctan(2/3*s 
qrt(3)*(b*x^3 + a)^(1/3)*(-d/(b*c - a*d))^(1/3) + 1/3*sqrt(3)) + ((4*b^3*c 
^2 - a*b^2*c*d - 3*a^2*b*d^2)*x^6 + (4*a*b^2*c^2 - a^2*b*c*d - 3*a^3*d^2)* 
x^3)*(-a)^(2/3)*log((b*x^3 + a)^(2/3) - (b*x^3 + a)^(1/3)*(-a)^(1/3) + (-a 
)^(2/3)) - 2*((4*b^3*c^2 - a*b^2*c*d - 3*a^2*b*d^2)*x^6 + (4*a*b^2*c^2 - a 
^2*b*c*d - 3*a^3*d^2)*x^3)*(-a)^(2/3)*log((b*x^3 + a)^(1/3) + (-a)^(1/3)) 
+ 3*(a^3*b*d^2*x^6 + a^4*d^2*x^3)*(-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)) - 6*(a^3*b*d^2*x^6 + a^4*d^2*x^3)*(-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) - 
6*(a^2*b*c^2 - a^3*c*d + (4*a*b^2*c^2 - a^2*b*c*d)*x^3)*(b*x^3 + a)^(2/3)) 
/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^6 + (a^4*b*c^3 - a^5*c^2*d)*x^3), -1/18*(6 
*sqrt(1/3)*((4*a*b^3*c^2 - a^2*b^2*c*d - 3*a^3*b*d^2)*x^6 + (4*a^2*b^2*c^2 
 - a^3*b*c*d - 3*a^4*d^2)*x^3)*sqrt(-(-a)^(1/3)/a)*arctan(sqrt(1/3)*(2*(b* 
x^3 + a)^(1/3) - (-a)^(1/3))*sqrt(-(-a)^(1/3)/a)) + 6*sqrt(3)*(a^3*b*d^2*x 
^6 + a^4*d^2*x^3)*(-d/(b*c - a*d))^(1/3)*arctan(2/3*sqrt(3)*(b*x^3 + a)...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x^4/(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^4), x)
 

Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\frac {d^{3} \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^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} d \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^{4} - 2 \, \sqrt {3} a b c^{3} d + \sqrt {3} a^{2} c^{2} d^{2}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} d \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^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac {4 \, {\left (b x^{3} + a\right )} b^{2} c - 3 \, a b^{2} c - {\left (b x^{3} + a\right )} a b d}{3 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} {\left ({\left (b x^{3} + a\right )}^{\frac {4}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} a\right )}} - \frac {\sqrt {3} {\left (4 \, b c + 3 \, a d\right )} \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 )}{9 \, a^{\frac {7}{3}} c^{2}} + \frac {{\left (4 \, b c + 3 \, a d\right )} \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 )}{18 \, a^{\frac {7}{3}} c^{2}} - \frac {{\left (4 \, a^{\frac {1}{3}} b c + 3 \, a^{\frac {4}{3}} d\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{9 \, a^{\frac {8}{3}} c^{2}} \] Input:

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

Output:

1/3*d^3*(-(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^4 - 2*a*b*c^3*d + a^2*c^2*d^2) + (-b*c*d^2 + a*d^3)^(2/3) 
*d*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^4 - 2*sqrt(3)*a*b*c^3*d + sqrt(3)*a^2*c^ 
2*d^2) - 1/6*(-b*c*d^2 + a*d^3)^(2/3)*d*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^4 - 2*a*b* 
c^3*d + a^2*c^2*d^2) - 1/3*(4*(b*x^3 + a)*b^2*c - 3*a*b^2*c - (b*x^3 + a)* 
a*b*d)/((a^2*b*c^2 - a^3*c*d)*((b*x^3 + a)^(4/3) - (b*x^3 + a)^(1/3)*a)) - 
 1/9*sqrt(3)*(4*b*c + 3*a*d)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^( 
1/3))/a^(1/3))/(a^(7/3)*c^2) + 1/18*(4*b*c + 3*a*d)*log((b*x^3 + a)^(2/3) 
+ (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/(a^(7/3)*c^2) - 1/9*(4*a^(1/3)*b*c 
+ 3*a^(4/3)*d)*log(abs((b*x^3 + a)^(1/3) - a^(1/3)))/(a^(8/3)*c^2)
 

Mupad [B] (verification not implemented)

Time = 5.12 (sec) , antiderivative size = 5875, normalized size of antiderivative = 17.33 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

log((d^7/(27*b^4*c^10 + 27*a^4*c^6*d^4 - 108*a^3*b*c^7*d^3 + 162*a^2*b^2*c 
^8*d^2 - 108*a*b^3*c^9*d))^(2/3)*(419904*a^13*b^17*c^20*d^4 - ((a + b*x^3) 
^(1/3)*(8975448*a^15*b^16*c^21*d^4 - 944784*a^14*b^17*c^22*d^3 - 36905625* 
a^16*b^15*c^20*d^5 + 83790531*a^17*b^14*c^19*d^6 - 107173935*a^18*b^13*c^1 
8*d^7 + 56509893*a^19*b^12*c^17*d^8 + 42338133*a^20*b^11*c^16*d^9 - 937107 
63*a^21*b^10*c^15*d^10 + 55092717*a^22*b^9*c^14*d^11 + 12105045*a^23*b^8*c 
^13*d^12 - 38736144*a^24*b^7*c^12*d^13 + 25745364*a^25*b^6*c^11*d^14 - 814 
8762*a^26*b^5*c^10*d^15 + 1062882*a^27*b^4*c^9*d^16) + (d^7/(27*b^4*c^10 + 
 27*a^4*c^6*d^4 - 108*a^3*b*c^7*d^3 + 162*a^2*b^2*c^8*d^2 - 108*a*b^3*c^9* 
d))^(2/3)*(4782969*a^19*b^15*c^24*d^3 - 57395628*a^20*b^14*c^23*d^4 + 3108 
92985*a^21*b^13*c^22*d^5 - 1004423490*a^22*b^12*c^21*d^6 + 2152336050*a^23 
*b^11*c^20*d^7 - 3214155168*a^24*b^10*c^19*d^8 + 3415039866*a^25*b^9*c^18* 
d^9 - 2582803260*a^26*b^8*c^17*d^10 + 1363146165*a^27*b^7*c^16*d^11 - 4782 
96900*a^28*b^6*c^15*d^12 + 100442349*a^29*b^5*c^14*d^13 - 9565938*a^30*b^4 
*c^13*d^14))*(d^7/(27*b^4*c^10 + 27*a^4*c^6*d^4 - 108*a^3*b*c^7*d^3 + 162* 
a^2*b^2*c^8*d^2 - 108*a*b^3*c^9*d))^(1/3) - 3254256*a^14*b^16*c^19*d^5 + 1 
0156428*a^15*b^15*c^18*d^6 - 14781933*a^16*b^14*c^17*d^7 + 4920750*a^17*b^ 
13*c^16*d^8 + 15529887*a^18*b^12*c^15*d^9 - 22182741*a^19*b^11*c^14*d^10 + 
 5412825*a^20*b^10*c^13*d^11 + 13404123*a^21*b^9*c^12*d^12 - 15713595*a^22 
*b^8*c^11*d^13 + 7801029*a^23*b^7*c^10*d^14 - 1889568*a^24*b^6*c^9*d^15...
 

Reduce [F]

\[ \int \frac {1}{x^4 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a c \,x^{4}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} a d \,x^{7}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b c \,x^{7}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b d \,x^{10}}d x \] Input:

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

Output:

int(1/((a + b*x**3)**(1/3)*a*c*x**4 + (a + b*x**3)**(1/3)*a*d*x**7 + (a + 
b*x**3)**(1/3)*b*c*x**7 + (a + b*x**3)**(1/3)*b*d*x**10),x)