Integrand size = 13, antiderivative size = 111 \[ \int \frac {1}{x^{5/3} (a+b x)} \, dx=-\frac {3}{2 a x^{2/3}}+\frac {\sqrt {3} b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3}}-\frac {3 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{5/3}}+\frac {b^{2/3} \log (a+b x)}{2 a^{5/3}} \]
-3/2/a/x^(2/3)-3/2*b^(2/3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/a^(5/3)+1/2*b^(2/3) *ln(b*x+a)/a^(5/3)+b^(2/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3))/a^(1/3)* 3^(1/2))*3^(1/2)/a^(5/3)
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^{5/3} (a+b x)} \, dx=\frac {-\frac {3 a^{2/3}}{x^{2/3}}+2 \sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{2 a^{5/3}} \]
((-3*a^(2/3))/x^(2/3) + 2*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/ a^(1/3))/Sqrt[3]] - 2*b^(2/3)*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)])/(2*a^(5/3))
Time = 0.20 (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 = {61, 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.
\(\displaystyle \int \frac {1}{x^{5/3} (a+b x)} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {b \int \frac {1}{x^{2/3} (a+b x)}dx}{a}-\frac {3}{2 a x^{2/3}}\) |
\(\Big \downarrow \) 70 |
\(\displaystyle -\frac {b \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {3}{2 a x^{2/3}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\frac {b \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {3}{2 a x^{2/3}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {b \left (\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {3}{2 a x^{2/3}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {b \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac {\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {3}{2 a x^{2/3}}\) |
-3/(2*a*x^(2/3)) - (b*(-((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))))/a
3.7.81.3.1 Defintions of rubi rules used
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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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 = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(-\frac {3}{2 a \,x^{\frac {2}{3}}}-\frac {3 \left (\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a}\) | \(112\) |
default | \(-\frac {3}{2 a \,x^{\frac {2}{3}}}-\frac {3 \left (\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a}\) | \(112\) |
-3/2/a/x^(2/3)-3*(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)))*b/a
Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^{5/3} (a+b x)} \, dx=\frac {2 \, \sqrt {3} x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x^{\frac {1}{3}} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{\frac {2}{3}} + a b x^{\frac {1}{3}} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{\frac {1}{3}} - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3 \, x^{\frac {1}{3}}}{2 \, a x} \]
1/2*(2*sqrt(3)*x*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x^(1/3)*(-b^2/a^ 2)^(2/3) - sqrt(3)*b)/b) - x*(-b^2/a^2)^(1/3)*log(b^2*x^(2/3) + a*b*x^(1/3 )*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) + 2*x*(-b^2/a^2)^(1/3)*log(b*x^ (1/3) - a*(-b^2/a^2)^(1/3)) - 3*x^(1/3))/(a*x)
Time = 52.37 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^{5/3} (a+b x)} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{5 b x^{\frac {5}{3}}} & \text {for}\: a = 0 \\- \frac {3}{2 a x^{\frac {2}{3}}} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{a \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 a \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{a \left (- \frac {a}{b}\right )^{\frac {2}{3}}} - \frac {3}{2 a x^{\frac {2}{3}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(5/3), Eq(a, 0) & Eq(b, 0)), (-3/(5*b*x**(5/3)), Eq(a, 0 )), (-3/(2*a*x**(2/3)), Eq(b, 0)), (-log(x**(1/3) - (-a/b)**(1/3))/(a*(-a/ b)**(2/3)) + log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/ (2*a*(-a/b)**(2/3)) + sqrt(3)*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(a*(-a/b)**(2/3)) - 3/(2*a*x**(2/3)), True))
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{5/3} (a+b x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {3}{2 \, a x^{\frac {2}{3}}} \]
-sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(a*(a/b )^(2/3)) + 1/2*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(a*(a/b)^( 2/3)) - log(x^(1/3) + (a/b)^(1/3))/(a*(a/b)^(2/3)) - 3/2/(a*x^(2/3))
Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^{5/3} (a+b x)} \, dx=\frac {b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, a^{2}} - \frac {3}{2 \, a x^{\frac {2}{3}}} \]
b*(-a/b)^(1/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^2 - 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^2 - 1/2*( -a*b^2)^(1/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/a^2 - 3/2 /(a*x^(2/3))
Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^{5/3} (a+b x)} \, dx=\frac {b^{2/3}\,\ln \left (9\,{\left (-a\right )}^{7/3}\,b^{8/3}-9\,a^2\,b^3\,x^{1/3}\right )}{{\left (-a\right )}^{5/3}}-\frac {3}{2\,a\,x^{2/3}}+\frac {b^{2/3}\,\ln \left (9\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-9\,a^2\,b^3\,x^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{{\left (-a\right )}^{5/3}}-\frac {b^{2/3}\,\ln \left (9\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+9\,a^2\,b^3\,x^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{{\left (-a\right )}^{5/3}} \]
(b^(2/3)*log(9*(-a)^(7/3)*b^(8/3) - 9*a^2*b^3*x^(1/3)))/(-a)^(5/3) - 3/(2* a*x^(2/3)) + (b^(2/3)*log(9*(-a)^(7/3)*b^(8/3)*((3^(1/2)*1i)/2 - 1/2) - 9* a^2*b^3*x^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(-a)^(5/3) - (b^(2/3)*log(9*(-a)^ (7/3)*b^(8/3)*((3^(1/2)*1i)/2 + 1/2) + 9*a^2*b^3*x^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(-a)^(5/3)
Time = 0.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^{5/3} (a+b x)} \, dx=\frac {2 x^{\frac {2}{3}} a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b +x^{\frac {2}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-x^{\frac {1}{3}} b^{\frac {1}{3}} a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {2}{3}}\right ) b -2 x^{\frac {2}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {1}{3}} b^{\frac {1}{3}}\right ) b -3 b^{\frac {1}{3}} a}{2 x^{\frac {2}{3}} b^{\frac {1}{3}} a^{2}} \]