Optimal. Leaf size=313 \[ \frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a (b e+a h)-b x (5 b c+a f+2 (2 b d+a g) x)}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (5 b^{4/3} c+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{5/3}} \]
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Rubi [A]
time = 0.29, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {1872, 1868,
1874, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} g+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+5 b^{4/3} c\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{54 a^{8/3} b^{5/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+5 b c)-\sqrt [3]{a} (a g+2 b d)\right )}{27 a^{8/3} b^{5/3}}-\frac {3 a (a h+b e)-b x (2 x (a g+2 b d)+a f+5 b c)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a b \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1868
Rule 1872
Rule 1874
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^3\right )^3} \, dx &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {\int \frac {-b (5 b c+a f)-2 b (2 b d+a g) x-3 b (b e+a h) x^2}{\left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a (b e+a h)-b x (5 b c+a f+2 (2 b d+a g) x)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\int \frac {2 b (5 b c+a f)+2 b (2 b d+a g) x}{a+b x^3} \, dx}{18 a^2 b^2}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a (b e+a h)-b x (5 b c+a f+2 (2 b d+a g) x)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\int \frac {\sqrt [3]{a} \left (4 b^{4/3} (5 b c+a f)+2 \sqrt [3]{a} b (2 b d+a g)\right )+\sqrt [3]{b} \left (-2 b^{4/3} (5 b c+a f)+2 \sqrt [3]{a} b (2 b d+a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{8/3} b^{7/3}}+\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{8/3} b^{4/3}}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a (b e+a h)-b x (5 b c+a f+2 (2 b d+a g) x)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{5/3}}+\frac {\left (5 b^{4/3} c+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{7/3} b^{4/3}}-\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\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^{8/3} b^{5/3}}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a (b e+a h)-b x (5 b c+a f+2 (2 b d+a g) x)}{18 a^2 b^2 \left (a+b x^3\right )}+\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{5/3}}+\frac {\left (5 b^{4/3} c+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\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^{8/3} b^{5/3}}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{6 a b \left (a+b x^3\right )^2}-\frac {3 a (b e+a h)-b x (5 b c+a f+2 (2 b d+a g) x)}{18 a^2 b^2 \left (a+b x^3\right )}-\frac {\left (5 b^{4/3} c+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (5 b c+a f)-\sqrt [3]{a} (2 b d+a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{5/3}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 295, normalized size = 0.94 \begin {gather*} \frac {\frac {3 a^{2/3} \left (-6 a^2 h+b^2 x (5 c+4 d x)+a b x (f+2 g x)\right )}{a+b x^3}+\frac {9 a^{5/3} \left (a^2 h+b^2 x (c+d x)-a b (e+x (f+g x))\right )}{\left (a+b x^3\right )^2}-2 \sqrt {3} \sqrt [3]{b} \left (5 b^{4/3} c+2 \sqrt [3]{a} b d+a \sqrt [3]{b} f+a^{4/3} g\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{b} \left (5 b^{4/3} c-2 \sqrt [3]{a} b d+a \sqrt [3]{b} f-a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{b} \left (-5 b^{4/3} c+2 \sqrt [3]{a} b d-a \sqrt [3]{b} f+a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 309, normalized size = 0.99
method | result | size |
risch | \(\frac {\frac {\left (a g +2 b d \right ) x^{5}}{9 a^{2}}+\frac {\left (a f +5 b c \right ) x^{4}}{18 a^{2}}-\frac {h \,x^{3}}{3 b}-\frac {\left (a g -7 b d \right ) x^{2}}{18 a b}-\frac {\left (a f -4 b c \right ) x}{9 a b}-\frac {a h +b e}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\left (a g +2 b d \right ) \textit {\_R} +a f +5 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 a^{2} b^{2}}\) | \(149\) |
default | \(\frac {\frac {\left (a g +2 b d \right ) x^{5}}{9 a^{2}}+\frac {\left (a f +5 b c \right ) x^{4}}{18 a^{2}}-\frac {h \,x^{3}}{3 b}-\frac {\left (a g -7 b d \right ) x^{2}}{18 a b}-\frac {\left (a f -4 b c \right ) x}{9 a b}-\frac {a h +b e}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (a f +5 b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a g +2 b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a^{2} b}\) | \(309\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 328, normalized size = 1.05 \begin {gather*} -\frac {6 \, a^{2} b h x^{3} - 2 \, {\left (2 \, b^{3} d + a b^{2} g\right )} x^{5} - {\left (5 \, b^{3} c + a b^{2} f\right )} x^{4} + 3 \, a^{3} h + 3 \, a^{2} b e - {\left (7 \, a b^{2} d - a^{2} b g\right )} x^{2} - 2 \, {\left (4 \, a b^{2} c - a^{2} b f\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 d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, b c + 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 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, b c - 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 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, b c - a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b^{2} \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 = 1.92, size = 6984, normalized size = 22.31 \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 = 0.48, size = 330, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {3} {\left (5 \, b^{2} c + a b f - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - \left (-a b^{2}\right )^{\frac {1}{3}} 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 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (5 \, b^{2} c + a b f + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + \left (-a b^{2}\right )^{\frac {1}{3}} 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 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (2 \, b d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, b c + a f\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} + \frac {4 \, b^{3} d x^{5} + 2 \, a b^{2} g x^{5} + 5 \, b^{3} c x^{4} + a b^{2} f x^{4} - 6 \, a^{2} b h x^{3} + 7 \, a b^{2} d x^{2} - a^{2} b g x^{2} + 8 \, a b^{2} c x - 2 \, a^{2} b f x - 3 \, a^{3} h - 3 \, a^{2} b e}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 630, normalized size = 2.01 \begin {gather*} \frac {\frac {x^4\,\left (5\,b\,c+a\,f\right )}{18\,a^2}-\frac {h\,x^3}{3\,b}-\frac {b\,e+a\,h}{6\,b^2}+\frac {x^5\,\left (2\,b\,d+a\,g\right )}{9\,a^2}+\frac {x\,\left (4\,b\,c-a\,f\right )}{9\,a\,b}+\frac {x^2\,\left (7\,b\,d-a\,g\right )}{18\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^8\,b^5\,z^3+81\,a^5\,b^2\,f\,g\,z+405\,a^4\,b^3\,c\,g\,z+162\,a^4\,b^3\,d\,f\,z+810\,a^3\,b^4\,c\,d\,z+6\,a^3\,b\,d\,g^2-75\,a\,b^3\,c^2\,f+12\,a^2\,b^2\,d^2\,g-15\,a^2\,b^2\,c\,f^2+8\,a\,b^3\,d^3+a^4\,g^3-125\,b^4\,c^3-a^3\,b\,f^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^8\,b^5\,z^3+81\,a^5\,b^2\,f\,g\,z+405\,a^4\,b^3\,c\,g\,z+162\,a^4\,b^3\,d\,f\,z+810\,a^3\,b^4\,c\,d\,z+6\,a^3\,b\,d\,g^2-75\,a\,b^3\,c^2\,f+12\,a^2\,b^2\,d^2\,g-15\,a^2\,b^2\,c\,f^2+8\,a\,b^3\,d^3+a^4\,g^3-125\,b^4\,c^3-a^3\,b\,f^3,z,k\right )\,a\,b^2\,9+\frac {x\,\left (27\,f\,a^3\,b^2+135\,c\,a^2\,b^3\right )}{81\,a^4\,b}\right )+\frac {10\,b^2\,c\,d+a^2\,f\,g+5\,a\,b\,c\,g+2\,a\,b\,d\,f}{81\,a^4\,b}+\frac {x\,\left (a^2\,g^2+4\,a\,b\,d\,g+4\,b^2\,d^2\right )}{81\,a^4\,b}\right )\,\mathrm {root}\left (19683\,a^8\,b^5\,z^3+81\,a^5\,b^2\,f\,g\,z+405\,a^4\,b^3\,c\,g\,z+162\,a^4\,b^3\,d\,f\,z+810\,a^3\,b^4\,c\,d\,z+6\,a^3\,b\,d\,g^2-75\,a\,b^3\,c^2\,f+12\,a^2\,b^2\,d^2\,g-15\,a^2\,b^2\,c\,f^2+8\,a\,b^3\,d^3+a^4\,g^3-125\,b^4\,c^3-a^3\,b\,f^3,z,k\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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