Integrand size = 13, antiderivative size = 140 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {\sqrt [3]{x}}{2 a (a+b x)^2}+\frac {5 \sqrt [3]{x}}{6 a^2 (a+b x)}-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac {5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}} \] Output:
1/2*x^(1/3)/a/(b*x+a)^2+5/6*x^(1/3)/a^2/(b*x+a)-5/9*arctan(1/3*(a^(1/3)-2* b^(1/3)*x^(1/3))*3^(1/2)/a^(1/3))*3^(1/2)/a^(8/3)/b^(1/3)+5/6*ln(a^(1/3)+b ^(1/3)*x^(1/3))/a^(8/3)/b^(1/3)-5/18*ln(b*x+a)/a^(8/3)/b^(1/3)
Time = 0.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {\frac {3 a^{2/3} \sqrt [3]{x} (8 a+5 b x)}{(a+b x)^2}-\frac {10 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{\sqrt [3]{b}}}{18 a^{8/3}} \] Input:
Integrate[1/(x^(2/3)*(a + b*x)^3),x]
Output:
((3*a^(2/3)*x^(1/3)*(8*a + 5*b*x))/(a + b*x)^2 - (10*Sqrt[3]*ArcTan[(1 - ( 2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/b^(1/3) + (10*Log[a^(1/3) + b^(1/3)* x^(1/3)])/b^(1/3) - (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^( 2/3)])/b^(1/3))/(18*a^(8/3))
Time = 0.20 (sec) , antiderivative size = 147, 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 = {52, 52, 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.
| \(\displaystyle \int \frac {1}{x^{2/3} (a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {5 \int \frac {1}{x^{2/3} (a+b x)^2}dx}{6 a}+\frac {\sqrt [3]{x}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {5 \left (\frac {2 \int \frac {1}{x^{2/3} (a+b x)}dx}{3 a}+\frac {\sqrt [3]{x}}{a (a+b x)}\right )}{6 a}+\frac {\sqrt [3]{x}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\sqrt [3]{x}}{a (a+b x)}\right )}{6 a}+\frac {\sqrt [3]{x}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\sqrt [3]{x}}{a (a+b x)}\right )}{6 a}+\frac {\sqrt [3]{x}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\sqrt [3]{x}}{a (a+b x)}\right )}{6 a}+\frac {\sqrt [3]{x}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {\sqrt [3]{x}}{a (a+b x)}\right )}{6 a}+\frac {\sqrt [3]{x}}{2 a (a+b x)^2}\) |
Input:
Int[1/(x^(2/3)*(a + b*x)^3),x]
Output:
x^(1/3)/(2*a*(a + b*x)^2) + (5*(x^(1/3)/(a*(a + b*x)) + (2*(-((Sqrt[3]*Arc Tan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/(a^(2/3)*b^(1/3))) + (3*Lo g[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*a)))/(6*a)
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_))^(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 = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {x^{\frac {1}{3}}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {5 x^{\frac {1}{3}}}{6 a \left (b x +a \right )}+\frac {5 \left (\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{2 a}}{a}\) | \(139\) |
| default | \(\frac {x^{\frac {1}{3}}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {5 x^{\frac {1}{3}}}{6 a \left (b x +a \right )}+\frac {5 \left (\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{2 a}}{a}\) | \(139\) |
Input:
int(1/x^(2/3)/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/2*x^(1/3)/a/(b*x+a)^2+5/2/a*(1/3*x^(1/3)/a/(b*x+a)+2/3/a*(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. 230 vs. \(2 (99) = 198\).
Time = 0.09 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.56 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\left [\frac {15 \, \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 ) - 5 \, {\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 ) + 10 \, {\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 (5 \, a^{2} b^{2} x + 8 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{4} b^{3} x^{2} + 2 \, a^{5} b^{2} x + a^{6} b\right )}}, \frac {30 \, \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 ) - 5 \, {\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 ) + 10 \, {\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 (5 \, a^{2} b^{2} x + 8 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{4} b^{3} x^{2} + 2 \, a^{5} b^{2} x + a^{6} b\right )}}\right ] \] Input:
integrate(1/x^(2/3)/(b*x+a)^3,x, algorithm="fricas")
Output:
[1/18*(15*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)) - 5*(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)) + 10*(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*(5*a^2*b^2*x + 8*a^3*b)*x^(1 /3))/(a^4*b^3*x^2 + 2*a^5*b^2*x + a^6*b), 1/18*(30*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) - 5*(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)) + 10*(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*(5*a^2*b^2*x + 8*a^3*b)*x^(1/3))/(a^4*b^3*x^2 + 2*a^5 *b^2*x + a^6*b)]
Leaf count of result is larger than twice the leaf count of optimal. 853 vs. \(2 (133) = 266\).
Time = 153.24 (sec) , antiderivative size = 853, normalized size of antiderivative = 6.09 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx =\text {Too large to display} \] Input:
integrate(1/x**(2/3)/(b*x+a)**3,x)
Output:
Piecewise((zoo/x**(8/3), Eq(a, 0) & Eq(b, 0)), (3*x**(1/3)/a**3, Eq(b, 0)) , (-3/(8*b**3*x**(8/3)), Eq(a, 0)), (24*a**2*x**(1/3)/(18*a**5 + 36*a**4*b *x + 18*a**3*b**2*x**2) - 10*a**2*(-a/b)**(1/3)*log(x**(1/3) - (-a/b)**(1/ 3))/(18*a**5 + 36*a**4*b*x + 18*a**3*b**2*x**2) + 5*a**2*(-a/b)**(1/3)*log (4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(18*a**5 + 36*a* *4*b*x + 18*a**3*b**2*x**2) + 10*sqrt(3)*a**2*(-a/b)**(1/3)*atan(2*sqrt(3) *x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(18*a**5 + 36*a**4*b*x + 18*a**3* b**2*x**2) - 10*a**2*(-a/b)**(1/3)*log(2)/(18*a**5 + 36*a**4*b*x + 18*a**3 *b**2*x**2) + 15*a*b*x**(4/3)/(18*a**5 + 36*a**4*b*x + 18*a**3*b**2*x**2) - 20*a*b*x*(-a/b)**(1/3)*log(x**(1/3) - (-a/b)**(1/3))/(18*a**5 + 36*a**4* b*x + 18*a**3*b**2*x**2) + 10*a*b*x*(-a/b)**(1/3)*log(4*x**(2/3) + 4*x**(1 /3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(18*a**5 + 36*a**4*b*x + 18*a**3*b**2 *x**2) + 20*sqrt(3)*a*b*x*(-a/b)**(1/3)*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)* *(1/3)) + sqrt(3)/3)/(18*a**5 + 36*a**4*b*x + 18*a**3*b**2*x**2) - 20*a*b* x*(-a/b)**(1/3)*log(2)/(18*a**5 + 36*a**4*b*x + 18*a**3*b**2*x**2) - 10*b* *2*x**2*(-a/b)**(1/3)*log(x**(1/3) - (-a/b)**(1/3))/(18*a**5 + 36*a**4*b*x + 18*a**3*b**2*x**2) + 5*b**2*x**2*(-a/b)**(1/3)*log(4*x**(2/3) + 4*x**(1 /3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(18*a**5 + 36*a**4*b*x + 18*a**3*b**2 *x**2) + 10*sqrt(3)*b**2*x**2*(-a/b)**(1/3)*atan(2*sqrt(3)*x**(1/3)/(3*(-a /b)**(1/3)) + sqrt(3)/3)/(18*a**5 + 36*a**4*b*x + 18*a**3*b**2*x**2) - ...
Time = 0.13 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {5 \, b x^{\frac {4}{3}} + 8 \, a x^{\frac {1}{3}}}{6 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(1/x^(2/3)/(b*x+a)^3,x, algorithm="maxima")
Output:
1/6*(5*b*x^(4/3) + 8*a*x^(1/3))/(a^2*b^2*x^2 + 2*a^3*b*x + a^4) + 5/9*sqrt (3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b*(a/b) ^(2/3)) - 5/18*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/ b)^(2/3)) + 5/9*log(x^(1/3) + (a/b)^(1/3))/(a^2*b*(a/b)^(2/3))
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=-\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3}} + \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b} + \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} b} + \frac {5 \, b x^{\frac {4}{3}} + 8 \, a x^{\frac {1}{3}}}{6 \, {\left (b x + a\right )}^{2} a^{2}} \] Input:
integrate(1/x^(2/3)/(b*x+a)^3,x, algorithm="giac")
Output:
-5/9*(-a/b)^(1/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^3 + 5/9*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^3 *b) + 5/18*(-a*b^2)^(1/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3 ))/(a^3*b) + 1/6*(5*b*x^(4/3) + 8*a*x^(1/3))/((b*x + a)^2*a^2)
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {\frac {4\,x^{1/3}}{3\,a}+\frac {5\,b\,x^{4/3}}{6\,a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {5\,\ln \left (\frac {5\,b^{5/3}}{a^{5/3}}+\frac {5\,b^2\,x^{1/3}}{a^2}\right )}{9\,a^{8/3}\,b^{1/3}}+\frac {\ln \left (\frac {5\,b^2\,x^{1/3}}{a^2}+\frac {b^{5/3}\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{2\,a^{5/3}}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{18\,a^{8/3}\,b^{1/3}}-\frac {\ln \left (\frac {5\,b^2\,x^{1/3}}{a^2}-\frac {b^{5/3}\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{2\,a^{5/3}}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{18\,a^{8/3}\,b^{1/3}} \] Input:
int(1/(x^(2/3)*(a + b*x)^3),x)
Output:
((4*x^(1/3))/(3*a) + (5*b*x^(4/3))/(6*a^2))/(a^2 + b^2*x^2 + 2*a*b*x) + (5 *log((5*b^(5/3))/a^(5/3) + (5*b^2*x^(1/3))/a^2))/(9*a^(8/3)*b^(1/3)) + (lo g((5*b^2*x^(1/3))/a^2 + (b^(5/3)*(3^(1/2)*5i - 5))/(2*a^(5/3)))*(3^(1/2)*5 i - 5))/(18*a^(8/3)*b^(1/3)) - (log((5*b^2*x^(1/3))/a^2 - (b^(5/3)*(3^(1/2 )*5i + 5))/(2*a^(5/3)))*(3^(1/2)*5i + 5))/(18*a^(8/3)*b^(1/3))
Time = 0.16 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.07 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {-10 a^{\frac {7}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right )-20 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b x -10 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{2}-5 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right )-10 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b x -5 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b^{2} x^{2}+10 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right )+20 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b x +10 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b^{2} x^{2}+24 x^{\frac {1}{3}} b^{\frac {1}{3}} a^{2}+15 x^{\frac {4}{3}} b^{\frac {4}{3}} a}{18 b^{\frac {1}{3}} a^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:
int(1/x^(2/3)/(b*x+a)^3,x)
Output:
( - 10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sq rt(3)))*a**2 - 20*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/( a**(1/3)*sqrt(3)))*a*b*x - 10*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 - 5*a**(1/3)*log(a**(2/3) - x**(1 /3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*a**2 - 10*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 - 5*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 + 10* a**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*a**2 + 20*a**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*a*b*x + 10*a**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3) )*b**2*x**2 + 24*x**(1/3)*b**(1/3)*a**2 + 15*x**(1/3)*b**(1/3)*a*b*x)/(18* b**(1/3)*a**3*(a**2 + 2*a*b*x + b**2*x**2))