Integrand size = 36, antiderivative size = 323 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{7/3}}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{7/3}}+\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{7/3}} \]
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Time = 0.32 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1842, 1872, 1874, 31, 648, 631, 210, 642} \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e+2 a^{5/3} h+a b^{2/3} f+2 b^{5/3} c\right )}{9 \sqrt {3} a^{7/3} b^{7/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{54 a^{7/3} b^{7/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (a f+2 b c)-a^{2/3} (2 a h+b e)\right )}{27 a^{7/3} b^{7/3}}+\frac {x \left (2 b x (a f+2 b c)+3 b x^2 (a g+b d)+a (b e-7 a h)\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a b^2 \left (a+b x^3\right )^2} \]
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Rule 31
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1842
Rule 1872
Rule 1874
Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-a (b e-a h)-2 b (2 b c+a f) x-3 b (b d+a g) x^2-6 a b h x^3}{\left (a+b x^3\right )^2} \, dx}{6 a b^2} \\ & = -\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\int \frac {2 a b (b e+2 a h)+2 b^2 (2 b c+a f) x}{a+b x^3} \, dx}{18 a^2 b^3} \\ & = -\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{a} b^2 (2 b c+a f)+4 a b^{4/3} (b e+2 a h)\right )+\sqrt [3]{b} \left (2 \sqrt [3]{a} b^2 (2 b c+a f)-2 a b^{4/3} (b e+2 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{8/3} b^{10/3}}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} b^2} \\ & = -\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{7/3}}+\frac {\left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^2 b^2}+\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 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^{7/3} b^{7/3}} \\ & = -\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{7/3}}+\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{7/3}}+\frac {\left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} 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^{7/3} b^{7/3}} \\ & = -\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a b^2 \left (a+b x^3\right )^2}+\frac {x \left (a (b e-7 a h)+2 b (2 b c+a f) x+3 b (b d+a g) x^2\right )}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{7/3}}-\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{7/3}}+\frac {\left (b^{2/3} (2 b c+a f)-a^{2/3} (b e+2 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{7/3}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\frac {-\frac {3 \sqrt [3]{a} \sqrt [3]{b} \left (-4 b^2 c x^2-a b x (e+2 f x)+a^2 (6 g+7 h x)\right )}{a+b x^3}+\frac {9 a^{4/3} \sqrt [3]{b} \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{\left (a+b x^3\right )^2}-2 \sqrt {3} \left (2 b^{5/3} c+a^{2/3} b e+a b^{2/3} f+2 a^{5/3} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \left (-2 b^{5/3} c+a^{2/3} b e-a b^{2/3} f+2 a^{5/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\left (2 b^{5/3} c-a^{2/3} b e+a b^{2/3} f-2 a^{5/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{7/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.67 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.49
method | result | size |
risch | \(\frac {\frac {\left (a f +2 b c \right ) x^{5}}{9 a^{2}}-\frac {\left (7 a h -b e \right ) x^{4}}{18 a b}-\frac {g \,x^{3}}{3 b}-\frac {\left (a f -7 b c \right ) x^{2}}{18 a b}-\frac {\left (2 a h +b e \right ) x}{9 b^{2}}-\frac {a g +b d}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\frac {\left (a f +2 b c \right ) \textit {\_R}}{a}+\frac {2 a h +b e}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 a \,b^{2}}\) | \(158\) |
default | \(\frac {\frac {\left (a f +2 b c \right ) x^{5}}{9 a^{2}}-\frac {\left (7 a h -b e \right ) x^{4}}{18 a b}-\frac {g \,x^{3}}{3 b}-\frac {\left (a f -7 b c \right ) x^{2}}{18 a b}-\frac {\left (2 a h +b e \right ) x}{9 b^{2}}-\frac {a g +b d}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (2 a^{2} h +a e b \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a f b +2 b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a^{2} b^{2}}\) | \(316\) |
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Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 7190, normalized size of antiderivative = 22.26 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.07 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {6 \, a^{2} b g x^{3} - 2 \, {\left (2 \, b^{3} c + a b^{2} f\right )} x^{5} - {\left (a b^{2} e - 7 \, a^{2} b h\right )} x^{4} + 3 \, a^{2} b d + 3 \, a^{3} g - {\left (7 \, a b^{2} c - a^{2} b f\right )} x^{2} + 2 \, {\left (a^{2} b e + 2 \, a^{3} h\right )} x}{18 \, {\left (a^{2} b^{4} x^{6} + 2 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b e + 2 \, a^{2} h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b e - 2 \, a^{2} h\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b e - 2 \, a^{2} h\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.28 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.04 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (a b e + 2 \, a^{2} h - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (a b e + 2 \, a^{2} h + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c + \left (-a b^{2}\right )^{\frac {1}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (2 \, b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b e + 2 \, a^{2} h\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} b^{2}} + \frac {4 \, b^{3} c x^{5} + 2 \, a b^{2} f x^{5} + a b^{2} e x^{4} - 7 \, a^{2} b h x^{4} - 6 \, a^{2} b g x^{3} + 7 \, a b^{2} c x^{2} - a^{2} b f x^{2} - 2 \, a^{2} b e x - 4 \, a^{3} h x - 3 \, a^{2} b d - 3 \, a^{3} g}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \]
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Time = 9.43 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.98 \[ \int \frac {x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^3} \, dx=\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\,a\,b^2\,9+\frac {x\,\left (54\,h\,a^4\,b+27\,e\,a^3\,b^2\right )}{81\,a^4\,b}\right )+\frac {2\,b^2\,c\,e+2\,a^2\,f\,h+4\,a\,b\,c\,h+a\,b\,e\,f}{81\,a^3\,b^2}+\frac {x\,\left (a^2\,f^2+4\,a\,b\,c\,f+4\,b^2\,c^2\right )}{81\,a^4\,b}\right )\,\mathrm {root}\left (19683\,a^7\,b^7\,z^3+162\,a^5\,b^3\,f\,h\,z+324\,a^4\,b^4\,c\,h\,z+81\,a^4\,b^4\,e\,f\,z+162\,a^3\,b^5\,c\,e\,z-12\,a^4\,b\,e\,h^2+12\,a\,b^4\,c^2\,f-6\,a^3\,b^2\,e^2\,h+6\,a^2\,b^3\,c\,f^2+a^3\,b^2\,f^3-8\,a^5\,h^3+8\,b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\right )-\frac {\frac {b\,d+a\,g}{6\,b^2}+\frac {x\,\left (b\,e+2\,a\,h\right )}{9\,b^2}+\frac {g\,x^3}{3\,b}-\frac {x^5\,\left (2\,b\,c+a\,f\right )}{9\,a^2}-\frac {x^2\,\left (7\,b\,c-a\,f\right )}{18\,a\,b}-\frac {x^4\,\left (b\,e-7\,a\,h\right )}{18\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \]
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