Integrand size = 15, antiderivative size = 136 \[ \int \frac {1}{x^3 \sqrt [3]{-a+b x}} \, dx=\frac {(-a+b x)^{2/3}}{2 a x^2}+\frac {2 b (-a+b x)^{2/3}}{3 a^2 x}-\frac {2 b^2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{-a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}+\frac {b^2 \log (x)}{9 a^{7/3}}-\frac {b^2 \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{3 a^{7/3}} \]
1/2*(b*x-a)^(2/3)/a/x^2+2/3*b*(b*x-a)^(2/3)/a^2/x+1/9*b^2*ln(x)/a^(7/3)-1/ 3*b^2*ln(a^(1/3)+(b*x-a)^(1/3))/a^(7/3)-2/9*b^2*arctan(1/3*(a^(1/3)-2*(b*x -a)^(1/3))/a^(1/3)*3^(1/2))/a^(7/3)*3^(1/2)
Time = 0.12 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^3 \sqrt [3]{-a+b x}} \, dx=\frac {(-a+b x)^{2/3} (7 a+4 (-a+b x))}{6 a^2 x^2}-\frac {2 b^2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {2 b^2 \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{9 a^{7/3}}+\frac {b^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{-a+b x}+(-a+b x)^{2/3}\right )}{9 a^{7/3}} \]
((-a + b*x)^(2/3)*(7*a + 4*(-a + b*x)))/(6*a^2*x^2) - (2*b^2*ArcTan[1/Sqrt [3] - (2*(-a + b*x)^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)) - (2*b^ 2*Log[a^(1/3) + (-a + b*x)^(1/3)])/(9*a^(7/3)) + (b^2*Log[a^(2/3) - a^(1/3 )*(-a + b*x)^(1/3) + (-a + b*x)^(2/3)])/(9*a^(7/3))
Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {52, 52, 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.
| \(\displaystyle \int \frac {1}{x^3 \sqrt [3]{b x-a}} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2 b \int \frac {1}{x^2 \sqrt [3]{b x-a}}dx}{3 a}+\frac {(b x-a)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2 b \left (\frac {b \int \frac {1}{x \sqrt [3]{b x-a}}dx}{3 a}+\frac {(b x-a)^{2/3}}{a x}\right )}{3 a}+\frac {(b x-a)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {2 b \left (\frac {b \left (\frac {3}{2} \int \frac {1}{a^{2/3}-\sqrt [3]{b x-a} \sqrt [3]{a}+(b x-a)^{2/3}}d\sqrt [3]{b x-a}-\frac {3 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b x-a}}d\sqrt [3]{b x-a}}{2 \sqrt [3]{a}}+\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}+\frac {(b x-a)^{2/3}}{a x}\right )}{3 a}+\frac {(b x-a)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 b \left (\frac {b \left (\frac {3}{2} \int \frac {1}{a^{2/3}-\sqrt [3]{b x-a} \sqrt [3]{a}+(b x-a)^{2/3}}d\sqrt [3]{b x-a}-\frac {3 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 \sqrt [3]{a}}+\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}+\frac {(b x-a)^{2/3}}{a x}\right )}{3 a}+\frac {(b x-a)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 b \left (\frac {b \left (\frac {3 \int \frac {1}{-(b x-a)^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b x-a}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 \sqrt [3]{a}}+\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}+\frac {(b x-a)^{2/3}}{a x}\right )}{3 a}+\frac {(b x-a)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 b \left (\frac {b \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b x-a}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 \sqrt [3]{a}}+\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}+\frac {(b x-a)^{2/3}}{a x}\right )}{3 a}+\frac {(b x-a)^{2/3}}{2 a x^2}\) |
(-a + b*x)^(2/3)/(2*a*x^2) + (2*b*((-a + b*x)^(2/3)/(a*x) + (b*(-((Sqrt[3] *ArcTan[(1 - (2*(-a + b*x)^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3)) + Log[x]/(2* a^(1/3)) - (3*Log[a^(1/3) + (-a + b*x)^(1/3)])/(2*a^(1/3))))/(3*a)))/(3*a)
3.5.5.3.1 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[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]
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 = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(-\frac {\left (-b x +a \right ) \left (4 b x +3 a \right )}{6 a^{2} x^{2} \left (b x -a \right )^{\frac {1}{3}}}-\frac {2 b^{2} \ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{9 a^{\frac {7}{3}}}+\frac {b^{2} \ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {7}{3}}}+\frac {2 b^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a^{\frac {7}{3}}}\) | \(124\) |
| pseudoelliptic | \(\frac {-4 \sqrt {3}\, \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 \left (b x -a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) b^{2} x^{2}-4 \ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right ) b^{2} x^{2}+2 \ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b^{2} x^{2}+12 b x \,a^{\frac {1}{3}} \left (b x -a \right )^{\frac {2}{3}}+9 \left (b x -a \right )^{\frac {2}{3}} a^{\frac {4}{3}}}{18 a^{\frac {7}{3}} x^{2}}\) | \(133\) |
| derivativedivides | \(3 b^{2} \left (\frac {\left (b x -a \right )^{\frac {2}{3}}}{6 a \,b^{2} x^{2}}+\frac {\frac {2 \left (b x -a \right )^{\frac {2}{3}}}{9 a b x}+\frac {2 \left (-\frac {\ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {1}{3}}}\right )}{9 a}}{a}\right )\) | \(141\) |
| default | \(3 b^{2} \left (\frac {\left (b x -a \right )^{\frac {2}{3}}}{6 a \,b^{2} x^{2}}+\frac {\frac {2 \left (b x -a \right )^{\frac {2}{3}}}{9 a b x}+\frac {2 \left (-\frac {\ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {1}{3}}}\right )}{9 a}}{a}\right )\) | \(141\) |
-1/6*(-b*x+a)*(4*b*x+3*a)/a^2/x^2/(b*x-a)^(1/3)-2/9*b^2*ln(a^(1/3)+(b*x-a) ^(1/3))/a^(7/3)+1/9*b^2/a^(7/3)*ln((b*x-a)^(2/3)-a^(1/3)*(b*x-a)^(1/3)+a^( 2/3))+2/9*b^2/a^(7/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x-a)^(1/3)- 1))
Time = 0.23 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.75 \[ \int \frac {1}{x^3 \sqrt [3]{-a+b x}} \, dx=\left [\frac {6 \, \sqrt {\frac {1}{3}} a b^{2} x^{2} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x - a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} - 3 \, a}{x}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 4 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x + 3 \, a^{2}\right )} {\left (b x - a\right )}^{\frac {2}{3}}}{18 \, a^{3} x^{2}}, \frac {12 \, \sqrt {\frac {1}{3}} a b^{2} x^{2} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 4 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x + 3 \, a^{2}\right )} {\left (b x - a\right )}^{\frac {2}{3}}}{18 \, a^{3} x^{2}}\right ] \]
[1/18*(6*sqrt(1/3)*a*b^2*x^2*sqrt((-a)^(1/3)/a)*log((2*b*x + 3*sqrt(1/3)*( 2*(b*x - a)^(2/3)*(-a)^(2/3) + (b*x - a)^(1/3)*a + (-a)^(1/3)*a)*sqrt((-a) ^(1/3)/a) - 3*(b*x - a)^(1/3)*(-a)^(2/3) - 3*a)/x) + 2*(-a)^(2/3)*b^2*x^2* log((b*x - a)^(2/3) + (b*x - a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) - 4*(-a)^(2 /3)*b^2*x^2*log((b*x - a)^(1/3) - (-a)^(1/3)) + 3*(4*a*b*x + 3*a^2)*(b*x - a)^(2/3))/(a^3*x^2), 1/18*(12*sqrt(1/3)*a*b^2*x^2*sqrt(-(-a)^(1/3)/a)*arc tan(sqrt(1/3)*(2*(b*x - a)^(1/3) + (-a)^(1/3))*sqrt(-(-a)^(1/3)/a)) + 2*(- a)^(2/3)*b^2*x^2*log((b*x - a)^(2/3) + (b*x - a)^(1/3)*(-a)^(1/3) + (-a)^( 2/3)) - 4*(-a)^(2/3)*b^2*x^2*log((b*x - a)^(1/3) - (-a)^(1/3)) + 3*(4*a*b* x + 3*a^2)*(b*x - a)^(2/3))/(a^3*x^2)]
Result contains complex when optimal does not.
Time = 3.86 (sec) , antiderivative size = 2744, normalized size of antiderivative = 20.18 \[ \int \frac {1}{x^3 \sqrt [3]{-a+b x}} \, dx=\text {Too large to display} \]
-4*a**(14/3)*b**(10/3)*(-a/b + x)**(4/3)*log(1 - b**(1/3)*(-a/b + x)**(1/3 )*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/ 3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3)*exp(2*I*p i/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*pi/3)*gamma (5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 4 *a**(14/3)*b**(10/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(-a/ b + x)**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a/b + x)**(7/3) *exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(10/3)*exp(2*I*p i/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp(2*I*pi/3)*gamma (5/3)) - 4*a**(14/3)*b**(10/3)*(-a/b + x)**(4/3)*exp(-2*I*pi/3)*log(1 - b* *(1/3)*(-a/b + x)**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7 *b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(- a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**( 10/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*exp( 2*I*pi/3)*gamma(5/3)) - 12*a**(11/3)*b**(13/3)*(-a/b + x)**(7/3)*log(1 - b **(1/3)*(-a/b + x)**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7* b**(4/3)*(-a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**6*b**(7/3)*(-a /b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(-a/b + x)**(1 0/3)*exp(2*I*pi/3)*gamma(5/3) + 27*a**4*b**(13/3)*(-a/b + x)**(13/3)*ex...
Time = 0.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^3 \sqrt [3]{-a+b x}} \, dx=\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {7}{3}}} + \frac {b^{2} \log \left ({\left (b x - a\right )}^{\frac {2}{3}} - {\left (b x - a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {7}{3}}} - \frac {2 \, b^{2} \log \left ({\left (b x - a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}{9 \, a^{\frac {7}{3}}} + \frac {4 \, {\left (b x - a\right )}^{\frac {5}{3}} b^{2} + 7 \, {\left (b x - a\right )}^{\frac {2}{3}} a b^{2}}{6 \, {\left ({\left (b x - a\right )}^{2} a^{2} + 2 \, {\left (b x - a\right )} a^{3} + a^{4}\right )}} \]
2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x - a)^(1/3) - a^(1/3))/a^(1/3))/ a^(7/3) + 1/9*b^2*log((b*x - a)^(2/3) - (b*x - a)^(1/3)*a^(1/3) + a^(2/3)) /a^(7/3) - 2/9*b^2*log((b*x - a)^(1/3) + a^(1/3))/a^(7/3) + 1/6*(4*(b*x - a)^(5/3)*b^2 + 7*(b*x - a)^(2/3)*a*b^2)/((b*x - a)^2*a^2 + 2*(b*x - a)*a^3 + a^4)
Time = 0.54 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^3 \sqrt [3]{-a+b x}} \, dx=\frac {\frac {4 \, \sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right )}}{3 \, \left (-a\right )^{\frac {1}{3}}}\right )}{\left (-a\right )^{\frac {1}{3}} a^{2}} - \frac {2 \, b^{3} \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right )}{\left (-a\right )^{\frac {1}{3}} a^{2}} - \frac {4 \, \left (-a\right )^{\frac {2}{3}} b^{3} \log \left ({\left | {\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}} \right |}\right )}{a^{3}} + \frac {3 \, {\left (4 \, {\left (b x - a\right )}^{\frac {5}{3}} b^{3} + 7 \, {\left (b x - a\right )}^{\frac {2}{3}} a b^{3}\right )}}{a^{2} b^{2} x^{2}}}{18 \, b} \]
1/18*(4*sqrt(3)*b^3*arctan(1/3*sqrt(3)*(2*(b*x - a)^(1/3) + (-a)^(1/3))/(- a)^(1/3))/((-a)^(1/3)*a^2) - 2*b^3*log((b*x - a)^(2/3) + (b*x - a)^(1/3)*( -a)^(1/3) + (-a)^(2/3))/((-a)^(1/3)*a^2) - 4*(-a)^(2/3)*b^3*log(abs((b*x - a)^(1/3) - (-a)^(1/3)))/a^3 + 3*(4*(b*x - a)^(5/3)*b^3 + 7*(b*x - a)^(2/3 )*a*b^3)/(a^2*b^2*x^2))/b
Time = 0.25 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.59 \[ \int \frac {1}{x^3 \sqrt [3]{-a+b x}} \, dx=\frac {\frac {7\,b^2\,{\left (b\,x-a\right )}^{2/3}}{6\,a}+\frac {2\,b^2\,{\left (b\,x-a\right )}^{5/3}}{3\,a^2}}{{\left (a-b\,x\right )}^2-2\,a\,\left (a-b\,x\right )+a^2}-\frac {\ln \left (\frac {4\,b^4\,{\left (b\,x-a\right )}^{1/3}}{9\,a^4}-\frac {{\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}^2}{9\,{\left (-a\right )}^{11/3}}\right )\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{7/3}}+\frac {2\,b^2\,\ln \left (\frac {4\,b^4\,{\left (b\,x-a\right )}^{1/3}}{9\,a^4}-\frac {4\,b^4}{9\,{\left (-a\right )}^{11/3}}\right )}{9\,{\left (-a\right )}^{7/3}}+\frac {b^2\,\ln \left (\frac {4\,b^4\,{\left (b\,x-a\right )}^{1/3}}{9\,a^4}-\frac {9\,b^4\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2}{{\left (-a\right )}^{11/3}}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}{{\left (-a\right )}^{7/3}} \]
((7*b^2*(b*x - a)^(2/3))/(6*a) + (2*b^2*(b*x - a)^(5/3))/(3*a^2))/((a - b* x)^2 - 2*a*(a - b*x) + a^2) - (log((4*b^4*(b*x - a)^(1/3))/(9*a^4) - (3^(1 /2)*b^2*1i + b^2)^2/(9*(-a)^(11/3)))*(3^(1/2)*b^2*1i + b^2))/(9*(-a)^(7/3) ) + (2*b^2*log((4*b^4*(b*x - a)^(1/3))/(9*a^4) - (4*b^4)/(9*(-a)^(11/3)))) /(9*(-a)^(7/3)) + (b^2*log((4*b^4*(b*x - a)^(1/3))/(9*a^4) - (9*b^4*((3^(1 /2)*1i)/9 - 1/9)^2)/(-a)^(11/3))*((3^(1/2)*1i)/9 - 1/9))/(-a)^(7/3)
Time = 0.01 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x^3 \sqrt [3]{-a+b x}} \, dx=\frac {4 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x -a \right )^{\frac {1}{6}}-a^{\frac {1}{6}} \sqrt {3}}{a^{\frac {1}{6}}}\right ) b^{2} x^{2}-4 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x -a \right )^{\frac {1}{6}}+a^{\frac {1}{6}} \sqrt {3}}{a^{\frac {1}{6}}}\right ) b^{2} x^{2}+9 a^{\frac {4}{3}} \left (b x -a \right )^{\frac {2}{3}}+12 a^{\frac {1}{3}} \left (b x -a \right )^{\frac {2}{3}} b x -4 \,\mathrm {log}\left (\left (b x -a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b^{2} x^{2}+2 \,\mathrm {log}\left (-a^{\frac {1}{6}} \left (b x -a \right )^{\frac {1}{6}} \sqrt {3}+\left (b x -a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b^{2} x^{2}+2 \,\mathrm {log}\left (a^{\frac {1}{6}} \left (b x -a \right )^{\frac {1}{6}} \sqrt {3}+\left (b x -a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b^{2} x^{2}}{18 a^{\frac {7}{3}} x^{2}} \]
(4*sqrt(3)*atan((2*( - a + b*x)**(1/6) - a**(1/6)*sqrt(3))/a**(1/6))*b**2* x**2 - 4*sqrt(3)*atan((2*( - a + b*x)**(1/6) + a**(1/6)*sqrt(3))/a**(1/6)) *b**2*x**2 + 9*a**(1/3)*( - a + b*x)**(2/3)*a + 12*a**(1/3)*( - a + b*x)** (2/3)*b*x - 4*log(( - a + b*x)**(1/3) + a**(1/3))*b**2*x**2 + 2*log( - a** (1/6)*( - a + b*x)**(1/6)*sqrt(3) + ( - a + b*x)**(1/3) + a**(1/3))*b**2*x **2 + 2*log(a**(1/6)*( - a + b*x)**(1/6)*sqrt(3) + ( - a + b*x)**(1/3) + a **(1/3))*b**2*x**2)/(18*a**(1/3)*a**2*x**2)