Integrand size = 28, antiderivative size = 268 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (a d-b d x^3\right )} \, dx=\frac {b \sqrt [3]{a+b x^3}}{3 a^2 d}-\frac {\left (a+b x^3\right )^{4/3}}{3 a^2 d x^3}-\frac {4 b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} d}+\frac {\sqrt [3]{2} b \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} d}-\frac {2 b \log (x)}{3 a^{5/3} d}+\frac {b \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{5/3} d}+\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 a^{5/3} d}-\frac {b \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{5/3} d} \]
1/3*b*(b*x^3+a)^(1/3)/a^2/d-1/3*(b*x^3+a)^(4/3)/a^2/d/x^3-2/3*b*ln(x)/a^(5 /3)/d+1/6*b*ln(-b*x^3+a)*2^(1/3)/a^(5/3)/d+2/3*b*ln(a^(1/3)-(b*x^3+a)^(1/3 ))/a^(5/3)/d-1/2*b*ln(2^(1/3)*a^(1/3)-(b*x^3+a)^(1/3))*2^(1/3)/a^(5/3)/d-4 /9*b*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/a^(5/3)/d*3^( 1/2)+1/3*2^(1/3)*b*arctan(1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3))/a^(1/3)*3^ (1/2))/a^(5/3)/d*3^(1/2)
Time = 0.62 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (a d-b d x^3\right )} \, dx=-\frac {6 a^{2/3} \sqrt [3]{a+b x^3}+8 \sqrt {3} b x^3 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-6 \sqrt [3]{2} \sqrt {3} b x^3 \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-8 b x^3 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )+6 \sqrt [3]{2} b x^3 \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )+4 b x^3 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-3 \sqrt [3]{2} b x^3 \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{18 a^{5/3} d x^3} \]
-1/18*(6*a^(2/3)*(a + b*x^3)^(1/3) + 8*Sqrt[3]*b*x^3*ArcTan[(1 + (2*(a + b *x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 6*2^(1/3)*Sqrt[3]*b*x^3*ArcTan[(1 + (2^(2 /3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 8*b*x^3*Log[-a^(1/3) + (a + b*x ^3)^(1/3)] + 6*2^(1/3)*b*x^3*Log[-2*a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/3)] + 4*b*x^3*Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] - 3* 2^(1/3)*b*x^3*Log[2*a^(2/3) + 2^(2/3)*a^(1/3)*(a + b*x^3)^(1/3) + 2^(1/3)* (a + b*x^3)^(2/3)])/(a^(5/3)*d*x^3)
Time = 0.38 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {948, 27, 114, 27, 174, 60, 69, 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.
| \(\displaystyle \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (a d-b d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{b x^3+a}}{d x^6 \left (a-b x^3\right )}dx^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{b x^3+a}}{x^6 \left (a-b x^3\right )}dx^3}{3 d}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {-\frac {\int -\frac {b \left (4 a-b x^3\right ) \sqrt [3]{b x^3+a}}{3 x^3 \left (a-b x^3\right )}dx^3}{a^2}-\frac {\left (a+b x^3\right )^{4/3}}{a^2 x^3}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \int \frac {\left (4 a-b x^3\right ) \sqrt [3]{b x^3+a}}{x^3 \left (a-b x^3\right )}dx^3}{3 a^2}-\frac {\left (a+b x^3\right )^{4/3}}{a^2 x^3}}{3 d}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {\frac {b \left (4 \int \frac {\sqrt [3]{b x^3+a}}{x^3}dx^3+3 b \int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx^3\right )}{3 a^2}-\frac {\left (a+b x^3\right )^{4/3}}{a^2 x^3}}{3 d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {b \left (4 \left (a \int \frac {1}{x^3 \left (b x^3+a\right )^{2/3}}dx^3+3 \sqrt [3]{a+b x^3}\right )+3 b \left (2 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx^3-\frac {3 \sqrt [3]{a+b x^3}}{b}\right )\right )}{3 a^2}-\frac {\left (a+b x^3\right )^{4/3}}{a^2 x^3}}{3 d}\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {\frac {b \left (4 \left (a \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+3 b \left (2 a \left (\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2\ 2^{2/3} a^{2/3} b}+\frac {3 \int \frac {1}{x^6+2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{2} \sqrt [3]{a} b}+\frac {\log \left (a-b x^3\right )}{2\ 2^{2/3} a^{2/3} b}\right )-\frac {3 \sqrt [3]{a+b x^3}}{b}\right )\right )}{3 a^2}-\frac {\left (a+b x^3\right )^{4/3}}{a^2 x^3}}{3 d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {b \left (4 \left (a \left (-\frac {3 \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+3 b \left (2 a \left (\frac {3 \int \frac {1}{x^6+2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{2} \sqrt [3]{a} b}+\frac {\log \left (a-b x^3\right )}{2\ 2^{2/3} a^{2/3} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2\ 2^{2/3} a^{2/3} b}\right )-\frac {3 \sqrt [3]{a+b x^3}}{b}\right )\right )}{3 a^2}-\frac {\left (a+b x^3\right )^{4/3}}{a^2 x^3}}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {b \left (4 \left (a \left (\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+3 b \left (2 a \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2^{2/3} \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{2^{2/3} a^{2/3} b}+\frac {\log \left (a-b x^3\right )}{2\ 2^{2/3} a^{2/3} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2\ 2^{2/3} a^{2/3} b}\right )-\frac {3 \sqrt [3]{a+b x^3}}{b}\right )\right )}{3 a^2}-\frac {\left (a+b x^3\right )^{4/3}}{a^2 x^3}}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {b \left (4 \left (a \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3}}-\frac {\log \left (x^3\right )}{2 a^{2/3}}\right )+3 \sqrt [3]{a+b x^3}\right )+3 b \left (2 a \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{2^{2/3} a^{2/3} b}+\frac {\log \left (a-b x^3\right )}{2\ 2^{2/3} a^{2/3} b}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2\ 2^{2/3} a^{2/3} b}\right )-\frac {3 \sqrt [3]{a+b x^3}}{b}\right )\right )}{3 a^2}-\frac {\left (a+b x^3\right )^{4/3}}{a^2 x^3}}{3 d}\) |
(-((a + b*x^3)^(4/3)/(a^2*x^3)) + (b*(4*(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)))) + 3*b*( (-3*(a + b*x^3)^(1/3))/b + 2*a*((Sqrt[3]*ArcTan[(1 + (2^(2/3)*(a + b*x^3)^ (1/3))/a^(1/3))/Sqrt[3]])/(2^(2/3)*a^(2/3)*b) + Log[a - b*x^3]/(2*2^(2/3)* a^(2/3)*b) - (3*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])/(2*2^(2/3)*a^(2/ 3)*b)))))/(3*a^2))/(3*d)
3.6.72.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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])
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]
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])
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]]
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]
Time = 4.86 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(-\frac {-2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\left (a^{\frac {1}{3}}+2^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) b \,x^{3}+2^{\frac {1}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right ) b \,x^{3}+\frac {4 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b \,x^{3}}{3}-\frac {2^{\frac {1}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right ) b \,x^{3}}{2}-\frac {4 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b \,x^{3}}{3}+\frac {2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b \,x^{3}}{3}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{\frac {2}{3}}}{3 a^{\frac {5}{3}} x^{3} d}\) | \(222\) |
-1/3*(-2^(1/3)*3^(1/2)*arctan(1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3))/a^(1/3 )*3^(1/2))*b*x^3+2^(1/3)*ln((b*x^3+a)^(1/3)-2^(1/3)*a^(1/3))*b*x^3+4/3*arc tan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)*b*x^3-1/2*2^( 1/3)*ln((b*x^3+a)^(2/3)+2^(1/3)*a^(1/3)*(b*x^3+a)^(1/3)+2^(2/3)*a^(2/3))*b *x^3-4/3*ln((b*x^3+a)^(1/3)-a^(1/3))*b*x^3+2/3*ln((b*x^3+a)^(2/3)+a^(1/3)* (b*x^3+a)^(1/3)+a^(2/3))*b*x^3+(b*x^3+a)^(1/3)*a^(2/3))/a^(5/3)/x^3/d
Time = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (a d-b d x^3\right )} \, dx=-\frac {6 \, \sqrt {3} 2^{\frac {1}{3}} a^{2} b x^{3} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} 2^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a \left (-\frac {1}{a^{2}}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 3 \cdot 2^{\frac {1}{3}} a^{2} b x^{3} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{2} \left (-\frac {1}{a^{2}}\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) - 6 \cdot 2^{\frac {1}{3}} a^{2} b x^{3} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} a \left (-\frac {1}{a^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 8 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a b x^{3} \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\left (\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, a^{2}}\right ) + 4 \, {\left (a^{2}\right )}^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right ) - 8 \, {\left (a^{2}\right )}^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2}}{18 \, a^{3} d x^{3}} \]
-1/18*(6*sqrt(3)*2^(1/3)*a^2*b*x^3*(-1/a^2)^(1/3)*arctan(1/3*sqrt(3)*2^(2/ 3)*(b*x^3 + a)^(1/3)*a*(-1/a^2)^(2/3) + 1/3*sqrt(3)) + 3*2^(1/3)*a^2*b*x^3 *(-1/a^2)^(1/3)*log(2^(2/3)*a^2*(-1/a^2)^(2/3) - 2^(1/3)*(b*x^3 + a)^(1/3) *a*(-1/a^2)^(1/3) + (b*x^3 + a)^(2/3)) - 6*2^(1/3)*a^2*b*x^3*(-1/a^2)^(1/3 )*log(2^(1/3)*a*(-1/a^2)^(1/3) + (b*x^3 + a)^(1/3)) + 8*sqrt(3)*(a^2)^(1/6 )*a*b*x^3*arctan(1/3*(a^2)^(1/6)*(sqrt(3)*(a^2)^(1/3)*a + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(a^2)^(2/3))/a^2) + 4*(a^2)^(2/3)*b*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)) - 8*(a^2)^(2/3)*b*x^3*l og((b*x^3 + a)^(1/3)*a - (a^2)^(2/3)) + 6*(b*x^3 + a)^(1/3)*a^2)/(a^3*d*x^ 3)
\[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (a d-b d x^3\right )} \, dx=- \frac {\int \frac {\sqrt [3]{a + b x^{3}}}{- a x^{4} + b x^{7}}\, dx}{d} \]
\[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (a d-b d x^3\right )} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (b d x^{3} - a d\right )} x^{4}} \,d x } \]
Time = 0.98 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (a d-b d x^3\right )} \, dx=\frac {\sqrt {3} 2^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {5}{3}} d} - \frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {5}{3}} d} + \frac {2^{\frac {1}{3}} b \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{6 \, a^{\frac {5}{3}} d} - \frac {2^{\frac {1}{3}} b \log \left ({\left | -2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {5}{3}} d} - \frac {2 \, b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {5}{3}} d} + \frac {4 \, b \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{9 \, a^{\frac {5}{3}} d} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{3 \, a d x^{3}} \]
1/3*sqrt(3)*2^(1/3)*b*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(b*x ^3 + a)^(1/3))/a^(1/3))/(a^(5/3)*d) - 4/9*sqrt(3)*b*arctan(1/3*sqrt(3)*(2* (b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3))/(a^(5/3)*d) + 1/6*2^(1/3)*b*log(2^(2 /3)*a^(2/3) + 2^(1/3)*(b*x^3 + a)^(1/3)*a^(1/3) + (b*x^3 + a)^(2/3))/(a^(5 /3)*d) - 1/3*2^(1/3)*b*log(abs(-2^(1/3)*a^(1/3) + (b*x^3 + a)^(1/3)))/(a^( 5/3)*d) - 2/9*b*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3 ))/(a^(5/3)*d) + 4/9*b*log(abs((b*x^3 + a)^(1/3) - a^(1/3)))/(a^(5/3)*d) - 1/3*(b*x^3 + a)^(1/3)/(a*d*x^3)
Time = 9.29 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^4 \left (a d-b d x^3\right )} \, dx=\frac {4\,\ln \left (b\,{\left (b\,x^3+a\right )}^{1/3}-a^2\,d\,{\left (\frac {b^3}{a^5\,d^3}\right )}^{1/3}\right )\,{\left (\frac {b^3}{a^5\,d^3}\right )}^{1/3}}{9}+\ln \left (b\,{\left (b\,x^3+a\right )}^{1/3}+2^{1/3}\,a^2\,d\,{\left (-\frac {b^3}{a^5\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {2\,b^3}{27\,a^5\,d^3}\right )}^{1/3}+\ln \left (2\,b\,{\left (b\,x^3+a\right )}^{1/3}+a^2\,d\,{\left (\frac {b^3}{a^5\,d^3}\right )}^{1/3}-\sqrt {3}\,a^2\,d\,{\left (\frac {b^3}{a^5\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {64\,b^3}{729\,a^5\,d^3}\right )}^{1/3}-\ln \left (2\,b\,{\left (b\,x^3+a\right )}^{1/3}+a^2\,d\,{\left (\frac {b^3}{a^5\,d^3}\right )}^{1/3}+\sqrt {3}\,a^2\,d\,{\left (\frac {b^3}{a^5\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {64\,b^3}{729\,a^5\,d^3}\right )}^{1/3}-\ln \left (2^{1/3}\,a^2\,d\,{\left (-\frac {b^3}{a^5\,d^3}\right )}^{1/3}-2\,b\,{\left (b\,x^3+a\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,a^2\,d\,{\left (-\frac {b^3}{a^5\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2\,b^3}{27\,a^5\,d^3}\right )}^{1/3}+\ln \left (2\,b\,{\left (b\,x^3+a\right )}^{1/3}-2^{1/3}\,a^2\,d\,{\left (-\frac {b^3}{a^5\,d^3}\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,a^2\,d\,{\left (-\frac {b^3}{a^5\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2\,b^3}{27\,a^5\,d^3}\right )}^{1/3}-\frac {b\,{\left (b\,x^3+a\right )}^{1/3}}{3\,a\,\left (d\,\left (b\,x^3+a\right )-a\,d\right )} \]
(4*log(b*(a + b*x^3)^(1/3) - a^2*d*(b^3/(a^5*d^3))^(1/3))*(b^3/(a^5*d^3))^ (1/3))/9 + log(b*(a + b*x^3)^(1/3) + 2^(1/3)*a^2*d*(-b^3/(a^5*d^3))^(1/3)) *(-(2*b^3)/(27*a^5*d^3))^(1/3) + log(2*b*(a + b*x^3)^(1/3) + a^2*d*(b^3/(a ^5*d^3))^(1/3) - 3^(1/2)*a^2*d*(b^3/(a^5*d^3))^(1/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*((64*b^3)/(729*a^5*d^3))^(1/3) - log(2*b*(a + b*x^3)^(1/3) + a^2*d*( b^3/(a^5*d^3))^(1/3) + 3^(1/2)*a^2*d*(b^3/(a^5*d^3))^(1/3)*1i)*((3^(1/2)*1 i)/2 + 1/2)*((64*b^3)/(729*a^5*d^3))^(1/3) - log(2^(1/3)*a^2*d*(-b^3/(a^5* d^3))^(1/3) - 2*b*(a + b*x^3)^(1/3) + 2^(1/3)*3^(1/2)*a^2*d*(-b^3/(a^5*d^3 ))^(1/3)*1i)*((3^(1/2)*1i)/2 + 1/2)*(-(2*b^3)/(27*a^5*d^3))^(1/3) + log(2* b*(a + b*x^3)^(1/3) - 2^(1/3)*a^2*d*(-b^3/(a^5*d^3))^(1/3) + 2^(1/3)*3^(1/ 2)*a^2*d*(-b^3/(a^5*d^3))^(1/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*(-(2*b^3)/(27*a ^5*d^3))^(1/3) - (b*(a + b*x^3)^(1/3))/(3*a*(d*(a + b*x^3) - a*d))