Integrand size = 13, antiderivative size = 152 \[ \int \frac {1}{x^{4/3} (a+b x)^3} \, dx=-\frac {14}{3 a^3 \sqrt [3]{x}}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}+\frac {7}{6 a^2 \sqrt [3]{x} (a+b x)}+\frac {14 \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {7 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{10/3}}-\frac {7 \sqrt [3]{b} \log (a+b x)}{9 a^{10/3}} \]
-14/3/a^3/x^(1/3)+1/2/a/x^(1/3)/(b*x+a)^2+7/6/a^2/x^(1/3)/(b*x+a)+7/3*b^(1 /3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/a^(10/3)-7/9*b^(1/3)*ln(b*x+a)/a^(10/3)+14 /9*b^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3))/a^(1/3)*3^(1/2))/a^(10/3 )*3^(1/2)
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{4/3} (a+b x)^3} \, dx=\frac {-\frac {3 \sqrt [3]{a} \left (18 a^2+49 a b x+28 b^2 x^2\right )}{\sqrt [3]{x} (a+b x)^2}+28 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+28 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-14 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{18 a^{10/3}} \]
((-3*a^(1/3)*(18*a^2 + 49*a*b*x + 28*b^2*x^2))/(x^(1/3)*(a + b*x)^2) + 28* Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]] + 28*b^( 1/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)] - 14*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^( 1/3)*x^(1/3) + b^(2/3)*x^(2/3)])/(18*a^(10/3))
Time = 0.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {52, 52, 61, 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.
| \(\displaystyle \int \frac {1}{x^{4/3} (a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 \int \frac {1}{x^{4/3} (a+b x)^2}dx}{6 a}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 \left (\frac {4 \int \frac {1}{x^{4/3} (a+b x)}dx}{3 a}+\frac {1}{a \sqrt [3]{x} (a+b x)}\right )}{6 a}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 \left (\frac {4 \left (-\frac {b \int \frac {1}{\sqrt [3]{x} (a+b x)}dx}{a}-\frac {3}{a \sqrt [3]{x}}\right )}{3 a}+\frac {1}{a \sqrt [3]{x} (a+b x)}\right )}{6 a}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {3}{a \sqrt [3]{x}}\right )}{3 a}+\frac {1}{a \sqrt [3]{x} (a+b x)}\right )}{6 a}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {3}{a \sqrt [3]{x}}\right )}{3 a}+\frac {1}{a \sqrt [3]{x} (a+b x)}\right )}{6 a}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {3}{a \sqrt [3]{x}}\right )}{3 a}+\frac {1}{a \sqrt [3]{x} (a+b x)}\right )}{6 a}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {7 \left (\frac {4 \left (-\frac {b \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {3}{a \sqrt [3]{x}}\right )}{3 a}+\frac {1}{a \sqrt [3]{x} (a+b x)}\right )}{6 a}+\frac {1}{2 a \sqrt [3]{x} (a+b x)^2}\) |
1/(2*a*x^(1/3)*(a + b*x)^2) + (7*(1/(a*x^(1/3)*(a + b*x)) + (4*(-3/(a*x^(1 /3)) - (b*(-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/( a^(1/3)*b^(2/3))) - (3*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(2*a^(1/3)*b^(2/3)) + Log[a + b*x]/(2*a^(1/3)*b^(2/3))))/a))/(3*a)))/(6*a)
3.7.96.3.1 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_))^(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 = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(-\frac {3}{a^{3} x^{\frac {1}{3}}}-\frac {3 b \left (\frac {\frac {5 b \,x^{\frac {5}{3}}}{9}+\frac {13 a \,x^{\frac {2}{3}}}{18}}{\left (b x +a \right )^{2}}-\frac {14 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {7 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{3}}\) | \(133\) |
| default | \(-\frac {3}{a^{3} x^{\frac {1}{3}}}-\frac {3 b \left (\frac {\frac {5 b \,x^{\frac {5}{3}}}{9}+\frac {13 a \,x^{\frac {2}{3}}}{18}}{\left (b x +a \right )^{2}}-\frac {14 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {7 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{3}}\) | \(133\) |
| risch | \(-\frac {3}{a^{3} x^{\frac {1}{3}}}-\frac {b \left (\frac {\frac {5 b \,x^{\frac {5}{3}}}{3}+\frac {13 a \,x^{\frac {2}{3}}}{6}}{\left (b x +a \right )^{2}}-\frac {14 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {7 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{3}}\) | \(134\) |
-3/a^3/x^(1/3)-3/a^3*b*((5/9*b*x^(5/3)+13/18*a*x^(2/3))/(b*x+a)^2-14/27/b/ (a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/3))+7/27/b/(a/b)^(1/3)*ln(x^(2/3)-(a/b)^(1 /3)*x^(1/3)+(a/b)^(2/3))+14/27*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2 /(a/b)^(1/3)*x^(1/3)-1)))
Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^{4/3} (a+b x)^3} \, dx=-\frac {28 \, \sqrt {3} {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 14 \, {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (-a x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 28 \, {\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (a \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {1}{3}}\right ) + 3 \, {\left (28 \, b^{2} x^{2} + 49 \, a b x + 18 \, a^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \]
-1/18*(28*sqrt(3)*(b^2*x^3 + 2*a*b*x^2 + a^2*x)*(b/a)^(1/3)*arctan(2/3*sqr t(3)*x^(1/3)*(b/a)^(1/3) - 1/3*sqrt(3)) + 14*(b^2*x^3 + 2*a*b*x^2 + a^2*x) *(b/a)^(1/3)*log(-a*x^(1/3)*(b/a)^(2/3) + b*x^(2/3) + a*(b/a)^(1/3)) - 28* (b^2*x^3 + 2*a*b*x^2 + a^2*x)*(b/a)^(1/3)*log(a*(b/a)^(2/3) + b*x^(1/3)) + 3*(28*b^2*x^2 + 49*a*b*x + 18*a^2)*x^(2/3))/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x)
Timed out. \[ \int \frac {1}{x^{4/3} (a+b x)^3} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{4/3} (a+b x)^3} \, dx=-\frac {28 \, b^{2} x^{2} + 49 \, a b x + 18 \, a^{2}}{6 \, {\left (a^{3} b^{2} x^{\frac {7}{3}} + 2 \, a^{4} b x^{\frac {4}{3}} + a^{5} x^{\frac {1}{3}}\right )}} - \frac {14 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {7 \, \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {14 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
-1/6*(28*b^2*x^2 + 49*a*b*x + 18*a^2)/(a^3*b^2*x^(7/3) + 2*a^4*b*x^(4/3) + a^5*x^(1/3)) - 14/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/ (a/b)^(1/3))/(a^3*(a/b)^(1/3)) - 7/9*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + ( a/b)^(2/3))/(a^3*(a/b)^(1/3)) + 14/9*log(x^(1/3) + (a/b)^(1/3))/(a^3*(a/b) ^(1/3))
Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^{4/3} (a+b x)^3} \, dx=\frac {14 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{4}} + \frac {14 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4} b} - \frac {3}{a^{3} x^{\frac {1}{3}}} - \frac {7 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{4} b} - \frac {10 \, b^{2} x^{\frac {5}{3}} + 13 \, a b x^{\frac {2}{3}}}{6 \, {\left (b x + a\right )}^{2} a^{3}} \]
14/9*b*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^4 + 14/9*sqrt(3)*(- a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/( a^4*b) - 3/(a^3*x^(1/3)) - 7/9*(-a*b^2)^(2/3)*log(x^(2/3) + x^(1/3)*(-a/b) ^(1/3) + (-a/b)^(2/3))/(a^4*b) - 1/6*(10*b^2*x^(5/3) + 13*a*b*x^(2/3))/((b *x + a)^2*a^3)
Time = 0.09 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^{4/3} (a+b x)^3} \, dx=\frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}+588\,a^3\,b^3\,x^{1/3}\right )}{9\,a^{10/3}}-\frac {\frac {3}{a}+\frac {14\,b^2\,x^2}{3\,a^3}+\frac {49\,b\,x}{6\,a^2}}{a^2\,x^{1/3}+b^2\,x^{7/3}+2\,a\,b\,x^{4/3}}+\frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+588\,a^3\,b^3\,x^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{10/3}}-\frac {14\,b^{1/3}\,\ln \left (588\,a^{10/3}\,b^{8/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+588\,a^3\,b^3\,x^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{10/3}} \]
(14*b^(1/3)*log(588*a^(10/3)*b^(8/3) + 588*a^3*b^3*x^(1/3)))/(9*a^(10/3)) - (3/a + (14*b^2*x^2)/(3*a^3) + (49*b*x)/(6*a^2))/(a^2*x^(1/3) + b^2*x^(7/ 3) + 2*a*b*x^(4/3)) + (14*b^(1/3)*log(588*a^(10/3)*b^(8/3)*((3^(1/2)*1i)/2 - 1/2)^2 + 588*a^3*b^3*x^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(9*a^(10/3)) - (1 4*b^(1/3)*log(588*a^(10/3)*b^(8/3)*((3^(1/2)*1i)/2 + 1/2)^2 + 588*a^3*b^3* x^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(9*a^(10/3))
Time = 0.01 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x^{4/3} (a+b x)^3} \, dx=\frac {28 x^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} b +56 x^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a \,b^{2}+28 x^{\frac {7}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{3}-54 b^{\frac {2}{3}} a^{\frac {7}{3}}-147 b^{\frac {5}{3}} a^{\frac {4}{3}} x -84 b^{\frac {8}{3}} a^{\frac {1}{3}} x^{2}-14 x^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) a^{2} b -28 x^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) a \,b^{2}-14 x^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b^{3}+28 x^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) a^{2} b +56 x^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) a \,b^{2}+28 x^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b^{3}}{18 x^{\frac {1}{3}} b^{\frac {2}{3}} a^{\frac {10}{3}} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \]
(28*x**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqrt( 3)))*a**2*b + 56*x**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a **(1/3)*sqrt(3)))*a*b**2*x + 28*x**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/ 3)*b**(1/3))/(a**(1/3)*sqrt(3)))*b**3*x**2 - 54*b**(2/3)*a**(1/3)*a**2 - 1 47*b**(2/3)*a**(1/3)*a*b*x - 84*b**(2/3)*a**(1/3)*b**2*x**2 - 14*x**(1/3)* log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*a**2*b - 28 *x**(1/3)*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*a *b**2*x - 14*x**(1/3)*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3) *b**(2/3))*b**3*x**2 + 28*x**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*a**2* b + 56*x**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*a*b**2*x + 28*x**(1/3)*l og(a**(1/3) + x**(1/3)*b**(1/3))*b**3*x**2)/(18*x**(1/3)*b**(2/3)*a**(1/3) *a**3*(a**2 + 2*a*b*x + b**2*x**2))