Integrand size = 13, antiderivative size = 140 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx=\frac {x^{2/3}}{2 a (a+b x)^2}+\frac {2 x^{2/3}}{3 a^2 (a+b x)}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{7/3} b^{2/3}}+\frac {\log (a+b x)}{9 a^{7/3} b^{2/3}} \] Output:
1/2*x^(2/3)/a/(b*x+a)^2+2/3*x^(2/3)/a^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^(7/3)/b^(2/3)-1/3*ln(a^(1/3)+b ^(1/3)*x^(1/3))/a^(7/3)/b^(2/3)+1/9*ln(b*x+a)/a^(7/3)/b^(2/3)
Time = 0.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx=\frac {\frac {3 \sqrt [3]{a} x^{2/3} (7 a+4 b x)}{(a+b x)^2}-\frac {4 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{2/3}}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{b^{2/3}}}{18 a^{7/3}} \] Input:
Integrate[1/(x^(1/3)*(a + b*x)^3),x]
Output:
((3*a^(1/3)*x^(2/3)*(7*a + 4*b*x))/(a + b*x)^2 - (4*Sqrt[3]*ArcTan[(1 - (2 *b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]])/b^(2/3) - (4*Log[a^(1/3) + b^(1/3)*x^ (1/3)])/b^(2/3) + (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/ 3)])/b^(2/3))/(18*a^(7/3))
Time = 0.21 (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, 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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2 \int \frac {1}{\sqrt [3]{x} (a+b x)^2}dx}{3 a}+\frac {x^{2/3}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {1}{\sqrt [3]{x} (a+b x)}dx}{3 a}+\frac {x^{2/3}}{a (a+b x)}\right )}{3 a}+\frac {x^{2/3}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^{2/3}}{a (a+b x)}\right )}{3 a}+\frac {x^{2/3}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^{2/3}}{a (a+b x)}\right )}{3 a}+\frac {x^{2/3}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^{2/3}}{a (a+b x)}\right )}{3 a}+\frac {x^{2/3}}{2 a (a+b x)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^{2/3}}{a (a+b x)}\right )}{3 a}+\frac {x^{2/3}}{2 a (a+b x)^2}\) |
Input:
Int[1/(x^(1/3)*(a + b*x)^3),x]
Output:
x^(2/3)/(2*a*(a + b*x)^2) + (2*(x^(2/3)/(a*(a + b*x)) + (-((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)))/(3*a)))/(3*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_))^(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 = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {x^{\frac {2}{3}}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {2 x^{\frac {2}{3}}}{3 a \left (b x +a \right )}+\frac {2 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 a}}{a}\) | \(139\) |
| default | \(\frac {x^{\frac {2}{3}}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {2 x^{\frac {2}{3}}}{3 a \left (b x +a \right )}+\frac {2 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 a}}{a}\) | \(139\) |
Input:
int(1/x^(1/3)/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/2*x^(2/3)/a/(b*x+a)^2+2/a*(1/3*x^(2/3)/a/(b*x+a)+1/3/a*(-1/3/b/(a/b)^(1/ 3)*ln(x^(1/3)+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^(2/3)-(a/b)^(1/3)*x^(1/3 )+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*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. 233 vs. \(2 (99) = 198\).
Time = 0.09 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.64 \[ \int \frac {1}{\sqrt [3]{x} (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 b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} a + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{\frac {2}{3}}\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{\frac {1}{3}}}{b x + a}\right ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{\frac {2}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b^{3} x + 7 \, a^{2} b^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\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 b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{\frac {2}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b^{3} x + 7 \, a^{2} b^{2}\right )} x^{\frac {2}{3}}}{18 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}\right ] \] Input:
integrate(1/x^(1/3)/(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*b^2)^(1/3)/a )*log((2*b^2*x - a*b + 3*sqrt(1/3)*(a*b*x^(1/3) + (-a*b^2)^(1/3)*a + 2*(-a *b^2)^(2/3)*x^(2/3))*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x^(1/3))/(b *x + a)) + 2*(b^2*x^2 + 2*a*b*x + a^2)*(-a*b^2)^(2/3)*log(b^2*x^(2/3) + (- a*b^2)^(1/3)*b*x^(1/3) + (-a*b^2)^(2/3)) - 4*(b^2*x^2 + 2*a*b*x + a^2)*(-a *b^2)^(2/3)*log(b*x^(1/3) - (-a*b^2)^(1/3)) + 3*(4*a*b^3*x + 7*a^2*b^2)*x^ (2/3))/(a^3*b^4*x^2 + 2*a^4*b^3*x + a^5*b^2), 1/18*(12*sqrt(1/3)*(a*b^3*x^ 2 + 2*a^2*b^2*x + a^3*b)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x^( 1/3) + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + 2*(b^2*x^2 + 2*a*b*x + a^2)*(-a*b^2)^(2/3)*log(b^2*x^(2/3) + (-a*b^2)^(1/3)*b*x^(1/3) + (-a*b^2) ^(2/3)) - 4*(b^2*x^2 + 2*a*b*x + a^2)*(-a*b^2)^(2/3)*log(b*x^(1/3) - (-a*b ^2)^(1/3)) + 3*(4*a*b^3*x + 7*a^2*b^2)*x^(2/3))/(a^3*b^4*x^2 + 2*a^4*b^3*x + a^5*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (129) = 258\).
Time = 106.40 (sec) , antiderivative size = 1175, normalized size of antiderivative = 8.39 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx=\text {Too large to display} \] Input:
integrate(1/x**(1/3)/(b*x+a)**3,x)
Output:
Piecewise((zoo/x**(7/3), Eq(a, 0) & Eq(b, 0)), (3*x**(2/3)/(2*a**3), Eq(b, 0)), (-3/(7*b**3*x**(7/3)), Eq(a, 0)), (4*a**2*log(x**(1/3) - (-a/b)**(1/ 3))/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3 *x**2*(-a/b)**(1/3)) - 2*a**2*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 4*sqrt(3)*a**2*atan(2*sqrt(3)*x**(1/3) /(3*(-a/b)**(1/3)) + sqrt(3)/3)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x* (-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 4*a**2*log(2)/(18*a**4* b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)* *(1/3)) + 21*a*b*x**(2/3)*(-a/b)**(1/3)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3 *b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 8*a*b*x*log(x** (1/3) - (-a/b)**(1/3))/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**( 1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) - 4*a*b*x*log(4*x**(2/3) + 4*x**(1 /3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b* *2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 8*sqrt(3)*a*b*x*at an(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(18*a**4*b*(-a/b)**(1 /3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)**(1/3)) + 8* a*b*x*log(2)/(18*a**4*b*(-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18* a**2*b**3*x**2*(-a/b)**(1/3)) + 12*b**2*x**(5/3)*(-a/b)**(1/3)/(18*a**4*b* (-a/b)**(1/3) + 36*a**3*b**2*x*(-a/b)**(1/3) + 18*a**2*b**3*x**2*(-a/b)...
Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx=\frac {4 \, b x^{\frac {5}{3}} + 7 \, a x^{\frac {2}{3}}}{6 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(1/x^(1/3)/(b*x+a)^3,x, algorithm="maxima")
Output:
1/6*(4*b*x^(5/3) + 7*a*x^(2/3))/(a^2*b^2*x^2 + 2*a^3*b*x + a^4) + 2/9*sqrt (3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b*(a/b) ^(1/3)) + 1/9*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b )^(1/3)) - 2/9*log(x^(1/3) + (a/b)^(1/3))/(a^2*b*(a/b)^(1/3))
Time = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx=-\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3}} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b^{2}} + \frac {4 \, b x^{\frac {5}{3}} + 7 \, a x^{\frac {2}{3}}}{6 \, {\left (b x + a\right )}^{2} a^{2}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{3} b^{2}} \] Input:
integrate(1/x^(1/3)/(b*x+a)^3,x, algorithm="giac")
Output:
-2/9*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^3 - 2/9*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^3 *b^2) + 1/6*(4*b*x^(5/3) + 7*a*x^(2/3))/((b*x + a)^2*a^2) + 1/9*(-a*b^2)^( 2/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2)
Time = 0.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx=\frac {\frac {7\,x^{2/3}}{6\,a}+\frac {2\,b\,x^{5/3}}{3\,a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {2\,\ln \left (\frac {4\,b\,x^{1/3}}{9\,a^4}-\frac {4\,b^{2/3}}{9\,{\left (-a\right )}^{11/3}}\right )}{9\,{\left (-a\right )}^{7/3}\,b^{2/3}}+\frac {\ln \left (\frac {4\,b\,x^{1/3}}{9\,a^4}-\frac {b^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,{\left (-a\right )}^{11/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{7/3}\,b^{2/3}}-\frac {\ln \left (\frac {4\,b\,x^{1/3}}{9\,a^4}-\frac {b^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,{\left (-a\right )}^{11/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{7/3}\,b^{2/3}} \] Input:
int(1/(x^(1/3)*(a + b*x)^3),x)
Output:
((7*x^(2/3))/(6*a) + (2*b*x^(5/3))/(3*a^2))/(a^2 + b^2*x^2 + 2*a*b*x) + (2 *log((4*b*x^(1/3))/(9*a^4) - (4*b^(2/3))/(9*(-a)^(11/3))))/(9*(-a)^(7/3)*b ^(2/3)) + (log((4*b*x^(1/3))/(9*a^4) - (b^(2/3)*(3^(1/2)*1i - 1)^2)/(9*(-a )^(11/3)))*(3^(1/2)*1i - 1))/(9*(-a)^(7/3)*b^(2/3)) - (log((4*b*x^(1/3))/( 9*a^4) - (b^(2/3)*(3^(1/2)*1i + 1)^2)/(9*(-a)^(11/3)))*(3^(1/2)*1i + 1))/( 9*(-a)^(7/3)*b^(2/3))
Time = 0.16 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^3} \, dx=\frac {-4 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}-8 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a b x -4 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{2}+21 x^{\frac {2}{3}} b^{\frac {2}{3}} a^{\frac {4}{3}}+12 x^{\frac {5}{3}} b^{\frac {5}{3}} a^{\frac {1}{3}}+2 \,\mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) a^{2}+4 \,\mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) a b x +2 \,\mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b^{2} x^{2}-4 \,\mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) a^{2}-8 \,\mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) a b x -4 \,\mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b^{2} x^{2}}{18 b^{\frac {2}{3}} a^{\frac {7}{3}} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:
int(1/x^(1/3)/(b*x+a)^3,x)
Output:
( - 4*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqrt(3)))*a* *2 - 8*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqrt(3)))*a *b*x - 4*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqrt(3))) *b**2*x**2 + 21*x**(2/3)*b**(2/3)*a**(1/3)*a + 12*x**(2/3)*b**(2/3)*a**(1/ 3)*b*x + 2*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))* a**2 + 4*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*a* b*x + 2*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*b** 2*x**2 - 4*log(a**(1/3) + x**(1/3)*b**(1/3))*a**2 - 8*log(a**(1/3) + x**(1 /3)*b**(1/3))*a*b*x - 4*log(a**(1/3) + x**(1/3)*b**(1/3))*b**2*x**2)/(18*b **(2/3)*a**(1/3)*a**2*(a**2 + 2*a*b*x + b**2*x**2))