Integrand size = 13, antiderivative size = 136 \[ \int \frac {x^4}{a+\frac {b}{x^3}} \, dx=-\frac {b x^2}{2 a^2}+\frac {x^5}{5 a}-\frac {b^{5/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{8/3}}-\frac {b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{8/3}}+\frac {b^{5/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{8/3}} \]
-1/2*b*x^2/a^2+1/5*x^5/a-1/3*b^(5/3)*ln(b^(1/3)+a^(1/3)*x)/a^(8/3)+1/6*b^( 5/3)*ln(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(8/3)-1/3*b^(5/3)*arctan( 1/3*(b^(1/3)-2*a^(1/3)*x)/b^(1/3)*3^(1/2))/a^(8/3)*3^(1/2)
Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90 \[ \int \frac {x^4}{a+\frac {b}{x^3}} \, dx=\frac {-15 a^{2/3} b x^2+6 a^{5/3} x^5-10 \sqrt {3} b^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-10 b^{5/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+5 b^{5/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{30 a^{8/3}} \]
(-15*a^(2/3)*b*x^2 + 6*a^(5/3)*x^5 - 10*Sqrt[3]*b^(5/3)*ArcTan[(1 - (2*a^( 1/3)*x)/b^(1/3))/Sqrt[3]] - 10*b^(5/3)*Log[b^(1/3) + a^(1/3)*x] + 5*b^(5/3 )*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(30*a^(8/3))
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {795, 831, 2009}
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.
\(\displaystyle \int \frac {x^4}{a+\frac {b}{x^3}} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {x^7}{a x^3+b}dx\) |
\(\Big \downarrow \) 831 |
\(\displaystyle \int \left (\frac {b^2 x}{a^2 \left (a x^3+b\right )}-\frac {b x}{a^2}+\frac {x^4}{a}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^{5/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{8/3}}+\frac {b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{8/3}}-\frac {b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{8/3}}-\frac {b x^2}{2 a^2}+\frac {x^5}{5 a}\) |
-1/2*(b*x^2)/a^2 + x^5/(5*a) - (b^(5/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sq rt[3]*b^(1/3))])/(Sqrt[3]*a^(8/3)) - (b^(5/3)*Log[b^(1/3) + a^(1/3)*x])/(3 *a^(8/3)) + (b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^ (8/3))
3.20.66.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x ^m, a + b*x^n, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && Gt Q[m, 2*n - 1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.35
method | result | size |
risch | \(\frac {x^{5}}{5 a}-\frac {b \,x^{2}}{2 a^{2}}+\frac {b^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{3 a^{3}}\) | \(48\) |
default | \(\frac {\frac {1}{5} x^{5} a -\frac {1}{2} b \,x^{2}}{a^{2}}+\frac {\left (-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right ) b^{2}}{a^{2}}\) | \(116\) |
Time = 0.31 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.03 \[ \int \frac {x^4}{a+\frac {b}{x^3}} \, dx=\frac {6 \, a x^{5} - 15 \, b x^{2} + 10 \, \sqrt {3} b \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + \sqrt {3} b}{3 \, b}\right ) - 5 \, b \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - b \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 10 \, b \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right )}{30 \, a^{2}} \]
1/30*(6*a*x^5 - 15*b*x^2 + 10*sqrt(3)*b*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqr t(3)*a*x*(-b^2/a^2)^(1/3) + sqrt(3)*b)/b) - 5*b*(-b^2/a^2)^(1/3)*log(b*x^2 - a*x*(-b^2/a^2)^(2/3) - b*(-b^2/a^2)^(1/3)) + 10*b*(-b^2/a^2)^(1/3)*log( b*x + a*(-b^2/a^2)^(2/3)))/a^2
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.32 \[ \int \frac {x^4}{a+\frac {b}{x^3}} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{8} + b^{5}, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{5}}{b^{3}} + x \right )} \right )\right )} + \frac {x^{5}}{5 a} - \frac {b x^{2}}{2 a^{2}} \]
RootSum(27*_t**3*a**8 + b**5, Lambda(_t, _t*log(9*_t**2*a**5/b**3 + x))) + x**5/(5*a) - b*x**2/(2*a**2)
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{a+\frac {b}{x^3}} \, dx=\frac {\sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {b^{2} \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{3} \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {b^{2} \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{3} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {2 \, a x^{5} - 5 \, b x^{2}}{10 \, a^{2}} \]
1/3*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*x - (b/a)^(1/3))/(b/a)^(1/3))/(a^3*( b/a)^(1/3)) + 1/6*b^2*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3))/(a^3*(b/a)^(1 /3)) - 1/3*b^2*log(x + (b/a)^(1/3))/(a^3*(b/a)^(1/3)) + 1/10*(2*a*x^5 - 5* b*x^2)/a^2
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.97 \[ \int \frac {x^4}{a+\frac {b}{x^3}} \, dx=-\frac {b \left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{4}} + \frac {\left (-a^{2} b\right )^{\frac {2}{3}} b \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{4}} + \frac {2 \, a^{4} x^{5} - 5 \, a^{3} b x^{2}}{10 \, a^{5}} \]
-1/3*b*(-b/a)^(2/3)*log(abs(x - (-b/a)^(1/3)))/a^2 - 1/3*sqrt(3)*(-a^2*b)^ (2/3)*b*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/a^4 + 1/6*(- a^2*b)^(2/3)*b*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^4 + 1/10*(2*a^4* x^5 - 5*a^3*b*x^2)/a^5
Time = 6.00 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.05 \[ \int \frac {x^4}{a+\frac {b}{x^3}} \, dx=\frac {x^5}{5\,a}-\frac {b\,x^2}{2\,a^2}+\frac {{\left (-b\right )}^{5/3}\,\ln \left (\frac {b^4\,x}{a^3}-\frac {{\left (-b\right )}^{13/3}}{a^{10/3}}\right )}{3\,a^{8/3}}-\frac {{\left (-b\right )}^{5/3}\,\ln \left (\frac {b^4\,x}{a^3}-\frac {{\left (-b\right )}^{13/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{a^{10/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{8/3}}+\frac {{\left (-b\right )}^{5/3}\,\ln \left (\frac {b^4\,x}{a^3}-\frac {9\,{\left (-b\right )}^{13/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{a^{10/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{8/3}} \]
x^5/(5*a) - (b*x^2)/(2*a^2) + ((-b)^(5/3)*log((b^4*x)/a^3 - (-b)^(13/3)/a^ (10/3)))/(3*a^(8/3)) - ((-b)^(5/3)*log((b^4*x)/a^3 - ((-b)^(13/3)*((3^(1/2 )*1i)/2 + 1/2)^2)/a^(10/3))*((3^(1/2)*1i)/2 + 1/2))/(3*a^(8/3)) + ((-b)^(5 /3)*log((b^4*x)/a^3 - (9*(-b)^(13/3)*((3^(1/2)*1i)/6 - 1/6)^2)/a^(10/3))*( (3^(1/2)*1i)/6 - 1/6))/a^(8/3)