Integrand size = 13, antiderivative size = 128 \[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx=-\frac {5}{2 a^2 x^{2/3}}+\frac {1}{a x^{2/3} (a+b x)}+\frac {5 b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{8/3}}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{8/3}}+\frac {5 b^{2/3} \log (a+b x)}{6 a^{8/3}} \]
-5/2/a^2/x^(2/3)+1/a/x^(2/3)/(b*x+a)-5/2*b^(2/3)*ln(a^(1/3)+b^(1/3)*x^(1/3 ))/a^(8/3)+5/6*b^(2/3)*ln(b*x+a)/a^(8/3)+5/3*b^(2/3)*arctan(1/3*(a^(1/3)-2 *b^(1/3)*x^(1/3))/a^(1/3)*3^(1/2))/a^(8/3)*3^(1/2)
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx=\frac {-\frac {3 a^{2/3} (3 a+5 b x)}{x^{2/3} (a+b x)}+10 \sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{6 a^{8/3}} \]
((-3*a^(2/3)*(3*a + 5*b*x))/(x^(2/3)*(a + b*x)) + 10*Sqrt[3]*b^(2/3)*ArcTa n[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]] - 10*b^(2/3)*Log[a^(1/3) + b^ (1/3)*x^(1/3)] + 5*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3) *x^(2/3)])/(6*a^(8/3))
Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {52, 61, 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.
| \(\displaystyle \int \frac {1}{x^{5/3} (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {5 \int \frac {1}{x^{5/3} (a+b x)}dx}{3 a}+\frac {1}{a x^{2/3} (a+b x)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {5 \left (-\frac {b \int \frac {1}{x^{2/3} (a+b x)}dx}{a}-\frac {3}{2 a x^{2/3}}\right )}{3 a}+\frac {1}{a x^{2/3} (a+b x)}\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {5 \left (-\frac {b \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {3}{2 a x^{2/3}}\right )}{3 a}+\frac {1}{a x^{2/3} (a+b x)}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {5 \left (-\frac {b \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {3}{2 a x^{2/3}}\right )}{3 a}+\frac {1}{a x^{2/3} (a+b x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (-\frac {b \left (\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {3}{2 a x^{2/3}}\right )}{3 a}+\frac {1}{a x^{2/3} (a+b x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (-\frac {b \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {3}{2 a x^{2/3}}\right )}{3 a}+\frac {1}{a x^{2/3} (a+b x)}\) |
1/(a*x^(2/3)*(a + b*x)) + (5*(-3/(2*a*x^(2/3)) - (b*(-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/(a^(2/3)*b^(1/3))) + (3*Log[a^(1/ 3) + b^(1/3)*x^(1/3)])/(2*a^(2/3)*b^(1/3)) - Log[a + b*x]/(2*a^(2/3)*b^(1/ 3))))/a))/(3*a)
3.7.89.3.1 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_.) + (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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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]
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]
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[((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 = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(-\frac {3}{2 a^{2} x^{\frac {2}{3}}}-\frac {3 b \left (\frac {x^{\frac {1}{3}}}{3 b x +3 a}+\frac {5 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {5 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{a^{2}}\) | \(124\) |
| default | \(-\frac {3}{2 a^{2} x^{\frac {2}{3}}}-\frac {3 b \left (\frac {x^{\frac {1}{3}}}{3 b x +3 a}+\frac {5 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {5 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{a^{2}}\) | \(124\) |
-3/2/a^2/x^(2/3)-3*b/a^2*(1/3*x^(1/3)/(b*x+a)+5/9/b/(a/b)^(2/3)*ln(x^(1/3) +(a/b)^(1/3))-5/18/b/(a/b)^(2/3)*ln(x^(2/3)-(a/b)^(1/3)*x^(1/3)+(a/b)^(2/3 ))+5/9/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/3)-1)) )
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (91) = 182\).
Time = 0.23 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx=\frac {10 \, \sqrt {3} {\left (b x^{2} + a x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x^{\frac {1}{3}} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 5 \, {\left (b x^{2} + a x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{\frac {2}{3}} + a b x^{\frac {1}{3}} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b x^{2} + a x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{\frac {1}{3}} - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3 \, {\left (5 \, b x + 3 \, a\right )} x^{\frac {1}{3}}}{6 \, {\left (a^{2} b x^{2} + a^{3} x\right )}} \]
1/6*(10*sqrt(3)*(b*x^2 + a*x)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x^( 1/3)*(-b^2/a^2)^(2/3) - sqrt(3)*b)/b) - 5*(b*x^2 + a*x)*(-b^2/a^2)^(1/3)*l og(b^2*x^(2/3) + a*b*x^(1/3)*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) + 10 *(b*x^2 + a*x)*(-b^2/a^2)^(1/3)*log(b*x^(1/3) - a*(-b^2/a^2)^(1/3)) - 3*(5 *b*x + 3*a)*x^(1/3))/(a^2*b*x^2 + a^3*x)
Timed out. \[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx=-\frac {5 \, b x + 3 \, a}{2 \, {\left (a^{2} b x^{\frac {5}{3}} + a^{3} x^{\frac {2}{3}}\right )}} - \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/2*(5*b*x + 3*a)/(a^2*b*x^(5/3) + a^3*x^(2/3)) - 5/3*sqrt(3)*arctan(1/3* sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*(a/b)^(2/3)) + 5/6*log (x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*(a/b)^(2/3)) - 5/3*log( x^(1/3) + (a/b)^(1/3))/(a^2*(a/b)^(2/3))
Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx=\frac {5 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{3}} - \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3}} - \frac {b x^{\frac {1}{3}}}{{\left (b x + a\right )} a^{2}} - \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{3}} - \frac {3}{2 \, a^{2} x^{\frac {2}{3}}} \]
5/3*b*(-a/b)^(1/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^3 - 5/3*sqrt(3)*(-a* b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/a^3 - b*x^(1/3)/((b*x + a)*a^2) - 5/6*(-a*b^2)^(1/3)*log(x^(2/3) + x^(1/3)*(- a/b)^(1/3) + (-a/b)^(2/3))/a^3 - 3/2/(a^2*x^(2/3))
Time = 0.16 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx=\frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left (15\,{\left (-1\right )}^{1/3}\,a^{13/3}\,b^{8/3}-15\,a^4\,b^3\,x^{1/3}\right )}{3\,a^{8/3}}-\frac {\frac {3}{2\,a}+\frac {5\,b\,x}{2\,a^2}}{a\,x^{2/3}+b\,x^{5/3}}+\frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left (15\,a^4\,b^3\,x^{1/3}-15\,{\left (-1\right )}^{1/3}\,a^{13/3}\,b^{8/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{8/3}}-\frac {5\,{\left (-1\right )}^{1/3}\,b^{2/3}\,\ln \left (15\,a^4\,b^3\,x^{1/3}+15\,{\left (-1\right )}^{1/3}\,a^{13/3}\,b^{8/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{8/3}} \]
(5*(-1)^(1/3)*b^(2/3)*log(15*(-1)^(1/3)*a^(13/3)*b^(8/3) - 15*a^4*b^3*x^(1 /3)))/(3*a^(8/3)) - (3/(2*a) + (5*b*x)/(2*a^2))/(a*x^(2/3) + b*x^(5/3)) + (5*(-1)^(1/3)*b^(2/3)*log(15*a^4*b^3*x^(1/3) - 15*(-1)^(1/3)*a^(13/3)*b^(8 /3)*((3^(1/2)*1i)/2 - 1/2))*((3^(1/2)*1i)/2 - 1/2))/(3*a^(8/3)) - (5*(-1)^ (1/3)*b^(2/3)*log(15*a^4*b^3*x^(1/3) + 15*(-1)^(1/3)*a^(13/3)*b^(8/3)*((3^ (1/2)*1i)/2 + 1/2))*((3^(1/2)*1i)/2 + 1/2))/(3*a^(8/3))
Time = 0.01 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.64 \[ \int \frac {1}{x^{5/3} (a+b x)^2} \, dx=\frac {10 x^{\frac {2}{3}} a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b +10 x^{\frac {5}{3}} a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2}+5 x^{\frac {2}{3}} a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b +5 x^{\frac {5}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b^{2}-10 x^{\frac {2}{3}} a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b -10 x^{\frac {5}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b^{2}-9 b^{\frac {1}{3}} a^{2}-15 b^{\frac {4}{3}} a x}{6 x^{\frac {2}{3}} b^{\frac {1}{3}} a^{3} \left (b x +a \right )} \]
(10*x**(2/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1 /3)*sqrt(3)))*a*b + 10*x**(2/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/ 3)*b**(1/3))/(a**(1/3)*sqrt(3)))*b**2*x + 5*x**(2/3)*a**(1/3)*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*a*b + 5*x**(2/3)*a**(1/ 3)*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*b**2*x - 10*x**(2/3)*a**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*a*b - 10*x**(2/3)* a**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*b**2*x - 9*b**(1/3)*a**2 - 15*b **(1/3)*a*b*x)/(6*x**(2/3)*b**(1/3)*a**3*(a + b*x))