\(\int \frac {(a+b x^3)^{4/3}}{x^4 (c+d x^3)} \, dx\) [715]

Optimal result

Integrand size = 24, antiderivative size = 340 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^4 \left (c+d x^3\right )} \, dx=\frac {b \sqrt [3]{a+b x^3}}{3 c}+\frac {b \left (a+b x^3\right )^{4/3}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{3 a c x^3}-\frac {\sqrt [3]{a} (4 b c-3 a d) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} c^2}-\frac {(b c-a d)^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c^2 \sqrt [3]{d}}-\frac {\sqrt [3]{a} (4 b c-3 a d) \log (x)}{6 c^2}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^2 \sqrt [3]{d}}+\frac {\sqrt [3]{a} (4 b c-3 a d) \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 c^2}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c^2 \sqrt [3]{d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^4 \left (c+d x^3\right )} \, dx=\frac {-\frac {6 a c \sqrt [3]{a+b x^3}}{x^3}+2 \sqrt {3} \sqrt [3]{a} (-4 b c+3 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\frac {6 \sqrt {3} (b c-a d)^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-2 \sqrt [3]{a} (-4 b c+3 a d) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )+\frac {6 (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{\sqrt [3]{d}}+\sqrt [3]{a} (-4 b c+3 a d) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-\frac {3 (b c-a d)^{4/3} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{\sqrt [3]{d}}}{18 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {948, 114, 27, 174, 60, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{3} \left (-\frac {\int -\frac {\left (b x^3+a\right )^{4/3} \left (4 b d x^3+4 b c-3 a d\right )}{3 x^3 \left (d x^3+c\right )}dx^3}{a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {\left (b x^3+a\right )^{4/3} \left (4 b d x^3+4 b c-3 a d\right )}{x^3 \left (d x^3+c\right )}dx^3}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{3} \left (\frac {\frac {3 a d^2 \int \frac {\left (b x^3+a\right )^{4/3}}{d x^3+c}dx^3}{c}+\frac {(4 b c-3 a d) \int \frac {\left (b x^3+a\right )^{4/3}}{x^3}dx^3}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{3} \left (\frac {\frac {3 a d^2 \left (\frac {3 \left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \int \frac {\sqrt [3]{b x^3+a}}{d x^3+c}dx^3}{d}\right )}{c}+\frac {(4 b c-3 a d) \left (a \int \frac {\sqrt [3]{b x^3+a}}{x^3}dx^3+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{3} \left (\frac {\frac {3 a d^2 \left (\frac {3 \left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{d}\right )}{d}\right )}{c}+\frac {(4 b c-3 a d) \left (a \left (a \int \frac {1}{x^3 \left (b x^3+a\right )^{2/3}}dx^3+3 \sqrt [3]{a+b x^3}\right )+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {1}{3} \left (\frac {\frac {(4 b c-3 a d) \left (a \left (a \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{c}+\frac {3 a d^2 \left (\frac {3 \left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{d}\right )}{d}\right )}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {\frac {(4 b c-3 a d) \left (a \left (a \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{c}+\frac {3 a d^2 \left (\frac {3 \left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \int \frac {1}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx^3}{d}\right )}{d}\right )}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 70

\(\displaystyle \frac {1}{3} \left (\frac {\frac {(4 b c-3 a d) \left (a \left (a \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{c}+\frac {3 a d^2 \left (\frac {3 \left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \left (\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{d}\right )}{d}\right )}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {\frac {(4 b c-3 a d) \left (a \left (a \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{c}+\frac {3 a d^2 \left (\frac {3 \left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \left (\frac {3 \int \frac {1}{x^6+\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} \sqrt [3]{b x^3+a}}{\sqrt [3]{d}}}d\sqrt [3]{b x^3+a}}{2 d^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{d}\right )}{d}\right )}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} \left (\frac {\frac {(4 b c-3 a d) \left (a \left (a \left (\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{c}+\frac {3 a d^2 \left (\frac {3 \left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \left (\frac {3 \int \frac {1}{-x^6-3}d\left (1-\frac {2 \sqrt [3]{d} \sqrt [3]{b x^3+a}}{\sqrt [3]{b c-a d}}\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{d}\right )}{d}\right )}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {\frac {(4 b c-3 a d) \left (a \left (a \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+\frac {3}{4} \left (a+b x^3\right )^{4/3}\right )}{c}+\frac {3 a d^2 \left (\frac {3 \left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \left (\frac {3 \sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\right )}{d}\right )}{d}\right )}{c}}{3 a c}-\frac {\left (a+b x^3\right )^{7/3}}{a c x^3}\right )\)

Input:

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

Output:

(-((a + b*x^3)^(7/3)/(a*c*x^3)) + (((4*b*c - 3*a*d)*((3*(a + b*x^3)^(4/3)) 
/4 + a*(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)^(4/3))/(4 
*d) - ((b*c - a*d)*((3*(a + b*x^3)^(1/3))/d - ((b*c - a*d)*(-((Sqrt[3]*Arc 
Tan[(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 
*Log[(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))/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 = 1.87 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.07

method	result	size
pseudoelliptic	\(\frac {-\frac {x^{3} \left (a d -b c \right )^{2} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}\right )}{2}-x^{3} \sqrt {3}\, \left (a d -b c \right )^{2} \arctan \left (\frac {\sqrt {3}\, \left (2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}}\right )+\frac {d \left (-\frac {4 a^{\frac {1}{3}} b c}{3}+d \,a^{\frac {4}{3}}\right ) \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} x^{3} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}+x^{3} \left (a d -b c \right )^{2} \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{3}}\right )-d \left (-\left (-\frac {4 a^{\frac {1}{3}} b c}{3}+d \,a^{\frac {4}{3}}\right ) \sqrt {3}\, x^{3} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right )+x^{3} \left (-\frac {4 a^{\frac {1}{3}} b c}{3}+d \,a^{\frac {4}{3}}\right ) \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )+\left (b \,x^{3}+a \right )^{\frac {1}{3}} a c \right ) \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}}}{3 \left (\frac {a d -b c}{d}\right )^{\frac {2}{3}} c^{2} d \,x^{3}}\)	\(364\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^4 \left (c+d x^3\right )} \, dx=\frac {6 \, \sqrt {3} {\left (b c - a d\right )} x^{3} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) + 2 \, \sqrt {3} {\left (4 \, b c - 3 \, a d\right )} \left (-a\right )^{\frac {1}{3}} x^{3} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + {\left (4 \, b c - 3 \, a d\right )} \left (-a\right )^{\frac {1}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 3 \, {\left (b c - a d\right )} x^{3} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (4 \, b c - 3 \, a d\right )} \left (-a\right )^{\frac {1}{3}} x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b c - a d\right )} x^{3} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a c}{18 \, c^{2} x^{3}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^4 \left (c+d x^3\right )} \, dx=-\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {\sqrt {3} {\left (4 \, a^{\frac {1}{3}} b c - 3 \, a^{\frac {4}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, c^{2}} - \frac {{\left (4 \, a^{\frac {1}{3}} b c - 3 \, a^{\frac {4}{3}} d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, c^{2}} + \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, c^{2} d} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, c^{2} d} + \frac {{\left (4 \, a b c - 3 \, a^{2} d\right )} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{9 \, a^{\frac {2}{3}} c^{2}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} a}{3 \, c x^{3}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 2047, normalized size of antiderivative = 6.02 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^4 \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

log(c^2*(-(a*(3*a*d - 4*b*c)^3)/c^6)^(1/3) + 3*a*d*(a + b*x^3)^(1/3) - 4*b 
*c*(a + b*x^3)^(1/3))*(-(27*a^4*d^3 - 64*a*b^3*c^3 + 144*a^2*b^2*c^2*d - 1 
08*a^3*b*c*d^2)/(729*c^6))^(1/3) + log(((((81*a*b^4*c^4*d^3*(2*a^2*d^2 + b 
^2*c^2 - 3*a*b*c*d)*((a*d - b*c)^4/(c^6*d))^(1/3) - 108*a*b^5*c^3*d^3*(a + 
 b*x^3)^(1/3)*(a*d - b*c)^2)*((a*d - b*c)^4/(c^6*d))^(2/3))/9 + (a*b^5*d^2 
*(27*a^5*d^5 - 27*b^5*c^5 - 341*a^2*b^3*c^3*d^2 + 332*a^3*b^2*c^2*d^3 + 16 
2*a*b^4*c^4*d - 153*a^4*b*c*d^4))/(3*c))*((a*d - b*c)^4/(c^6*d))^(1/3))/3 
- (a*b^4*d^2*(a + b*x^3)^(1/3)*(a*d - b*c)^2*(54*a^5*d^5 - 36*b^5*c^5 - 38 
8*a^2*b^3*c^3*d^2 + 450*a^3*b^2*c^2*d^3 + 171*a*b^4*c^4*d - 252*a^4*b*c*d^ 
4))/(9*c^4))*((a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a 
^3*b*c*d^3)/(27*c^6*d))^(1/3) + log((((3^(1/2)*1i)/2 - 1/2)*((((3^(1/2)*1i 
)/2 + 1/2)*(108*a*b^5*c^3*d^3*(a + b*x^3)^(1/3)*(a*d - b*c)^2 - 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)*((a*d - b* 
c)^4/(c^6*d))^(1/3))*((a*d - b*c)^4/(c^6*d))^(2/3))/9 + (a*b^5*d^2*(27*a^5 
*d^5 - 27*b^5*c^5 - 341*a^2*b^3*c^3*d^2 + 332*a^3*b^2*c^2*d^3 + 162*a*b^4* 
c^4*d - 153*a^4*b*c*d^4))/(3*c))*((a*d - b*c)^4/(c^6*d))^(1/3))/3 - (a*b^4 
*d^2*(a + b*x^3)^(1/3)*(a*d - b*c)^2*(54*a^5*d^5 - 36*b^5*c^5 - 388*a^2*b^ 
3*c^3*d^2 + 450*a^3*b^2*c^2*d^3 + 171*a*b^4*c^4*d - 252*a^4*b*c*d^4))/(9*c 
^4))*((3^(1/2)*1i)/2 - 1/2)*((a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a* 
b^3*c^3*d - 4*a^3*b*c*d^3)/(27*c^6*d))^(1/3) - log((a*b^4*d^2*(a + b*x^...
 

Reduce [F]

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

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

Output:

( - 3*(a + b*x**3)**(1/3)*a**2*d - 2*(a + b*x**3)**(1/3)*a*b*c - 3*(a + b* 
x**3)**(1/3)*a*b*d*x**3 + 6*(a + b*x**3)**(1/3)*b**2*c*x**3 - 27*int((a + 
b*x**3)**(1/3)/(3*a**2*c*d*x + 3*a**2*d**2*x**4 + 2*a*b*c**2*x + 5*a*b*c*d 
*x**4 + 3*a*b*d**2*x**7 + 2*b**2*c**2*x**4 + 2*b**2*c*d*x**7),x)*a**4*d**3 
*x**3 + 36*int((a + b*x**3)**(1/3)/(3*a**2*c*d*x + 3*a**2*d**2*x**4 + 2*a* 
b*c**2*x + 5*a*b*c*d*x**4 + 3*a*b*d**2*x**7 + 2*b**2*c**2*x**4 + 2*b**2*c* 
d*x**7),x)*a**2*b**2*c**2*d*x**3 + 16*int((a + b*x**3)**(1/3)/(3*a**2*c*d* 
x + 3*a**2*d**2*x**4 + 2*a*b*c**2*x + 5*a*b*c*d*x**4 + 3*a*b*d**2*x**7 + 2 
*b**2*c**2*x**4 + 2*b**2*c*d*x**7),x)*a*b**3*c**3*x**3 + 9*int(((a + b*x** 
3)**(1/3)*x**5)/(3*a**2*c*d + 3*a**2*d**2*x**3 + 2*a*b*c**2 + 5*a*b*c*d*x* 
*3 + 3*a*b*d**2*x**6 + 2*b**2*c**2*x**3 + 2*b**2*c*d*x**6),x)*a**2*b**2*d* 
*3*x**3 - 12*int(((a + b*x**3)**(1/3)*x**5)/(3*a**2*c*d + 3*a**2*d**2*x**3 
 + 2*a*b*c**2 + 5*a*b*c*d*x**3 + 3*a*b*d**2*x**6 + 2*b**2*c**2*x**3 + 2*b* 
*2*c*d*x**6),x)*a*b**3*c*d**2*x**3 - 12*int(((a + b*x**3)**(1/3)*x**5)/(3* 
a**2*c*d + 3*a**2*d**2*x**3 + 2*a*b*c**2 + 5*a*b*c*d*x**3 + 3*a*b*d**2*x** 
6 + 2*b**2*c**2*x**3 + 2*b**2*c*d*x**6),x)*b**4*c**2*d*x**3 - 18*int(((a + 
 b*x**3)**(1/3)*x**2)/(3*a**2*c*d + 3*a**2*d**2*x**3 + 2*a*b*c**2 + 5*a*b* 
c*d*x**3 + 3*a*b*d**2*x**6 + 2*b**2*c**2*x**3 + 2*b**2*c*d*x**6),x)*a**3*b 
*d**3*x**3 + 12*int(((a + b*x**3)**(1/3)*x**2)/(3*a**2*c*d + 3*a**2*d**2*x 
**3 + 2*a*b*c**2 + 5*a*b*c*d*x**3 + 3*a*b*d**2*x**6 + 2*b**2*c**2*x**3 ...