Integrand size = 38, antiderivative size = 297 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{8/3}}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}} \]
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Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1837, 1872, 1874, 31, 648, 631, 210, 642} \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} h+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{9 \sqrt {3} a^{5/3} b^{8/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{54 a^{5/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{27 a^{5/3} b^{8/3}}+\frac {x \left (x (2 b e-5 a h)-4 a g+b d+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2} \]
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Rule 31
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1837
Rule 1872
Rule 1874
Rubi steps \begin{align*} \text {integral}& = -\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}+\frac {\int \frac {d+2 e x+3 f x^2+4 g x^3+5 h x^4}{\left (a+b x^3\right )^2} \, dx}{6 b} \\ & = \frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}-\frac {\int \frac {-2 b (b d+2 a g)-2 b (b e+5 a h) x}{a+b x^3} \, dx}{18 a b^3} \\ & = \frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}-\frac {\int \frac {\sqrt [3]{a} \left (-4 b^{4/3} (b d+2 a g)-2 \sqrt [3]{a} b (b e+5 a h)\right )+\sqrt [3]{b} \left (2 b^{4/3} (b d+2 a g)-2 \sqrt [3]{a} b (b e+5 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{5/3} b^{10/3}}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{5/3} b^{7/3}} \\ & = \frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{4/3} b^{7/3}}-\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{5/3} b^{8/3}} \\ & = \frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{5/3} b^{8/3}} \\ & = \frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{8/3}}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {-\frac {9 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{\left (a+b x^3\right )^2}+\frac {3 b^{2/3} (b x (d+2 e x)-a (6 f+x (7 g+8 h x)))}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \left (b^{4/3} d+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \left (b^{4/3} d-\sqrt [3]{a} b e+2 a \sqrt [3]{b} g-5 a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {\left (-b^{4/3} d+\sqrt [3]{a} b e-2 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{54 b^{8/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.55 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {-\frac {\left (4 a h -b e \right ) x^{5}}{9 a b}-\frac {\left (7 a g -b d \right ) x^{4}}{18 a b}-\frac {f \,x^{3}}{3 b}-\frac {\left (5 a h +b e \right ) x^{2}}{18 b^{2}}-\frac {\left (2 a g +b d \right ) x}{9 b^{2}}-\frac {a f +b c}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\left (5 a h +b e \right ) \textit {\_R} +2 a g +b d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 a \,b^{3}}\) | \(151\) |
default | \(\frac {-\frac {\left (4 a h -b e \right ) x^{5}}{9 a b}-\frac {\left (7 a g -b d \right ) x^{4}}{18 a b}-\frac {f \,x^{3}}{3 b}-\frac {\left (5 a h +b e \right ) x^{2}}{18 b^{2}}-\frac {\left (2 a g +b d \right ) x}{9 b^{2}}-\frac {a f +b c}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (2 a g +b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (5 a h +b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a \,b^{2}}\) | \(311\) |
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Result contains complex when optimal does not.
Time = 1.47 (sec) , antiderivative size = 6926, normalized size of antiderivative = 23.32 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.04 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {6 \, a b f x^{3} - 2 \, {\left (b^{2} e - 4 \, a b h\right )} x^{5} - {\left (b^{2} d - 7 \, a b g\right )} x^{4} + 3 \, a b c + 3 \, a^{2} f + {\left (a b e + 5 \, a^{2} h\right )} x^{2} + 2 \, {\left (a b d + 2 \, a^{2} g\right )} x}{18 \, {\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}} + \frac {\sqrt {3} {\left (b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a h \left (\frac {a}{b}\right )^{\frac {1}{3}} + b d + 2 \, a g\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a h \left (\frac {a}{b}\right )^{\frac {1}{3}} - b d - 2 \, a g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a h \left (\frac {a}{b}\right )^{\frac {1}{3}} - b d - 2 \, a g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.06 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (b^{2} d + 2 \, a b g - \left (-a b^{2}\right )^{\frac {1}{3}} b e - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (b^{2} d + 2 \, a b g + \left (-a b^{2}\right )^{\frac {1}{3}} b e + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a h \left (-\frac {a}{b}\right )^{\frac {1}{3}} + b d + 2 \, a g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b^{2}} + \frac {2 \, b^{2} e x^{5} - 8 \, a b h x^{5} + b^{2} d x^{4} - 7 \, a b g x^{4} - 6 \, a b f x^{3} - a b e x^{2} - 5 \, a^{2} h x^{2} - 2 \, a b d x - 4 \, a^{2} g x - 3 \, a b c - 3 \, a^{2} f}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{2}} \]
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Time = 9.37 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.11 \[ \int \frac {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^5\,b^8\,z^3+810\,a^4\,b^3\,g\,h\,z+405\,a^3\,b^4\,d\,h\,z+162\,a^3\,b^4\,e\,g\,z+81\,a^2\,b^5\,d\,e\,z+75\,a^3\,b\,e\,h^2-6\,a\,b^3\,d^2\,g+15\,a^2\,b^2\,e^2\,h-12\,a^2\,b^2\,d\,g^2-8\,a^3\,b\,g^3+a\,b^3\,e^3+125\,a^4\,h^3-b^4\,d^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^5\,b^8\,z^3+810\,a^4\,b^3\,g\,h\,z+405\,a^3\,b^4\,d\,h\,z+162\,a^3\,b^4\,e\,g\,z+81\,a^2\,b^5\,d\,e\,z+75\,a^3\,b\,e\,h^2-6\,a\,b^3\,d^2\,g+15\,a^2\,b^2\,e^2\,h-12\,a^2\,b^2\,d\,g^2-8\,a^3\,b\,g^3+a\,b^3\,e^3+125\,a^4\,h^3-b^4\,d^3,z,k\right )\,a\,b^2\,9+\frac {x\,\left (54\,g\,a^2\,b^3+27\,d\,a\,b^4\right )}{81\,a^2\,b^3}\right )+\frac {b^2\,d\,e+10\,a^2\,g\,h+5\,a\,b\,d\,h+2\,a\,b\,e\,g}{81\,a^2\,b^3}+\frac {x\,\left (25\,a^2\,h^2+10\,a\,b\,e\,h+b^2\,e^2\right )}{81\,a^2\,b^3}\right )\,\mathrm {root}\left (19683\,a^5\,b^8\,z^3+810\,a^4\,b^3\,g\,h\,z+405\,a^3\,b^4\,d\,h\,z+162\,a^3\,b^4\,e\,g\,z+81\,a^2\,b^5\,d\,e\,z+75\,a^3\,b\,e\,h^2-6\,a\,b^3\,d^2\,g+15\,a^2\,b^2\,e^2\,h-12\,a^2\,b^2\,d\,g^2-8\,a^3\,b\,g^3+a\,b^3\,e^3+125\,a^4\,h^3-b^4\,d^3,z,k\right )\right )-\frac {\frac {b\,c+a\,f}{6\,b^2}+\frac {x\,\left (b\,d+2\,a\,g\right )}{9\,b^2}+\frac {f\,x^3}{3\,b}+\frac {x^2\,\left (b\,e+5\,a\,h\right )}{18\,b^2}-\frac {x^4\,\left (b\,d-7\,a\,g\right )}{18\,a\,b}-\frac {x^5\,\left (b\,e-4\,a\,h\right )}{9\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \]
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