Integrand size = 13, antiderivative size = 125 \[ \int \frac {x^{5/3}}{a+b x} \, dx=-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}-\frac {\sqrt {3} a^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{8/3}}-\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}} \]
-3/2*a*x^(2/3)/b^2+3/5*x^(5/3)/b-3/2*a^(5/3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/b ^(8/3)+1/2*a^(5/3)*ln(b*x+a)/b^(8/3)-a^(5/3)*arctan(1/3*(a^(1/3)-2*b^(1/3) *x^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)/b^(8/3)
Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.12 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\frac {-15 a b^{2/3} x^{2/3}+6 b^{5/3} x^{5/3}-10 \sqrt {3} a^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-10 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+5 a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{10 b^{8/3}} \]
(-15*a*b^(2/3)*x^(2/3) + 6*b^(5/3)*x^(5/3) - 10*Sqrt[3]*a^(5/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]] - 10*a^(5/3)*Log[a^(1/3) + b^(1/3) *x^(1/3)] + 5*a^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2 /3)])/(10*b^(8/3))
Time = 0.22 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {60, 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.
| \(\displaystyle \int \frac {x^{5/3}}{a+b x} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {3 x^{5/3}}{5 b}-\frac {a \int \frac {x^{2/3}}{a+b x}dx}{b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {3 x^{5/3}}{5 b}-\frac {a \left (\frac {3 x^{2/3}}{2 b}-\frac {a \int \frac {1}{\sqrt [3]{x} (a+b x)}dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {3 x^{5/3}}{5 b}-\frac {a \left (\frac {3 x^{2/3}}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 x^{5/3}}{5 b}-\frac {a \left (\frac {3 x^{2/3}}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 x^{5/3}}{5 b}-\frac {a \left (\frac {3 x^{2/3}}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 x^{5/3}}{5 b}-\frac {a \left (\frac {3 x^{2/3}}{2 b}-\frac {a \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\right )}{b}\) |
(3*x^(5/3))/(5*b) - (a*((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))/b
3.7.74.3.1 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/(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 = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(-\frac {3 \left (-2 b x +5 a \right ) x^{\frac {2}{3}}}{10 b^{2}}-\frac {a^{2} \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {a^{2} \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {a^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(121\) |
| derivativedivides | \(-\frac {3 \left (-\frac {b \,x^{\frac {5}{3}}}{5}+\frac {a \,x^{\frac {2}{3}}}{2}\right )}{b^{2}}+\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}}{b^{2}}\) | \(124\) |
| default | \(-\frac {3 \left (-\frac {b \,x^{\frac {5}{3}}}{5}+\frac {a \,x^{\frac {2}{3}}}{2}\right )}{b^{2}}+\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}}{b^{2}}\) | \(124\) |
-3/10*(-2*b*x+5*a)*x^(2/3)/b^2-a^2/b^3/(a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/3)) +1/2*a^2/b^3/(a/b)^(1/3)*ln(x^(2/3)-(a/b)^(1/3)*x^(1/3)+(a/b)^(2/3))+a^2/b ^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/3)-1))
Time = 0.22 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.18 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\frac {10 \, \sqrt {3} a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + \sqrt {3} a}{3 \, a}\right ) - 5 \, a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (-b x^{\frac {1}{3}} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) + 10 \, a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {1}{3}}\right ) + 3 \, {\left (2 \, b x - 5 \, a\right )} x^{\frac {2}{3}}}{10 \, b^{2}} \]
1/10*(10*sqrt(3)*a*(-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) - 5*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)) + 10*a*(-a^2/b^2)^(1/3)*log(b*(- a^2/b^2)^(2/3) + a*x^(1/3)) + 3*(2*b*x - 5*a)*x^(2/3))/b^2
Time = 58.03 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.44 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\begin {cases} \tilde {\infty } x^{\frac {5}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {8}{3}}}{8 a} & \text {for}\: b = 0 \\\frac {3 x^{\frac {5}{3}}}{5 b} & \text {for}\: a = 0 \\\frac {a^{2} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{b^{3} \sqrt [3]{- \frac {a}{b}}} - \frac {a^{2} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 b^{3} \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3} a^{2} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{b^{3} \sqrt [3]{- \frac {a}{b}}} - \frac {3 a x^{\frac {2}{3}}}{2 b^{2}} + \frac {3 x^{\frac {5}{3}}}{5 b} & \text {otherwise} \end {cases} \]
Piecewise((zoo*x**(5/3), Eq(a, 0) & Eq(b, 0)), (3*x**(8/3)/(8*a), Eq(b, 0) ), (3*x**(5/3)/(5*b), Eq(a, 0)), (a**2*log(x**(1/3) - (-a/b)**(1/3))/(b**3 *(-a/b)**(1/3)) - a**2*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b )**(2/3))/(2*b**3*(-a/b)**(1/3)) + sqrt(3)*a**2*atan(2*sqrt(3)*x**(1/3)/(3 *(-a/b)**(1/3)) + sqrt(3)/3)/(b**3*(-a/b)**(1/3)) - 3*a*x**(2/3)/(2*b**2) + 3*x**(5/3)/(5*b), True))
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.04 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\frac {\sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {a^{2} \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {a^{2} \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {3 \, {\left (2 \, b x^{\frac {5}{3}} - 5 \, a x^{\frac {2}{3}}\right )}}{10 \, b^{2}} \]
sqrt(3)*a^2*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(b^3 *(a/b)^(1/3)) + 1/2*a^2*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/( b^3*(a/b)^(1/3)) - a^2*log(x^(1/3) + (a/b)^(1/3))/(b^3*(a/b)^(1/3)) + 3/10 *(2*b*x^(5/3) - 5*a*x^(2/3))/b^2
Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10 \[ \int \frac {x^{5/3}}{a+b x} \, dx=-\frac {a \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{b^{2}} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{4}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} a \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{4}} + \frac {3 \, {\left (2 \, b^{4} x^{\frac {5}{3}} - 5 \, a b^{3} x^{\frac {2}{3}}\right )}}{10 \, b^{5}} \]
-a*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/b^2 - sqrt(3)*(-a*b^2)^(2 /3)*a*arctan(1/3*sqrt(3)*(2*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 + 1/ 2*(-a*b^2)^(2/3)*a*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/b^4 + 3/10*(2*b^4*x^(5/3) - 5*a*b^3*x^(2/3))/b^5
Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.21 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\frac {3\,x^{5/3}}{5\,b}+\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}}{b^{10/3}}\right )}{b^{8/3}}-\frac {3\,a\,x^{2/3}}{2\,b^2}+\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{10/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{8/3}}-\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{10/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{8/3}} \]
(3*x^(5/3))/(5*b) + ((-a)^(5/3)*log((9*a^4*x^(1/3))/b^3 - (9*(-a)^(13/3))/ b^(10/3)))/b^(8/3) - (3*a*x^(2/3))/(2*b^2) + ((-a)^(5/3)*log((9*a^4*x^(1/3 ))/b^3 - (9*(-a)^(13/3)*((3^(1/2)*1i)/2 - 1/2)^2)/b^(10/3))*((3^(1/2)*1i)/ 2 - 1/2))/b^(8/3) - ((-a)^(5/3)*log((9*a^4*x^(1/3))/b^3 - (9*(-a)^(13/3)*( (3^(1/2)*1i)/2 + 1/2)^2)/b^(10/3))*((3^(1/2)*1i)/2 + 1/2))/b^(8/3)