\(\int \frac {\sqrt [3]{a+b x^3}}{x^4 (c+d x^3)} \, dx\) [678]

Optimal result

Integrand size = 24, antiderivative size = 317 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (c+d x^3\right )} \, dx=\frac {b \sqrt [3]{a+b x^3}}{3 a c}-\frac {\left (a+b x^3\right )^{4/3}}{3 a c x^3}-\frac {(b c-3 a d) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} c^2}+\frac {d^{2/3} \sqrt [3]{b c-a d} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^2}-\frac {(b c-3 a d) \log (x)}{6 a^{2/3} c^2}+\frac {d^{2/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^2}+\frac {(b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3} c^2}-\frac {d^{2/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (c+d x^3\right )} \, dx=\frac {-\frac {6 c \sqrt [3]{a+b x^3}}{x^3}+\frac {2 \sqrt {3} (-b c+3 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+6 \sqrt {3} d^{2/3} \sqrt [3]{b c-a d} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )+\frac {2 (b c-3 a d) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )}{a^{2/3}}-6 d^{2/3} \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+\frac {(-b c+3 a d) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{a^{2/3}}+3 d^{2/3} \sqrt [3]{b c-a d} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{18 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.05, 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, 60, 69, 16, 70, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 70

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

(-((a + b*x^3)^(4/3)/(a*c*x^3)) + (((b*c - 3*a*d)*(3*(a + b*x^3)^(1/3) + a 
*(-((Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3)) 
 - Log[x^3]/(2*a^(2/3)) + (3*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(2*a^(2/3)) 
)))/c + (3*a*d^2*((3*(a + b*x^3)^(1/3))/d - ((b*c - a*d)*(-((Sqrt[3]*ArcTa 
n[(1 - (2*d^(1/3)*(a + b*x^3)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3]])/(d^(1/3) 
*(b*c - a*d)^(2/3))) - Log[c + d*x^3]/(2*d^(1/3)*(b*c - a*d)^(2/3)) + (3*L 
og[(b*c - a*d)^(1/3) + d^(1/3)*(a + b*x^3)^(1/3)])/(2*d^(1/3)*(b*c - a*d)^ 
(2/3))))/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 60

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

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   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 70

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) 
, x] + (Simp[3/(2*b*q)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] + Simp[3/(2*b*q^2)   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 = 1.94 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(-\frac {-\frac {x^{3} \left (a^{\frac {2}{3}} b c -a^{\frac {5}{3}} d \right ) \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}\right )}{2}-x^{3} \sqrt {3}\, \left (a^{\frac {2}{3}} b c -a^{\frac {5}{3}} d \right ) \arctan \left (\frac {\sqrt {3}\, \left (2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}}\right )-\frac {\left (a d -\frac {b c}{3}\right ) \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} x^{3} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}+x^{3} \left (a^{\frac {2}{3}} b c -a^{\frac {5}{3}} d \right ) \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )+\left (-\left (a d -\frac {b c}{3}\right ) \sqrt {3}\, x^{3} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right )+x^{3} \left (a d -\frac {b c}{3}\right ) \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )+a^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} c \right ) \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}}{3 \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} a^{\frac {2}{3}} c^{2} x^{3}}\)	\(357\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (c+d x^3\right )} \, dx=-\frac {6 \, \sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} a^{2} x^{3} \arctan \left (-\frac {2 \, \sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \sqrt {3} {\left (b c d - a d^{2}\right )}}{3 \, {\left (b c d - a d^{2}\right )}}\right ) + 3 \, {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} a^{2} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} d^{2} + {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d + {\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}}\right ) - 6 \, {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} a^{2} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} d - {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}}\right ) + 6 \, \sqrt {\frac {1}{3}} {\left (a b c - 3 \, a^{2} d\right )} x^{3} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-a^{2}\right )^{\frac {1}{3}} a - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{a^{2}}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} {\left (b c - 3 \, a d\right )} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} a - \left (-a^{2}\right )^{\frac {1}{3}} a + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} {\left (b c - 3 \, a d\right )} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - \left (-a^{2}\right )^{\frac {2}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} c}{18 \, a^{2} c^{2} x^{3}} \] Input:

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

Output:

-1/18*(6*sqrt(3)*(-b*c*d^2 + a*d^3)^(1/3)*a^2*x^3*arctan(-1/3*(2*sqrt(3)*( 
-b*c*d^2 + a*d^3)^(2/3)*(b*x^3 + a)^(1/3) - sqrt(3)*(b*c*d - a*d^2))/(b*c* 
d - a*d^2)) + 3*(-b*c*d^2 + a*d^3)^(1/3)*a^2*x^3*log((b*x^3 + a)^(2/3)*d^2 
 + (-b*c*d^2 + a*d^3)^(1/3)*(b*x^3 + a)^(1/3)*d + (-b*c*d^2 + a*d^3)^(2/3) 
) - 6*(-b*c*d^2 + a*d^3)^(1/3)*a^2*x^3*log((b*x^3 + a)^(1/3)*d - (-b*c*d^2 
 + a*d^3)^(1/3)) + 6*sqrt(1/3)*(a*b*c - 3*a^2*d)*x^3*sqrt(-(-a^2)^(1/3))*a 
rctan(-sqrt(1/3)*((-a^2)^(1/3)*a - 2*(b*x^3 + a)^(1/3)*(-a^2)^(2/3))*sqrt( 
-(-a^2)^(1/3))/a^2) + (-a^2)^(2/3)*(b*c - 3*a*d)*x^3*log((b*x^3 + a)^(2/3) 
*a - (-a^2)^(1/3)*a + (b*x^3 + a)^(1/3)*(-a^2)^(2/3)) - 2*(-a^2)^(2/3)*(b* 
c - 3*a*d)*x^3*log((b*x^3 + a)^(1/3)*a - (-a^2)^(2/3)) + 6*(b*x^3 + a)^(1/ 
3)*a^2*c)/(a^2*c^2*x^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (c+d x^3\right )} \, dx=\frac {{\left (b c d - a d^{2}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {\sqrt {3} {\left (b c - 3 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {2}{3}} c^{2}} - \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, c^{2}} - \frac {{\left (b c - 3 \, a d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, a^{\frac {2}{3}} c^{2}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, c^{2}} + \frac {{\left (b c - 3 \, a d\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{9 \, a^{\frac {2}{3}} c^{2}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{3 \, c x^{3}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 1917, normalized size of antiderivative = 6.05 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

log(- ((((27*b^5*c^3*d^3*(a + b*x^3)^(1/3)*(4*a^2*d^2 + b^2*c^2 - 5*a*b*c* 
d) - 27*a*b^4*c^4*d^3*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)*(-(3*a*d - b*c)^3/ 
(a^2*c^6))^(1/3))*(-(3*a*d - b*c)^3/(a^2*c^6))^(2/3))/81 - (b^5*d^4*(27*a^ 
3*d^3 + b^3*c^3 + 17*a*b^2*c^2*d - 45*a^2*b*c*d^2))/(3*c))*(-(3*a*d - b*c) 
^3/(a^2*c^6))^(1/3))/9 - (2*b^4*d^5*(a + b*x^3)^(1/3)*(27*a^4*d^4 + 5*b^4* 
c^4 + 72*a^2*b^2*c^2*d^2 - 32*a*b^3*c^3*d - 72*a^3*b*c*d^3))/(9*c^4))*(-(2 
7*a^3*d^3 - b^3*c^3 + 9*a*b^2*c^2*d - 27*a^2*b*c*d^2)/(729*a^2*c^6))^(1/3) 
 + log(- ((((27*b^5*c^3*d^3*(a + b*x^3)^(1/3)*(4*a^2*d^2 + b^2*c^2 - 5*a*b 
*c*d) - 81*a*b^4*c^4*d^3*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)*((d^2*(a*d - b* 
c))/c^6)^(1/3))*((d^2*(a*d - b*c))/c^6)^(2/3))/9 - (b^5*d^4*(27*a^3*d^3 + 
b^3*c^3 + 17*a*b^2*c^2*d - 45*a^2*b*c*d^2))/(3*c))*((d^2*(a*d - b*c))/c^6) 
^(1/3))/3 - (2*b^4*d^5*(a + b*x^3)^(1/3)*(27*a^4*d^4 + 5*b^4*c^4 + 72*a^2* 
b^2*c^2*d^2 - 32*a*b^3*c^3*d - 72*a^3*b*c*d^3))/(9*c^4))*((a*d^3 - b*c*d^2 
)/(27*c^6))^(1/3) + log((((3^(1/2)*1i)/2 - 1/2)*((((3^(1/2)*1i)/2 + 1/2)*( 
27*b^5*c^3*d^3*(a + b*x^3)^(1/3)*(4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d) - 81*a* 
b^4*c^4*d^3*((3^(1/2)*1i)/2 - 1/2)*(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)*((d^2 
*(a*d - b*c))/c^6)^(1/3))*((d^2*(a*d - b*c))/c^6)^(2/3))/9 + (b^5*d^4*(27* 
a^3*d^3 + b^3*c^3 + 17*a*b^2*c^2*d - 45*a^2*b*c*d^2))/(3*c))*((d^2*(a*d - 
b*c))/c^6)^(1/3))/3 - (2*b^4*d^5*(a + b*x^3)^(1/3)*(27*a^4*d^4 + 5*b^4*c^4 
 + 72*a^2*b^2*c^2*d^2 - 32*a*b^3*c^3*d - 72*a^3*b*c*d^3))/(9*c^4))*((3^...
 

Reduce [F]

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

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

Output:

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