Integrand size = 13, antiderivative size = 111 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {3 x^{2/3}}{2 b}+\frac {\sqrt {3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{5/3}}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}} \] Output:
3/2*x^(2/3)/b+3^(1/2)*a^(2/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3))*3^(1/ 2)/a^(1/3))/b^(5/3)+3/2*a^(2/3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/b^(5/3)-1/2*a^ (2/3)*ln(b*x+a)/b^(5/3)
Time = 0.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.14 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {3 b^{2/3} x^{2/3}+2 \sqrt {3} a^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{2 b^{5/3}} \] Input:
Integrate[x^(2/3)/(a + b*x),x]
Output:
(3*b^(2/3)*x^(2/3) + 2*Sqrt[3]*a^(2/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^( 1/3))/Sqrt[3]] + 2*a^(2/3)*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)])/(2*b^(5/3))
Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {60, 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.
| \(\displaystyle \int \frac {x^{2/3}}{a+b x} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {3 x^{2/3}}{2 b}-\frac {a \int \frac {1}{\sqrt [3]{x} (a+b x)}dx}{b}\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {3 x^{2/3}}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 x^{2/3}}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 x^{2/3}}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 x^{2/3}}{2 b}-\frac {a \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\) |
Input:
Int[x^(2/3)/(a + b*x),x]
Output:
(3*x^(2/3))/(2*b) - (a*(-((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))))/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 + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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 = 107, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(\frac {3 x^{\frac {2}{3}}}{2 b}+\frac {a \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {a \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {a \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(107\) |
| derivativedivides | \(\frac {3 x^{\frac {2}{3}}}{2 b}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a}{b}\) | \(112\) |
| default | \(\frac {3 x^{\frac {2}{3}}}{2 b}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a}{b}\) | \(112\) |
Input:
int(x^(2/3)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
3/2*x^(2/3)/b+a/b^2/(a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/3))-1/2*a/b^2/(a/b)^(1 /3)*ln(x^(2/3)-(a/b)^(1/3)*x^(1/3)+(a/b)^(2/3))-a/b^2*3^(1/2)/(a/b)^(1/3)* arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/3)-1))
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.15 \[ \int \frac {x^{2/3}}{a+b x} \, dx=-\frac {2 \, \sqrt {3} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (-b x^{\frac {1}{3}} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 2 \, \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {1}{3}}\right ) - 3 \, x^{\frac {2}{3}}}{2 \, b} \] Input:
integrate(x^(2/3)/(b*x+a),x, algorithm="fricas")
Output:
-1/2*(2*sqrt(3)*(a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x^(1/3)*(a^2/b^2)^ (1/3) - sqrt(3)*a)/a) + (a^2/b^2)^(1/3)*log(-b*x^(1/3)*(a^2/b^2)^(2/3) + a *x^(2/3) + a*(a^2/b^2)^(1/3)) - 2*(a^2/b^2)^(1/3)*log(b*(a^2/b^2)^(2/3) + a*x^(1/3)) - 3*x^(2/3))/b
Time = 2.73 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.46 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\begin {cases} \tilde {\infty } x^{\frac {2}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {5}{3}}}{5 a} & \text {for}\: b = 0 \\\frac {3 x^{\frac {2}{3}}}{2 b} & \text {for}\: a = 0 \\- \frac {a \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{b^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {a \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 b^{2} \sqrt [3]{- \frac {a}{b}}} - \frac {\sqrt {3} a \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{b^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {3 x^{\frac {2}{3}}}{2 b} & \text {otherwise} \end {cases} \] Input:
integrate(x**(2/3)/(b*x+a),x)
Output:
Piecewise((zoo*x**(2/3), Eq(a, 0) & Eq(b, 0)), (3*x**(5/3)/(5*a), Eq(b, 0) ), (3*x**(2/3)/(2*b), Eq(a, 0)), (-a*log(x**(1/3) - (-a/b)**(1/3))/(b**2*( -a/b)**(1/3)) + a*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2 /3))/(2*b**2*(-a/b)**(1/3)) - sqrt(3)*a*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)* *(1/3)) + sqrt(3)/3)/(b**2*(-a/b)**(1/3)) + 3*x**(2/3)/(2*b), True))
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.03 \[ \int \frac {x^{2/3}}{a+b x} \, dx=-\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} - \frac {a \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {a \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x^(2/3)/(b*x+a),x, algorithm="maxima")
Output:
-sqrt(3)*a*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(b^2* (a/b)^(1/3)) + 3/2*x^(2/3)/b - 1/2*a*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + ( a/b)^(2/3))/(b^2*(a/b)^(1/3)) + a*log(x^(1/3) + (a/b)^(1/3))/(b^2*(a/b)^(1 /3))
Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.06 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{b} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{3}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{3}} \] Input:
integrate(x^(2/3)/(b*x+a),x, algorithm="giac")
Output:
(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/b + 3/2*x^(2/3)/b + sqrt(3)* (-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3)) /b^3 - 1/2*(-a*b^2)^(2/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3 ))/b^3
Time = 0.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.17 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {3\,x^{2/3}}{2\,b}+\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )}{b^{5/3}}+\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{5/3}}-\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{5/3}} \] Input:
int(x^(2/3)/(a + b*x),x)
Output:
(3*x^(2/3))/(2*b) + (a^(2/3)*log((9*a^(7/3))/b^(4/3) + (9*a^2*x^(1/3))/b)) /b^(5/3) + (a^(2/3)*log((9*a^(7/3)*((3^(1/2)*1i)/2 - 1/2)^2)/b^(4/3) + (9* a^2*x^(1/3))/b)*((3^(1/2)*1i)/2 - 1/2))/b^(5/3) - (a^(2/3)*log((9*a^(7/3)* ((3^(1/2)*1i)/2 + 1/2)^2)/b^(4/3) + (9*a^2*x^(1/3))/b)*((3^(1/2)*1i)/2 + 1 /2))/b^(5/3)
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.78 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a +3 x^{\frac {2}{3}} b^{\frac {2}{3}} a^{\frac {1}{3}}-\mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) a +2 \,\mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) a}{2 b^{\frac {5}{3}} a^{\frac {1}{3}}} \] Input:
int(x^(2/3)/(b*x+a),x)
Output:
(2*sqrt(3)*atan((a**(1/3) - 2*x**(1/3)*b**(1/3))/(a**(1/3)*sqrt(3)))*a + 3 *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 + 2*log(a**(1/3) + x**(1/3)*b**(1/3))*a)/(2*b**(2/3)* a**(1/3)*b)