Integrand size = 13, antiderivative size = 116 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx=\frac {x^{2/3}}{a (a+b x)}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3} b^{2/3}}+\frac {\log (a+b x)}{6 a^{4/3} b^{2/3}} \] Output:
x^(2/3)/a/(b*x+a)-1/3*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3))*3^(1/2)/a^(1/ 3))*3^(1/2)/a^(4/3)/b^(2/3)-1/2*ln(a^(1/3)+b^(1/3)*x^(1/3))/a^(4/3)/b^(2/3 )+1/6*ln(b*x+a)/a^(4/3)/b^(2/3)
Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx=\frac {\frac {6 \sqrt [3]{a} x^{2/3}}{a+b x}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{b^{2/3}}}{6 a^{4/3}} \] Input:
Integrate[1/(x^(1/3)*(a + b*x)^2),x]
Output:
((6*a^(1/3)*x^(2/3))/(a + b*x) - (2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^(1/3) )/a^(1/3))/Sqrt[3]])/b^(2/3) - (2*Log[a^(1/3) + b^(1/3)*x^(1/3)])/b^(2/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)]/b^(2/3))/(6*a^( 4/3))
Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{x} (a+b x)}dx}{3 a}+\frac {x^{2/3}}{a (a+b x)}\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^{2/3}}{a (a+b x)}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^{2/3}}{a (a+b x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^{2/3}}{a (a+b x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^{2/3}}{a (a+b x)}\) |
Input:
Int[1/(x^(1/3)*(a + b*x)^2),x]
Output:
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)
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 = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {x^{\frac {2}{3}}}{a \left (b x +a \right )}+\frac {-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{a}\) | \(116\) |
| default | \(\frac {x^{\frac {2}{3}}}{a \left (b x +a \right )}+\frac {-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{a}\) | \(116\) |
Input:
int(1/x^(1/3)/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
x^(2/3)/a/(b*x+a)+1/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. 167 vs. \(2 (83) = 166\).
Time = 0.09 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.27 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx=\left [\frac {6 \, a b^{2} x^{\frac {2}{3}} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x^{\frac {1}{3}} - \left (a b^{2}\right )^{\frac {1}{3}} a + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} x^{\frac {2}{3}}\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac {2}{3}} x^{\frac {1}{3}}}{b x + a}\right ) + \left (a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (b^{2} x^{\frac {2}{3}} - \left (a b^{2}\right )^{\frac {1}{3}} b x^{\frac {1}{3}} + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (b x^{\frac {1}{3}} + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{6 \, {\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}, \frac {6 \, a b^{2} x^{\frac {2}{3}} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x^{\frac {1}{3}} - \left (a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + \left (a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (b^{2} x^{\frac {2}{3}} - \left (a b^{2}\right )^{\frac {1}{3}} b x^{\frac {1}{3}} + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (b x^{\frac {1}{3}} + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{6 \, {\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}\right ] \] Input:
integrate(1/x^(1/3)/(b*x+a)^2,x, algorithm="fricas")
Output:
[1/6*(6*a*b^2*x^(2/3) + 3*sqrt(1/3)*(a*b^2*x + a^2*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)) + (a*b^2)^(2/3)*(b*x + a)*log(b^2*x^(2/3) - (a*b^2)^(1/3)*b*x^(1/3) + (a*b^2)^(2/3)) - 2*(a*b^2)^(2/3)*(b*x + a)*log(b*x^(1/3) + (a*b^2)^(1/3 )))/(a^2*b^3*x + a^3*b^2), 1/6*(6*a*b^2*x^(2/3) + 6*sqrt(1/3)*(a*b^2*x + a ^2*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) + (a*b^2)^(2/3)*(b*x + a)*log(b^2*x^(2/3) - (a*b ^2)^(1/3)*b*x^(1/3) + (a*b^2)^(2/3)) - 2*(a*b^2)^(2/3)*(b*x + a)*log(b*x^( 1/3) + (a*b^2)^(1/3)))/(a^2*b^3*x + a^3*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (107) = 214\).
Time = 22.69 (sec) , antiderivative size = 544, normalized size of antiderivative = 4.69 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {4}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {2}{3}}}{2 a^{2}} & \text {for}\: b = 0 \\- \frac {3}{4 b^{2} x^{\frac {4}{3}}} & \text {for}\: a = 0 \\\frac {2 a \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} - \frac {a \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 \sqrt {3} a \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 a \log {\left (2 \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {6 b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 b x \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} - \frac {b x \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 \sqrt {3} b x \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 b x \log {\left (2 \right )}}{6 a^{2} b \sqrt [3]{- \frac {a}{b}} + 6 a b^{2} x \sqrt [3]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**(1/3)/(b*x+a)**2,x)
Output:
Piecewise((zoo/x**(4/3), Eq(a, 0) & Eq(b, 0)), (3*x**(2/3)/(2*a**2), Eq(b, 0)), (-3/(4*b**2*x**(4/3)), Eq(a, 0)), (2*a*log(x**(1/3) - (-a/b)**(1/3)) /(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) - a*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(6*a**2*b*(-a/b)**(1/3) + 6*a* b**2*x*(-a/b)**(1/3)) + 2*sqrt(3)*a*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/ 3)) + sqrt(3)/3)/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 2*a *log(2)/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 6*b*x**(2/3) *(-a/b)**(1/3)/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 2*b*x *log(x**(1/3) - (-a/b)**(1/3))/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b) **(1/3)) - b*x*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3) )/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 2*sqrt(3)*b*x*atan (2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(6*a**2*b*(-a/b)**(1/3) + 6*a*b**2*x*(-a/b)**(1/3)) + 2*b*x*log(2)/(6*a**2*b*(-a/b)**(1/3) + 6*a* b**2*x*(-a/b)**(1/3)), True))
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx=\frac {x^{\frac {2}{3}}}{a b x + a^{2}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(1/x^(1/3)/(b*x+a)^2,x, algorithm="maxima")
Output:
x^(2/3)/(a*b*x + a^2) + 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^ (1/3))/(a/b)^(1/3))/(a*b*(a/b)^(1/3)) + 1/6*log(x^(2/3) - x^(1/3)*(a/b)^(1 /3) + (a/b)^(2/3))/(a*b*(a/b)^(1/3)) - 1/3*log(x^(1/3) + (a/b)^(1/3))/(a*b *(a/b)^(1/3))
Time = 0.13 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx=-\frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} + \frac {x^{\frac {2}{3}}}{{\left (b x + a\right )} a} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b^{2}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b^{2}} \] Input:
integrate(1/x^(1/3)/(b*x+a)^2,x, algorithm="giac")
Output:
-1/3*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^2 + x^(2/3)/((b*x + a )*a) - 1/3*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^2*b^2) + 1/6*(-a*b^2)^(2/3)*log(x^(2/3) + x^(1/3)*( -a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^2)
Time = 0.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx=\frac {x^{2/3}}{a\,\left (a+b\,x\right )}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {{\left (-1\right )}^{2/3}\,b^{2/3}}{a^{5/3}}+\frac {b\,x^{1/3}}{a^2}\right )}{3\,a^{4/3}\,b^{2/3}}-\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {b\,x^{1/3}}{a^2}+\frac {{\left (-1\right )}^{2/3}\,b^{2/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{a^{5/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,b^{2/3}}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {b\,x^{1/3}}{a^2}+\frac {9\,{\left (-1\right )}^{2/3}\,b^{2/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{a^{5/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}\,b^{2/3}} \] Input:
int(1/(x^(1/3)*(a + b*x)^2),x)
Output:
x^(2/3)/(a*(a + b*x)) + ((-1)^(1/3)*log(((-1)^(2/3)*b^(2/3))/a^(5/3) + (b* x^(1/3))/a^2))/(3*a^(4/3)*b^(2/3)) - ((-1)^(1/3)*log((b*x^(1/3))/a^2 + ((- 1)^(2/3)*b^(2/3)*((3^(1/2)*1i)/2 + 1/2)^2)/a^(5/3))*((3^(1/2)*1i)/2 + 1/2) )/(3*a^(4/3)*b^(2/3)) + ((-1)^(1/3)*log((b*x^(1/3))/a^2 + (9*(-1)^(2/3)*b^ (2/3)*((3^(1/2)*1i)/6 - 1/6)^2)/a^(5/3))*((3^(1/2)*1i)/6 - 1/6))/(a^(4/3)* b^(2/3))
Time = 0.16 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt [3]{x} (a+b x)^2} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a -2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b x +6 x^{\frac {2}{3}} b^{\frac {2}{3}} a^{\frac {1}{3}}+\mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) a +\mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b x -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) a -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b x}{6 b^{\frac {2}{3}} a^{\frac {4}{3}} \left (b x +a \right )} \] Input:
int(1/x^(1/3)/(b*x+a)^2,x)
Output:
( - 2*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqrt(3)))*a - 2*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqrt(3)))*b*x + 6*x**(2/3)*b**(2/3)*a**(1/3) + log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*a + log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**( 2/3)*b**(2/3))*b*x - 2*log(a**(1/3) + x**(1/3)*b**(1/3))*a - 2*log(a**(1/3 ) + x**(1/3)*b**(1/3))*b*x)/(6*b**(2/3)*a**(1/3)*a*(a + b*x))