Integrand size = 13, antiderivative size = 117 \[ \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx=-\frac {\sqrt [3]{x}}{b (a+b x)}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} b^{4/3}}-\frac {\log (a+b x)}{6 a^{2/3} b^{4/3}} \] Output:
-x^(1/3)/b/(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^(2/3)/b^(4/3)+1/2*ln(a^(1/3)+b^(1/3)*x^(1/3))/a^(2/3)/b^(4/ 3)-1/6*ln(b*x+a)/a^(2/3)/b^(4/3)
Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx=\frac {-\frac {6 \sqrt [3]{b} \sqrt [3]{x}}{a+b x}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{2/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{a^{2/3}}}{6 b^{4/3}} \] Input:
Integrate[x^(1/3)/(a + b*x)^2,x]
Output:
((-6*b^(1/3)*x^(1/3))/(a + b*x) - (2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^(1/3 ))/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*Log[a^(1/3) + b^(1/3)*x^(1/3)])/a^(2/3) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)]/a^(2/3))/(6*b^ (4/3))
Time = 0.19 (sec) , antiderivative size = 121, 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 = {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.
| \(\displaystyle \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\int \frac {1}{x^{2/3} (a+b x)}dx}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {\sqrt [3]{x}}{b (a+b x)}\) |
Input:
Int[x^(1/3)/(a + b*x)^2,x]
Output:
-(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)
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 = 117, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(-\frac {x^{\frac {1}{3}}}{b \left (b x +a \right )}+\frac {\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{b}\) | \(117\) |
| default | \(-\frac {x^{\frac {1}{3}}}{b \left (b x +a \right )}+\frac {\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{b}\) | \(117\) |
Input:
int(x^(1/3)/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-x^(1/3)/b/(b*x+a)+1/b*(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. 175 vs. \(2 (84) = 168\).
Time = 0.08 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.32 \[ \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx=\left [-\frac {6 \, a^{2} b x^{\frac {1}{3}} - 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{\frac {2}{3}} - \left (a^{2} b\right )^{\frac {1}{3}} a + \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{\frac {1}{3}}}{b x + a}\right ) + \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) - 2 \, \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}, -\frac {6 \, a^{2} b x^{\frac {1}{3}} - 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (a^{2} b\right )^{\frac {1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) + \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) - 2 \, \left (a^{2} b\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}\right ] \] Input:
integrate(x^(1/3)/(b*x+a)^2,x, algorithm="fricas")
Output:
[-1/6*(6*a^2*b*x^(1/3) - 3*sqrt(1/3)*(a*b^2*x + a^2*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)) + (a^2*b)^(2/3)*(b*x + a)*log(a*b*x^(2/3) + (a^2*b)^(1/3)*a - (a ^2*b)^(2/3)*x^(1/3)) - 2*(a^2*b)^(2/3)*(b*x + a)*log(a*b*x^(1/3) + (a^2*b) ^(2/3)))/(a^2*b^3*x + a^3*b^2), -1/6*(6*a^2*b*x^(1/3) - 6*sqrt(1/3)*(a*b^2 *x + a^2*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) + (a^2*b)^(2/3)*(b*x + a) *log(a*b*x^(2/3) + (a^2*b)^(1/3)*a - (a^2*b)^(2/3)*x^(1/3)) - 2*(a^2*b)^(2 /3)*(b*x + a)*log(a*b*x^(1/3) + (a^2*b)^(2/3)))/(a^2*b^3*x + a^3*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (107) = 214\).
Time = 21.20 (sec) , antiderivative size = 450, normalized size of antiderivative = 3.85 \[ \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {2}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {4}{3}}}{4 a^{2}} & \text {for}\: b = 0 \\- \frac {3}{2 b^{2} x^{\frac {2}{3}}} & \text {for}\: a = 0 \\- \frac {6 a \sqrt [3]{x}}{6 a^{2} b + 6 a b^{2} x} - \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{2} b + 6 a b^{2} x} + \frac {a \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{6 a^{2} b + 6 a b^{2} x} + \frac {2 \sqrt {3} a \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a^{2} b + 6 a b^{2} x} - \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{6 a^{2} b + 6 a b^{2} x} - \frac {2 b x \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{2} b + 6 a b^{2} x} + \frac {b x \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{6 a^{2} b + 6 a b^{2} x} + \frac {2 \sqrt {3} b x \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{6 a^{2} b + 6 a b^{2} x} - \frac {2 b x \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{6 a^{2} b + 6 a b^{2} x} & \text {otherwise} \end {cases} \] Input:
integrate(x**(1/3)/(b*x+a)**2,x)
Output:
Piecewise((zoo/x**(2/3), Eq(a, 0) & Eq(b, 0)), (3*x**(4/3)/(4*a**2), Eq(b, 0)), (-3/(2*b**2*x**(2/3)), Eq(a, 0)), (-6*a*x**(1/3)/(6*a**2*b + 6*a*b** 2*x) - 2*a*(-a/b)**(1/3)*log(x**(1/3) - (-a/b)**(1/3))/(6*a**2*b + 6*a*b** 2*x) + a*(-a/b)**(1/3)*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b )**(2/3))/(6*a**2*b + 6*a*b**2*x) + 2*sqrt(3)*a*(-a/b)**(1/3)*atan(2*sqrt( 3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(6*a**2*b + 6*a*b**2*x) - 2*a*( -a/b)**(1/3)*log(2)/(6*a**2*b + 6*a*b**2*x) - 2*b*x*(-a/b)**(1/3)*log(x**( 1/3) - (-a/b)**(1/3))/(6*a**2*b + 6*a*b**2*x) + b*x*(-a/b)**(1/3)*log(4*x* *(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(6*a**2*b + 6*a*b**2* x) + 2*sqrt(3)*b*x*(-a/b)**(1/3)*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(6*a**2*b + 6*a*b**2*x) - 2*b*x*(-a/b)**(1/3)*log(2)/(6*a**2 *b + 6*a*b**2*x), True))
Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx=-\frac {x^{\frac {1}{3}}}{b^{2} x + a b} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(x^(1/3)/(b*x+a)^2,x, algorithm="maxima")
Output:
-x^(1/3)/(b^2*x + a*b) + 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b) ^(1/3))/(a/b)^(1/3))/(b^2*(a/b)^(2/3)) - 1/6*log(x^(2/3) - x^(1/3)*(a/b)^( 1/3) + (a/b)^(2/3))/(b^2*(a/b)^(2/3)) + 1/3*log(x^(1/3) + (a/b)^(1/3))/(b^ 2*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx=-\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{2}} - \frac {x^{\frac {1}{3}}}{{\left (b x + a\right )} b} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}} \] Input:
integrate(x^(1/3)/(b*x+a)^2,x, algorithm="giac")
Output:
-1/3*(-a/b)^(1/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/(a*b) + 1/3*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^2) - x^(1/3)/((b*x + a)*b) + 1/6*(-a*b^2)^(1/3)*log(x^(2/3) + x^(1/3)*( -a/b)^(1/3) + (-a/b)^(2/3))/(a*b^2)
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx=\frac {\ln \left (3\,b\,x^{1/3}+3\,a^{1/3}\,b^{2/3}\right )}{3\,a^{2/3}\,b^{4/3}}-\frac {x^{1/3}}{b\,\left (a+b\,x\right )}+\frac {\ln \left (3\,b\,x^{1/3}+\frac {3\,a^{1/3}\,b^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{4/3}}-\frac {\ln \left (3\,b\,x^{1/3}-\frac {3\,a^{1/3}\,b^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,b^{4/3}} \] Input:
int(x^(1/3)/(a + b*x)^2,x)
Output:
log(3*b*x^(1/3) + 3*a^(1/3)*b^(2/3))/(3*a^(2/3)*b^(4/3)) - x^(1/3)/(b*(a + b*x)) + (log(3*b*x^(1/3) + (3*a^(1/3)*b^(2/3)*(3^(1/2)*1i - 1))/2)*(3^(1/ 2)*1i - 1))/(6*a^(2/3)*b^(4/3)) - (log(3*b*x^(1/3) - (3*a^(1/3)*b^(2/3)*(3 ^(1/2)*1i + 1))/2)*(3^(1/2)*1i + 1))/(6*a^(2/3)*b^(4/3))
Time = 0.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt [3]{x}}{(a+b x)^2} \, dx=\frac {-2 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right )-2 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b x -a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right )-a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b x +2 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right )+2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b x -6 x^{\frac {1}{3}} b^{\frac {1}{3}} a}{6 b^{\frac {4}{3}} a \left (b x +a \right )} \] Input:
int(x^(1/3)/(b*x+a)^2,x)
Output:
( - 2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqr t(3)))*a - 2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1 /3)*sqrt(3)))*b*x - a**(1/3)*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x **(2/3)*b**(2/3))*a - a**(1/3)*log(a**(2/3) - x**(1/3)*b**(1/3)*a**(1/3) + x**(2/3)*b**(2/3))*b*x + 2*a**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*a + 2*a**(1/3)*log(a**(1/3) + x**(1/3)*b**(1/3))*b*x - 6*x**(1/3)*b**(1/3)*a) /(6*b**(1/3)*a*b*(a + b*x))