Optimal. Leaf size=276 \[ -\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} b^{2/3}}-\frac {(b c-a f) \log (x)}{a^2}-\frac {\left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} b^{2/3}}+\frac {\left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} b^{2/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2} \]
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Rubi [A]
time = 0.30, antiderivative size = 274, normalized size of antiderivative = 0.99, number of steps
used = 10, number of rules used = 9, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {1848, 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^{4/3} (-h)+\sqrt [3]{a} b e-a \sqrt [3]{b} g+b^{4/3} d\right )}{\sqrt {3} a^{5/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}-a g+b d\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-a g)-\sqrt [3]{a} (b e-a h)\right )}{3 a^{5/3} b^{2/3}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac {\log (x) (b c-a f)}{a^2}-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1848
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}{x^4 \left (a+b x^3\right )} \, dx &=\int \left (\frac {c}{a x^4}+\frac {d}{a x^3}+\frac {e}{a x^2}+\frac {-b c+a f}{a^2 x}+\frac {-a (b d-a g)-a (b e-a h) x+b (b c-a f) x^2}{a^2 \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}+\frac {\int \frac {-a (b d-a g)-a (b e-a h) x+b (b c-a f) x^2}{a+b x^3} \, dx}{a^2}\\ &=-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}+\frac {\int \frac {-a (b d-a g)-a (b e-a h) x}{a+b x^3} \, dx}{a^2}+\frac {(b (b c-a f)) \int \frac {x^2}{a+b x^3} \, dx}{a^2}\\ &=-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}+\frac {\int \frac {\sqrt [3]{a} \left (-2 a \sqrt [3]{b} (b d-a g)-a^{4/3} (b e-a h)\right )+\sqrt [3]{b} \left (a \sqrt [3]{b} (b d-a g)-a^{4/3} (b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{8/3} \sqrt [3]{b}}-\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{5/3}}\\ &=-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}-\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt [3]{b}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{4/3} \sqrt [3]{b}}+\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\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}{6 a^{5/3} \sqrt [3]{b}}\\ &=-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {(b c-a f) \log (x)}{a^2}-\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt [3]{b}}+\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt [3]{b}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-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 )}{a^{5/3} b^{2/3}}\\ &=-\frac {c}{3 a x^3}-\frac {d}{2 a x^2}-\frac {e}{a x}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e-a \sqrt [3]{b} g-a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} b^{2/3}}-\frac {(b c-a f) \log (x)}{a^2}-\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt [3]{b}}+\frac {\left (b d-a g-\frac {\sqrt [3]{a} (b e-a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt [3]{b}}+\frac {(b c-a f) \log \left (a+b x^3\right )}{3 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 264, normalized size = 0.96 \begin {gather*} -\frac {\frac {2 a c}{x^3}+\frac {3 a d}{x^2}+\frac {6 a e}{x}+\frac {2 \sqrt {3} \sqrt [3]{a} \left (-b^{4/3} d-\sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+6 (b c-a f) \log (x)+\frac {2 \sqrt [3]{a} \left (b^{4/3} d-\sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {\sqrt [3]{a} \left (b^{4/3} d-\sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}-2 (b c-a f) \log \left (a+b x^3\right )}{6 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 276, normalized size = 1.00
method | result | size |
default | \(\frac {\left (a^{2} g -a b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a^{2} h -a b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (-a b f +b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{a^{2}}-\frac {e}{a x}-\frac {c}{3 a \,x^{3}}-\frac {d}{2 a \,x^{2}}+\frac {\left (a f -b c \right ) \ln \left (x \right )}{a^{2}}\) | \(276\) |
risch | \(\frac {-\frac {e \,x^{2}}{a}-\frac {x d}{2 a}-\frac {c}{3 a}}{x^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{6} b^{2} \textit {\_Z}^{3}+\left (3 a^{5} b^{2} f -3 a^{4} b^{3} c \right ) \textit {\_Z}^{2}+\left (3 a^{5} b g h -3 a^{4} b^{2} d h -3 a^{4} b^{2} e g +3 a^{4} b^{2} f^{2}-6 a^{3} b^{3} c f +3 a^{3} b^{3} d e +3 a^{2} b^{4} c^{2}\right ) \textit {\_Z} +a^{5} h^{3}-3 a^{4} b e \,h^{2}+3 a^{4} b f g h -a^{4} b \,g^{3}-3 a^{3} b^{2} c g h -3 a^{3} b^{2} d f h +3 a^{3} b^{2} d \,g^{2}+3 a^{3} b^{2} e^{2} h -3 a^{3} b^{2} e f g +a^{3} b^{2} f^{3}+3 a^{2} b^{3} c d h +3 a^{2} b^{3} c e g -3 a^{2} b^{3} c \,f^{2}-3 a^{2} b^{3} d^{2} g +3 a^{2} b^{3} d e f -a^{2} b^{3} e^{3}+3 a \,b^{4} c^{2} f -3 a \,b^{4} c d e +a \,b^{4} d^{3}-b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{5} b^{2}+\left (-8 a^{4} b^{2} f +8 c \,a^{3} b^{3}\right ) \textit {\_R}^{2}+\left (-10 a^{4} b g h +10 a^{3} b^{2} d h +10 a^{3} b^{2} e g -4 a^{3} b^{2} f^{2}+8 a^{2} b^{3} c f -10 a^{2} b^{3} d e -4 a \,b^{4} c^{2}\right ) \textit {\_R} -3 a^{4} h^{3}+9 a^{3} b e \,h^{2}-6 a^{3} b f g h +3 a^{3} b \,g^{3}+6 a^{2} b^{2} c g h +6 a^{2} b^{2} d f h -9 a^{2} b^{2} d \,g^{2}-9 a^{2} b^{2} e^{2} h +6 a^{2} b^{2} e f g -6 a \,b^{3} c d h -6 a \,b^{3} c e g +9 a \,b^{3} d^{2} g -6 a \,b^{3} d e f +3 a \,b^{3} e^{3}+6 b^{4} c d e -3 b^{4} d^{3}\right ) x +\left (a^{5} b h -e \,a^{4} b^{2}\right ) \textit {\_R}^{2}+\left (-2 a^{4} b f h -a^{4} b \,g^{2}+2 a^{3} b^{2} c h +2 a^{3} b^{2} d g +2 a^{3} b^{2} e f -2 a^{2} b^{3} c e -a^{2} b^{3} d^{2}\right ) \textit {\_R} -3 a^{3} b \,f^{2} h +3 a^{3} b f \,g^{2}+6 a^{2} b^{2} c f h -3 a^{2} b^{2} c \,g^{2}-6 a^{2} b^{2} d f g +3 a^{2} b^{2} e \,f^{2}-3 a \,b^{3} c^{2} h +6 a \,b^{3} c d g -6 a \,b^{3} c e f +3 a \,b^{3} d^{2} f +3 b^{4} c^{2} e -3 b^{4} c \,d^{2}\right )\right )}{3}+\frac {\ln \left (-x \right ) f}{a}-\frac {\ln \left (-x \right ) b c}{a^{2}}\) | \(847\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 306, normalized size = 1.11 \begin {gather*} -\frac {{\left (b c - a f\right )} \log \left (x\right )}{a^{2}} + \frac {\sqrt {3} {\left (a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b \left (\frac {a}{b}\right )^{\frac {2}{3}} e - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} g \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 )}{3 \, a^{3}} + \frac {{\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e + a b d - a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - a b d + a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {6 \, x^{2} e + 3 \, d x + 2 \, c}{6 \, a x^{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 = 64.23, size = 15204, normalized size = 55.09 \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 = 0.54, size = 291, normalized size = 1.05 \begin {gather*} \frac {\sqrt {3} {\left (b^{2} d - a b g + \left (-a b^{2}\right )^{\frac {1}{3}} a h - \left (-a b^{2}\right )^{\frac {1}{3}} b 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 )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} + \frac {{\left (b^{2} d - a b g - \left (-a b^{2}\right )^{\frac {1}{3}} a h + \left (-a b^{2}\right )^{\frac {1}{3}} 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 )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} + \frac {{\left (b c - a f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} - \frac {{\left (b c - a f\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {{\left (a^{4} b h \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e - a^{3} b^{2} d + a^{4} b g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{5} b} - \frac {6 \, a x^{2} e + 3 \, a d x + 2 \, a c}{6 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.87, size = 1842, normalized size = 6.67 \begin {gather*} \left (\sum _{k=1}^3\ln \left (-\frac {h\,a^3\,b^2\,f^2-a^3\,b^2\,f\,g^2-2\,h\,a^2\,b^3\,c\,f+a^2\,b^3\,c\,g^2+2\,a^2\,b^3\,d\,f\,g-e\,a^2\,b^3\,f^2+h\,a\,b^4\,c^2-2\,a\,b^4\,c\,d\,g+2\,e\,a\,b^4\,c\,f-a\,b^4\,d^2\,f-e\,b^5\,c^2+b^5\,c\,d^2}{a^3}-\mathrm {root}\left (27\,a^6\,b^2\,z^3+27\,a^5\,b^2\,f\,z^2-27\,a^4\,b^3\,c\,z^2+9\,a^5\,b\,g\,h\,z-9\,a^4\,b^2\,e\,g\,z-9\,a^4\,b^2\,d\,h\,z-18\,a^3\,b^3\,c\,f\,z+9\,a^3\,b^3\,d\,e\,z+9\,a^4\,b^2\,f^2\,z+9\,a^2\,b^4\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3,z,k\right )\,\left (\frac {a^2\,b^4\,d^2+a^4\,b^2\,g^2+2\,a^2\,b^4\,c\,e-2\,a^3\,b^3\,c\,h-2\,a^3\,b^3\,d\,g-2\,a^3\,b^3\,e\,f+2\,a^4\,b^2\,f\,h}{a^3}+\mathrm {root}\left (27\,a^6\,b^2\,z^3+27\,a^5\,b^2\,f\,z^2-27\,a^4\,b^3\,c\,z^2+9\,a^5\,b\,g\,h\,z-9\,a^4\,b^2\,e\,g\,z-9\,a^4\,b^2\,d\,h\,z-18\,a^3\,b^3\,c\,f\,z+9\,a^3\,b^3\,d\,e\,z+9\,a^4\,b^2\,f^2\,z+9\,a^2\,b^4\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3,z,k\right )\,\left (\frac {3\,a^4\,b^3\,e-3\,a^5\,b^2\,h}{a^3}-\frac {x\,\left (24\,a^3\,b^4\,c-24\,a^4\,b^3\,f\right )}{a^3}+\mathrm {root}\left (27\,a^6\,b^2\,z^3+27\,a^5\,b^2\,f\,z^2-27\,a^4\,b^3\,c\,z^2+9\,a^5\,b\,g\,h\,z-9\,a^4\,b^2\,e\,g\,z-9\,a^4\,b^2\,d\,h\,z-18\,a^3\,b^3\,c\,f\,z+9\,a^3\,b^3\,d\,e\,z+9\,a^4\,b^2\,f^2\,z+9\,a^2\,b^4\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3,z,k\right )\,a^2\,b^3\,x\,36\right )+\frac {x\,\left (4\,a\,b^5\,c^2+4\,a^3\,b^3\,f^2-8\,a^2\,b^4\,c\,f+10\,a^2\,b^4\,d\,e-10\,a^3\,b^3\,d\,h-10\,a^3\,b^3\,e\,g+10\,a^4\,b^2\,g\,h\right )}{a^3}\right )-\frac {x\,\left (a^4\,b\,h^3-3\,a^3\,b^2\,e\,h^2-a^3\,b^2\,g^3+2\,f\,a^3\,b^2\,g\,h+3\,a^2\,b^3\,d\,g^2-2\,f\,a^2\,b^3\,d\,h+3\,a^2\,b^3\,e^2\,h-2\,f\,a^2\,b^3\,e\,g-2\,c\,a^2\,b^3\,g\,h-3\,a\,b^4\,d^2\,g+2\,f\,a\,b^4\,d\,e+2\,c\,a\,b^4\,d\,h-a\,b^4\,e^3+2\,c\,a\,b^4\,e\,g+b^5\,d^3-2\,c\,b^5\,d\,e\right )}{a^3}\right )\,\mathrm {root}\left (27\,a^6\,b^2\,z^3+27\,a^5\,b^2\,f\,z^2-27\,a^4\,b^3\,c\,z^2+9\,a^5\,b\,g\,h\,z-9\,a^4\,b^2\,e\,g\,z-9\,a^4\,b^2\,d\,h\,z-18\,a^3\,b^3\,c\,f\,z+9\,a^3\,b^3\,d\,e\,z+9\,a^4\,b^2\,f^2\,z+9\,a^2\,b^4\,c^2\,z+3\,a^4\,b\,f\,g\,h-3\,a\,b^4\,c\,d\,e-3\,a^3\,b^2\,e\,f\,g-3\,a^3\,b^2\,d\,f\,h-3\,a^3\,b^2\,c\,g\,h+3\,a^2\,b^3\,d\,e\,f+3\,a^2\,b^3\,c\,e\,g+3\,a^2\,b^3\,c\,d\,h-3\,a^4\,b\,e\,h^2+3\,a\,b^4\,c^2\,f+3\,a^3\,b^2\,e^2\,h+3\,a^3\,b^2\,d\,g^2-3\,a^2\,b^3\,d^2\,g-3\,a^2\,b^3\,c\,f^2-a^2\,b^3\,e^3-a^4\,b\,g^3-b^5\,c^3+a^3\,b^2\,f^3+a\,b^4\,d^3+a^5\,h^3,z,k\right )\right )-\frac {\frac {c}{3\,a}+\frac {e\,x^2}{a}+\frac {d\,x}{2\,a}}{x^3}-\frac {\ln \left (x\right )\,\left (b\,c-a\,f\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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