Integrand size = 13, antiderivative size = 141 \[ \int \frac {x^{4/3}}{(a+b x)^3} \, dx=\frac {a \sqrt [3]{x}}{2 b^2 (a+b x)^2}-\frac {7 \sqrt [3]{x}}{6 b^2 (a+b x)}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} b^{7/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac {\log (a+b x)}{9 a^{2/3} b^{7/3}} \] Output:
1/2*a*x^(1/3)/b^2/(b*x+a)^2-7/6*x^(1/3)/b^2/(b*x+a)-2/9*arctan(1/3*(a^(1/3 )-2*b^(1/3)*x^(1/3))*3^(1/2)/a^(1/3))*3^(1/2)/a^(2/3)/b^(7/3)+1/3*ln(a^(1/ 3)+b^(1/3)*x^(1/3))/a^(2/3)/b^(7/3)-1/9*ln(b*x+a)/a^(2/3)/b^(7/3)
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {x^{4/3}}{(a+b x)^3} \, dx=\frac {-\frac {3 \sqrt [3]{b} \sqrt [3]{x} (4 a+7 b x)}{(a+b x)^2}-\frac {4 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{2/3}}-\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{a^{2/3}}}{18 b^{7/3}} \] Input:
Integrate[x^(4/3)/(a + b*x)^3,x]
Output:
((-3*b^(1/3)*x^(1/3)*(4*a + 7*b*x))/(a + b*x)^2 - (4*Sqrt[3]*ArcTan[(1 - ( 2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3) + (4*Log[a^(1/3) + b^(1/3)*x ^(1/3)])/a^(2/3) - (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2 /3)])/a^(2/3))/(18*b^(7/3))
Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {51, 51, 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.
| \(\displaystyle \int \frac {x^{4/3}}{(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {2 \int \frac {\sqrt [3]{x}}{(a+b x)^2}dx}{3 b}-\frac {x^{4/3}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {1}{x^{2/3} (a+b x)}dx}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\right )}{3 b}-\frac {x^{4/3}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\right )}{3 b}-\frac {x^{4/3}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\right )}{3 b}-\frac {x^{4/3}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\right )}{3 b}-\frac {x^{4/3}}{2 b (a+b x)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\right )}{3 b}-\frac {x^{4/3}}{2 b (a+b x)^2}\) |
Input:
Int[x^(4/3)/(a + b*x)^3,x]
Output:
-1/2*x^(4/3)/(b*(a + b*x)^2) + (2*(-(x^(1/3)/(b*(a + b*x))) + (-((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)))/(3*b)))/(3*b)
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/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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 = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.92
| method | result | size |
| derivativedivides | \(\frac {-\frac {7 x^{\frac {4}{3}}}{6 b}-\frac {2 a \,x^{\frac {1}{3}}}{3 b^{2}}}{\left (b x +a \right )^{2}}+\frac {\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{b^{2}}\) | \(130\) |
| default | \(\frac {-\frac {7 x^{\frac {4}{3}}}{6 b}-\frac {2 a \,x^{\frac {1}{3}}}{3 b^{2}}}{\left (b x +a \right )^{2}}+\frac {\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{b^{2}}\) | \(130\) |
Input:
int(x^(4/3)/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
3*(-7/18*x^(4/3)/b-2/9*a*x^(1/3)/b^2)/(b*x+a)^2+2/3/b^2*(1/3/b/(a/b)^(2/3) *ln(x^(1/3)+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x^(2/3)-(a/b)^(1/3)*x^(1/3)+ (a/b)^(2/3))+1/3/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. 232 vs. \(2 (100) = 200\).
Time = 0.11 (sec) , antiderivative size = 503, normalized size of antiderivative = 3.57 \[ \int \frac {x^{4/3}}{(a+b x)^3} \, dx=\left [\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{\frac {2}{3}} - \left (a^{2} b\right )^{\frac {1}{3}} a + \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{\frac {1}{3}}}{b x + a}\right ) - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right ) - 3 \, {\left (7 \, a^{2} b^{2} x + 4 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}, \frac {12 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (a^{2} b\right )^{\frac {1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right ) - 3 \, {\left (7 \, a^{2} b^{2} x + 4 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\right ] \] Input:
integrate(x^(4/3)/(b*x+a)^3,x, algorithm="fricas")
Output:
[1/18*(6*sqrt(1/3)*(a*b^3*x^2 + 2*a^2*b^2*x + a^3*b)*sqrt(-(a^2*b)^(1/3)/b )*log((2*a*b*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^(2/3) - (a^2*b)^(1/3)*a + (a^2 *b)^(2/3)*x^(1/3))*sqrt(-(a^2*b)^(1/3)/b) - 3*(a^2*b)^(1/3)*a*x^(1/3))/(b* x + a)) - 2*(b^2*x^2 + 2*a*b*x + a^2)*(a^2*b)^(2/3)*log(a*b*x^(2/3) + (a^2 *b)^(1/3)*a - (a^2*b)^(2/3)*x^(1/3)) + 4*(b^2*x^2 + 2*a*b*x + a^2)*(a^2*b) ^(2/3)*log(a*b*x^(1/3) + (a^2*b)^(2/3)) - 3*(7*a^2*b^2*x + 4*a^3*b)*x^(1/3 ))/(a^2*b^5*x^2 + 2*a^3*b^4*x + a^4*b^3), 1/18*(12*sqrt(1/3)*(a*b^3*x^2 + 2*a^2*b^2*x + a^3*b)*sqrt((a^2*b)^(1/3)/b)*arctan(-sqrt(1/3)*((a^2*b)^(1/3 )*a - 2*(a^2*b)^(2/3)*x^(1/3))*sqrt((a^2*b)^(1/3)/b)/a^2) - 2*(b^2*x^2 + 2 *a*b*x + a^2)*(a^2*b)^(2/3)*log(a*b*x^(2/3) + (a^2*b)^(1/3)*a - (a^2*b)^(2 /3)*x^(1/3)) + 4*(b^2*x^2 + 2*a*b*x + a^2)*(a^2*b)^(2/3)*log(a*b*x^(1/3) + (a^2*b)^(2/3)) - 3*(7*a^2*b^2*x + 4*a^3*b)*x^(1/3))/(a^2*b^5*x^2 + 2*a^3* b^4*x + a^4*b^3)]
Timed out. \[ \int \frac {x^{4/3}}{(a+b x)^3} \, dx=\text {Timed out} \] Input:
integrate(x**(4/3)/(b*x+a)**3,x)
Output:
Timed out
Time = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01 \[ \int \frac {x^{4/3}}{(a+b x)^3} \, dx=-\frac {7 \, b x^{\frac {4}{3}} + 4 \, a x^{\frac {1}{3}}}{6 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^(4/3)/(b*x+a)^3,x, algorithm="maxima")
Output:
-1/6*(7*b*x^(4/3) + 4*a*x^(1/3))/(b^4*x^2 + 2*a*b^3*x + a^2*b^2) + 2/9*sqr t(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(b^3*(a/b)^ (2/3)) - 1/9*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^( 2/3)) + 2/9*log(x^(1/3) + (a/b)^(1/3))/(b^3*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04 \[ \int \frac {x^{4/3}}{(a+b x)^3} \, dx=-\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{2}} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{3}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a b^{3}} - \frac {7 \, b x^{\frac {4}{3}} + 4 \, a x^{\frac {1}{3}}}{6 \, {\left (b x + a\right )}^{2} b^{2}} \] Input:
integrate(x^(4/3)/(b*x+a)^3,x, algorithm="giac")
Output:
-2/9*(-a/b)^(1/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/(a*b^2) + 2/9*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*b^3) + 1/9*(-a*b^2)^(1/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^( 2/3))/(a*b^3) - 1/6*(7*b*x^(4/3) + 4*a*x^(1/3))/((b*x + a)^2*b^2)
Time = 0.04 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99 \[ \int \frac {x^{4/3}}{(a+b x)^3} \, dx=\frac {2\,\ln \left (2\,x^{1/3}+\frac {2\,a^{1/3}}{b^{1/3}}\right )}{9\,a^{2/3}\,b^{7/3}}-\frac {\frac {7\,x^{4/3}}{6\,b}+\frac {2\,a\,x^{1/3}}{3\,b^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\ln \left (2\,x^{1/3}+\frac {a^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^{1/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{2/3}\,b^{7/3}}-\frac {\ln \left (2\,x^{1/3}-\frac {a^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^{1/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{2/3}\,b^{7/3}} \] Input:
int(x^(4/3)/(a + b*x)^3,x)
Output:
(2*log(2*x^(1/3) + (2*a^(1/3))/b^(1/3)))/(9*a^(2/3)*b^(7/3)) - ((7*x^(4/3) )/(6*b) + (2*a*x^(1/3))/(3*b^2))/(a^2 + b^2*x^2 + 2*a*b*x) + (log(2*x^(1/3 ) + (a^(1/3)*(3^(1/2)*1i - 1))/b^(1/3))*(3^(1/2)*1i - 1))/(9*a^(2/3)*b^(7/ 3)) - (log(2*x^(1/3) - (a^(1/3)*(3^(1/2)*1i + 1))/b^(1/3))*(3^(1/2)*1i + 1 ))/(9*a^(2/3)*b^(7/3))
Time = 0.16 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.06 \[ \int \frac {x^{4/3}}{(a+b x)^3} \, dx=\frac {-4 a^{\frac {7}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right )-8 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b x -4 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{2}-2 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right )-4 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b x -2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b^{2} x^{2}+4 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right )+8 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b x +4 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b^{2} x^{2}-12 x^{\frac {1}{3}} b^{\frac {1}{3}} a^{2}-21 x^{\frac {4}{3}} b^{\frac {4}{3}} a}{18 b^{\frac {7}{3}} a \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:
int(x^(4/3)/(b*x+a)^3,x)
Output:
( - 4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqr t(3)))*a**2 - 8*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a* *(1/3)*sqrt(3)))*a*b*x - 4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b* *(1/3))/(a**(1/3)*sqrt(3)))*b**2*x**2 - 2*a**(1/3)*log(a**(2/3) - x**(1/3) *b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*a**2 - 4*a**(1/3)*log(a**(2/3) - x **(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*a*b*x - 2*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**2 + 4*a**(1 /3)*log(a**(1/3) + x**(1/3)*b**(1/3))*a**2 + 8*a**(1/3)*log(a**(1/3) + x** (1/3)*b**(1/3))*a*b*x + 4*a**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*b**2* x**2 - 12*x**(1/3)*b**(1/3)*a**2 - 21*x**(1/3)*b**(1/3)*a*b*x)/(18*b**(1/3 )*a*b**2*(a**2 + 2*a*b*x + b**2*x**2))