Optimal. Leaf size=276 \[ \frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a b \left (a+b x^3\right )}-\frac {\left (2 b^{4/3} c+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{5/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^2} \]
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Rubi [A]
time = 0.25, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1872, 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 (2 a^{4/3} g+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 b^{4/3} c\right )}{3 \sqrt {3} a^{5/3} 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+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{18 a^{5/3} b^{5/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a f+2 b c)-\sqrt [3]{a} (2 a g+b d)\right )}{9 a^{5/3} b^{5/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^2}+\frac {x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a b \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1872
Rule 1874
Rule 1885
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 )^2} \, dx &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a b \left (a+b x^3\right )}-\frac {\int \frac {-b (2 b c+a f)-b (b d+2 a g) x-3 a b h x^2}{a+b x^3} \, dx}{3 a b^2}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a b \left (a+b x^3\right )}-\frac {\int \frac {-b (2 b c+a f)-b (b d+2 a g) x}{a+b x^3} \, dx}{3 a b^2}+\frac {h \int \frac {x^2}{a+b x^3} \, dx}{b}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a b \left (a+b x^3\right )}+\frac {h \log \left (a+b x^3\right )}{3 b^2}-\frac {\int \frac {\sqrt [3]{a} \left (-2 b^{4/3} (2 b c+a f)-\sqrt [3]{a} b (b d+2 a g)\right )+\sqrt [3]{b} \left (b^{4/3} (2 b c+a f)-\sqrt [3]{a} b (b d+2 a g)\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 (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^{4/3}}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a b \left (a+b x^3\right )}+\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{5/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (2 b^{4/3} c+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 a^{4/3} g\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3} b^{4/3}}-\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 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}{18 a^{5/3} b^{5/3}}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a b \left (a+b x^3\right )}+\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{5/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (2 b^{4/3} c+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 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 )}{3 a^{5/3} b^{5/3}}\\ &=\frac {x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a b \left (a+b x^3\right )}-\frac {\left (2 b^{4/3} c+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 a^{4/3} g\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{5/3}}-\frac {\left (\sqrt [3]{b} (2 b c+a f)-\sqrt [3]{a} (b d+2 a g)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{5/3}}+\frac {h \log \left (a+b x^3\right )}{3 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 268, normalized size = 0.97 \begin {gather*} \frac {\frac {6 \left (a^2 h+b^2 x (c+d x)-a b (e+x (f+g x))\right )}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \sqrt [3]{b} \left (2 b^{4/3} c+\sqrt [3]{a} b d+a \sqrt [3]{b} f+2 a^{4/3} g\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \sqrt [3]{b} \left (2 b^{4/3} c-\sqrt [3]{a} b d+a \sqrt [3]{b} f-2 a^{4/3} g\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {\sqrt [3]{b} \left (-2 b^{4/3} c+\sqrt [3]{a} b d-a \sqrt [3]{b} f+2 a^{4/3} g\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+6 h \log \left (a+b x^3\right )}{18 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 283, normalized size = 1.03
method | result | size |
risch | \(\frac {-\frac {\left (a g -b d \right ) x^{2}}{3 a b}-\frac {\left (a f -b c \right ) x}{3 a b}+\frac {a h -b e}{3 b^{2}}}{b \,x^{3}+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (3 h \,\textit {\_R}^{2}+\frac {\left (2 a g +b d \right ) \textit {\_R}}{a}+\frac {a f +2 b c}{a}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{2}}\) | \(120\) |
default | \(\frac {-\frac {\left (a g -b d \right ) x^{2}}{3 a b}-\frac {\left (a f -b c \right ) x}{3 a b}+\frac {a h -b e}{3 b^{2}}}{b \,x^{3}+a}+\frac {\left (a f +2 b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (2 a g +b d \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {a h \ln \left (b \,x^{3}+a \right )}{b}}{3 b a}\) | \(283\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 291, normalized size = 1.05 \begin {gather*} \frac {a^{2} h + {\left (b^{2} d - a b g\right )} x^{2} - a b e + {\left (b^{2} c - a b f\right )} x}{3 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {\sqrt {3} {\left (b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b f \left (\frac {a}{b}\right )^{\frac {1}{3}}\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 h \left (\frac {a}{b}\right )^{\frac {2}{3}} + b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, 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 )}{18 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (3 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - b d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, b c + a f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a 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 = 1.79, size = 12636, normalized size = 45.78 \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 = 0.50, size = 302, normalized size = 1.09 \begin {gather*} \frac {h \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac {\sqrt {3} {\left (2 \, b^{2} c + a b f - \left (-a b^{2}\right )^{\frac {1}{3}} b d - 2 \, \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 )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (2 \, b^{2} c + a b f + \left (-a b^{2}\right )^{\frac {1}{3}} b d + 2 \, \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 )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} + \frac {{\left (b d - a g\right )} x^{2} + {\left (b c - a f\right )} x + \frac {a^{2} h - a b e}{b}}{3 \, {\left (b x^{3} + a\right )} a b} - \frac {{\left (a b^{3} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{2} b^{2} g \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b^{3} c + a^{2} b^{2} f\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^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.54, size = 835, normalized size = 3.03 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\frac {\mathrm {root}\left (729\,a^5\,b^6\,z^3-729\,a^5\,b^4\,h\,z^2+54\,a^4\,b^3\,f\,g\,z+108\,a^3\,b^4\,c\,g\,z+27\,a^3\,b^4\,d\,f\,z+54\,a^2\,b^5\,c\,d\,z+243\,a^5\,b^2\,h^2\,z-18\,a^4\,b\,f\,g\,h-36\,a^3\,b^2\,c\,g\,h-9\,a^3\,b^2\,d\,f\,h-18\,a^2\,b^3\,c\,d\,h-12\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,d\,g^2+6\,a^2\,b^3\,d^2\,g-6\,a^2\,b^3\,c\,f^2+8\,a^4\,b\,g^3+a\,b^4\,d^3-27\,a^5\,h^3-8\,b^5\,c^3-a^3\,b^2\,f^3,z,k\right )\,\left (-6\,a^2\,h+\mathrm {root}\left (729\,a^5\,b^6\,z^3-729\,a^5\,b^4\,h\,z^2+54\,a^4\,b^3\,f\,g\,z+108\,a^3\,b^4\,c\,g\,z+27\,a^3\,b^4\,d\,f\,z+54\,a^2\,b^5\,c\,d\,z+243\,a^5\,b^2\,h^2\,z-18\,a^4\,b\,f\,g\,h-36\,a^3\,b^2\,c\,g\,h-9\,a^3\,b^2\,d\,f\,h-18\,a^2\,b^3\,c\,d\,h-12\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,d\,g^2+6\,a^2\,b^3\,d^2\,g-6\,a^2\,b^3\,c\,f^2+8\,a^4\,b\,g^3+a\,b^4\,d^3-27\,a^5\,h^3-8\,b^5\,c^3-a^3\,b^2\,f^3,z,k\right )\,a^2\,b^2\,9+2\,b^2\,c\,x+a\,b\,f\,x\right )}{a}+\frac {9\,a^3\,h^2+2\,b^3\,c\,d+4\,a\,b^2\,c\,g+a\,b^2\,d\,f+2\,a^2\,b\,f\,g}{9\,a^2\,b^2}+\frac {x\,\left (4\,a^2\,g^2-3\,f\,h\,a^2+4\,a\,b\,d\,g-6\,c\,h\,a\,b+b^2\,d^2\right )}{9\,a^2\,b}\right )\,\mathrm {root}\left (729\,a^5\,b^6\,z^3-729\,a^5\,b^4\,h\,z^2+54\,a^4\,b^3\,f\,g\,z+108\,a^3\,b^4\,c\,g\,z+27\,a^3\,b^4\,d\,f\,z+54\,a^2\,b^5\,c\,d\,z+243\,a^5\,b^2\,h^2\,z-18\,a^4\,b\,f\,g\,h-36\,a^3\,b^2\,c\,g\,h-9\,a^3\,b^2\,d\,f\,h-18\,a^2\,b^3\,c\,d\,h-12\,a\,b^4\,c^2\,f+12\,a^3\,b^2\,d\,g^2+6\,a^2\,b^3\,d^2\,g-6\,a^2\,b^3\,c\,f^2+8\,a^4\,b\,g^3+a\,b^4\,d^3-27\,a^5\,h^3-8\,b^5\,c^3-a^3\,b^2\,f^3,z,k\right )\right )+\frac {\frac {x\,\left (b\,c-a\,f\right )}{3\,a\,b}-\frac {b\,e-a\,h}{3\,b^2}+\frac {x^2\,\left (b\,d-a\,g\right )}{3\,a\,b}}{b\,x^3+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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