Optimal. Leaf size=289 \[ \frac {h x}{b^2}-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac {\left (b^{5/3} c+a^{2/3} b e+2 a b^{2/3} f-4 a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{7/3}}-\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{7/3}}+\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{7/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^2} \]
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Rubi [A]
time = 0.34, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1842, 1901,
1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e-4 a^{5/3} h+2 a b^{2/3} f+b^{5/3} c\right )}{3 \sqrt {3} a^{4/3} 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} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{18 a^{4/3} b^{7/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{9 a^{4/3} b^{7/3}}-\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a b^2 \left (a+b x^3\right )}+\frac {g \log \left (a+b x^3\right )}{3 b^2}+\frac {h x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1842
Rule 1874
Rule 1885
Rule 1901
Rubi steps
\begin {align*} \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 )^2} \, dx &=-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac {\int \frac {-a (b e-a h)-b (b c+2 a f) x-3 a b g x^2-3 a b h x^3}{a+b x^3} \, dx}{3 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 )}{3 a b^2 \left (a+b x^3\right )}-\frac {\int \left (-3 a h-\frac {a (b e-4 a h)+b (b c+2 a f) x+3 a b g x^2}{a+b x^3}\right ) \, dx}{3 a b^2}\\ &=\frac {h x}{b^2}-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}+\frac {\int \frac {a (b e-4 a h)+b (b c+2 a f) x+3 a b g x^2}{a+b x^3} \, dx}{3 a b^2}\\ &=\frac {h x}{b^2}-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}+\frac {\int \frac {a (b e-4 a h)+b (b c+2 a f) x}{a+b x^3} \, dx}{3 a b^2}+\frac {g \int \frac {x^2}{a+b x^3} \, dx}{b}\\ &=\frac {h x}{b^2}-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}+\frac {g \log \left (a+b x^3\right )}{3 b^2}+\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{a} b (b c+2 a f)+2 a \sqrt [3]{b} (b e-4 a h)\right )+\sqrt [3]{b} \left (\sqrt [3]{a} b (b c+2 a f)-a \sqrt [3]{b} (b e-4 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{5/3} b^{7/3}}-\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{4/3} b^2}\\ &=\frac {h x}{b^2}-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{7/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (b^{5/3} c+a^{2/3} b e+2 a b^{2/3} f-4 a^{5/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a b^2}+\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 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}{18 a^{4/3} b^{7/3}}\\ &=\frac {h x}{b^2}-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{7/3}}+\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{7/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (b^{5/3} c+a^{2/3} b e+2 a b^{2/3} f-4 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 )}{3 a^{4/3} b^{7/3}}\\ &=\frac {h x}{b^2}-\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac {\left (b^{5/3} c+a^{2/3} b e+2 a b^{2/3} f-4 a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{7/3}}-\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{7/3}}+\frac {\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{7/3}}+\frac {g \log \left (a+b x^3\right )}{3 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 285, normalized size = 0.99 \begin {gather*} \frac {18 b^{2/3} h x+\frac {6 b^{2/3} \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \left (b^2 c+a^{2/3} b^{4/3} e+2 a b f-4 a^{5/3} \sqrt [3]{b} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-\frac {2 \left (b^2 c-a^{2/3} b^{4/3} e+2 a b f+4 a^{5/3} \sqrt [3]{b} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}+\frac {\left (b^2 c-a^{2/3} b^{4/3} e+2 a b f+4 a^{5/3} \sqrt [3]{b} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}+6 b^{2/3} g \log \left (a+b x^3\right )}{18 b^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 285, normalized size = 0.99
method | result | size |
risch | \(\frac {h x}{b^{2}}+\frac {-\frac {b \left (a f -b c \right ) x^{2}}{3 a}+\left (\frac {a h}{3}-\frac {b e}{3}\right ) x +\frac {a g}{3}-\frac {b d}{3}}{b^{2} \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (3 b g \,\textit {\_R}^{2}+\frac {b \left (2 a f +b c \right ) \textit {\_R}}{a}-4 a h +b e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{3}}\) | \(113\) |
default | \(\frac {h x}{b^{2}}-\frac {\frac {\frac {b \left (a f -b c \right ) x^{2}}{3 a}+\left (-\frac {a h}{3}+\frac {b e}{3}\right ) x -\frac {a g}{3}+\frac {b d}{3}}{b \,x^{3}+a}+\frac {\left (4 a^{2} h -a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (-2 a b f -b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-a g \ln \left (b \,x^{3}+a \right )}{3 a}}{b^{2}}\) | \(285\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 316, normalized size = 1.09 \begin {gather*} -\frac {a b d - a^{2} g - {\left (b^{2} c - a b f\right )} x^{2} - {\left (a^{2} h - a b e\right )} x}{3 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {h x}{b^{2}} + \frac {\sqrt {3} {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e\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 )}{9 \, a^{2} b^{2}} + \frac {{\left (6 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + 4 \, a^{2} h - a b e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (3 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a^{2} h + a b e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.91, size = 12617, normalized size = 43.66 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 318, normalized size = 1.10 \begin {gather*} \frac {h x}{b^{2}} + \frac {g \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac {\sqrt {3} {\left (4 \, a^{2} h - a b e + \left (-a b^{2}\right )^{\frac {1}{3}} b c + 2 \, \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 )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} + \frac {{\left (4 \, a^{2} h - a b e - \left (-a b^{2}\right )^{\frac {1}{3}} b c - 2 \, \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 )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {a b d - a^{2} g - {\left (b^{2} c - a b f\right )} x^{2} - {\left (a^{2} h - a b e\right )} x}{3 \, {\left (b x^{3} + a\right )} a b^{2}} - \frac {{\left (a b^{5} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} b^{4} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a^{3} b^{3} h + a^{2} b^{4} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.39, size = 827, normalized size = 2.86 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\frac {9\,a^2\,g^2+b^2\,c\,e-8\,a^2\,f\,h-4\,a\,b\,c\,h+2\,a\,b\,e\,f}{9\,a\,b^2}-\mathrm {root}\left (729\,a^4\,b^7\,z^3-729\,a^4\,b^5\,g\,z^2-216\,a^4\,b^3\,f\,h\,z-108\,a^3\,b^4\,c\,h\,z+54\,a^3\,b^4\,e\,f\,z+27\,a^2\,b^5\,c\,e\,z+243\,a^4\,b^3\,g^2\,z+72\,a^4\,b\,f\,g\,h+36\,a^3\,b^2\,c\,g\,h-18\,a^3\,b^2\,e\,f\,g-9\,a^2\,b^3\,c\,e\,g-48\,a^4\,b\,e\,h^2+6\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,e^2\,h+12\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-27\,a^4\,b\,g^3+64\,a^5\,h^3+b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\,\left (6\,a\,g-b\,e\,x+4\,a\,h\,x-\mathrm {root}\left (729\,a^4\,b^7\,z^3-729\,a^4\,b^5\,g\,z^2-216\,a^4\,b^3\,f\,h\,z-108\,a^3\,b^4\,c\,h\,z+54\,a^3\,b^4\,e\,f\,z+27\,a^2\,b^5\,c\,e\,z+243\,a^4\,b^3\,g^2\,z+72\,a^4\,b\,f\,g\,h+36\,a^3\,b^2\,c\,g\,h-18\,a^3\,b^2\,e\,f\,g-9\,a^2\,b^3\,c\,e\,g-48\,a^4\,b\,e\,h^2+6\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,e^2\,h+12\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-27\,a^4\,b\,g^3+64\,a^5\,h^3+b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\,a\,b^2\,9\right )+\frac {x\,\left (12\,g\,h\,a^3+4\,a^2\,b\,f^2-3\,e\,g\,a^2\,b+4\,a\,b^2\,c\,f+b^3\,c^2\right )}{9\,a^2\,b^2}\right )\,\mathrm {root}\left (729\,a^4\,b^7\,z^3-729\,a^4\,b^5\,g\,z^2-216\,a^4\,b^3\,f\,h\,z-108\,a^3\,b^4\,c\,h\,z+54\,a^3\,b^4\,e\,f\,z+27\,a^2\,b^5\,c\,e\,z+243\,a^4\,b^3\,g^2\,z+72\,a^4\,b\,f\,g\,h+36\,a^3\,b^2\,c\,g\,h-18\,a^3\,b^2\,e\,f\,g-9\,a^2\,b^3\,c\,e\,g-48\,a^4\,b\,e\,h^2+6\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,e^2\,h+12\,a^2\,b^3\,c\,f^2+8\,a^3\,b^2\,f^3-27\,a^4\,b\,g^3+64\,a^5\,h^3+b^5\,c^3-a^2\,b^3\,e^3,z,k\right )\right )-\frac {\frac {b\,d}{3}-\frac {a\,g}{3}+x\,\left (\frac {b\,e}{3}-\frac {a\,h}{3}\right )-\frac {b\,x^2\,\left (b\,c-a\,f\right )}{3\,a}}{b^3\,x^3+a\,b^2}+\frac {h\,x}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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