Optimal. Leaf size=337 \[ \frac {(b e-2 a h) x}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 b^2}+\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 b^3 \left (a+b x^3\right )}-\frac {\left (2 b^{5/3} c-4 a^{2/3} b e-5 a b^{2/3} f+7 a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{10/3}}-\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{10/3}}+\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{10/3}}+\frac {(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3} \]
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Rubi [A]
time = 0.46, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, 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 (-4 a^{2/3} b e+7 a^{5/3} h-5 a b^{2/3} f+2 b^{5/3} c\right )}{3 \sqrt {3} \sqrt [3]{a} b^{10/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{18 \sqrt [3]{a} b^{10/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (4 b e-7 a h)+b^{2/3} (2 b c-5 a f)\right )}{9 \sqrt [3]{a} b^{10/3}}+\frac {x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 b^3 \left (a+b x^3\right )}+\frac {(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}+\frac {x (b e-2 a h)}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 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^4 \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 b^3 \left (a+b x^3\right )}-\frac {\int \frac {a^2 (b e-a h)-2 a b (b c-a f) x-3 a b (b d-a g) x^2-3 a b (b e-a h) x^3-3 a b^2 f x^4-3 a b^2 g x^5-3 a b^2 h x^6}{a+b x^3} \, dx}{3 a b^3}\\ &=\frac {x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 b^3 \left (a+b x^3\right )}-\frac {\int \left (-3 a (b e-2 a h)-3 a b f x-3 a b g x^2-3 a b h x^3+\frac {a^2 (4 b e-7 a h)-a b (2 b c-5 a f) x-3 a b (b d-2 a g) x^2}{a+b x^3}\right ) \, dx}{3 a b^3}\\ &=\frac {(b e-2 a h) x}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 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 b^3 \left (a+b x^3\right )}-\frac {\int \frac {a^2 (4 b e-7 a h)-a b (2 b c-5 a f) x-3 a b (b d-2 a g) x^2}{a+b x^3} \, dx}{3 a b^3}\\ &=\frac {(b e-2 a h) x}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 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 b^3 \left (a+b x^3\right )}-\frac {\int \frac {a^2 (4 b e-7 a h)-a b (2 b c-5 a f) x}{a+b x^3} \, dx}{3 a b^3}+\frac {(b d-2 a g) \int \frac {x^2}{a+b x^3} \, dx}{b^2}\\ &=\frac {(b e-2 a h) x}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 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 b^3 \left (a+b x^3\right )}+\frac {(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}-\frac {\int \frac {\sqrt [3]{a} \left (-a^{4/3} b (2 b c-5 a f)+2 a^2 \sqrt [3]{b} (4 b e-7 a h)\right )+\sqrt [3]{b} \left (-a^{4/3} b (2 b c-5 a f)-a^2 \sqrt [3]{b} (4 b e-7 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^{10/3}}-\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^3}\\ &=\frac {(b e-2 a h) x}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 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 b^3 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{10/3}}+\frac {(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}+\frac {\left (2 b^{5/3} c-4 a^{2/3} b e-5 a b^{2/3} f+7 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 b^3}+\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 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 \sqrt [3]{a} b^{10/3}}\\ &=\frac {(b e-2 a h) x}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 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 b^3 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{10/3}}+\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{10/3}}+\frac {(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}+\frac {\left (2 b^{5/3} c-4 a^{2/3} b e-5 a b^{2/3} f+7 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 \sqrt [3]{a} b^{10/3}}\\ &=\frac {(b e-2 a h) x}{b^3}+\frac {f x^2}{2 b^2}+\frac {g x^3}{3 b^2}+\frac {h x^4}{4 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 b^3 \left (a+b x^3\right )}-\frac {\left (2 b^{5/3} c-4 a^{2/3} b e-5 a b^{2/3} f+7 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} \sqrt [3]{a} b^{10/3}}-\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{10/3}}+\frac {\left (b^{2/3} (2 b c-5 a f)+a^{2/3} (4 b e-7 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{10/3}}+\frac {(b d-2 a g) \log \left (a+b x^3\right )}{3 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 334, normalized size = 0.99 \begin {gather*} \frac {36 b^{2/3} (b e-2 a h) x+18 b^{5/3} f x^2+12 b^{5/3} g x^3+9 b^{5/3} h x^4-\frac {12 b^{2/3} \left (b^2 c x^2+a^2 (g+h x)-a b (d+x (e+f x))\right )}{a+b x^3}-\frac {4 \sqrt {3} \left (2 b^2 c-4 a^{2/3} b^{4/3} e-5 a b f+7 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 )}{\sqrt [3]{a}}+\frac {4 \left (-2 b^2 c-4 a^{2/3} b^{4/3} e+5 a b f+7 a^{5/3} \sqrt [3]{b} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {2 \left (2 b^2 c+4 a^{2/3} b^{4/3} e-5 a b f-7 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 )}{\sqrt [3]{a}}+12 b^{2/3} (b d-2 a g) \log \left (a+b x^3\right )}{36 b^{11/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 329, normalized size = 0.98
method | result | size |
risch | \(\frac {h \,x^{4}}{4 b^{2}}+\frac {g \,x^{3}}{3 b^{2}}+\frac {f \,x^{2}}{2 b^{2}}-\frac {2 a h x}{b^{3}}+\frac {e x}{b^{2}}+\frac {\left (\frac {1}{3} a b f -\frac {1}{3} b^{2} c \right ) x^{2}+\left (-\frac {1}{3} a^{2} h +\frac {1}{3} a b e \right ) x -\frac {a \left (a g -b d \right )}{3}}{b^{3} \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (3 b \left (-2 a g +b d \right ) \textit {\_R}^{2}+b \left (-5 a f +2 b c \right ) \textit {\_R} +7 a^{2} h -4 a b e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{4}}\) | \(162\) |
default | \(-\frac {-\frac {1}{4} b h \,x^{4}-\frac {1}{3} b g \,x^{3}-\frac {1}{2} b f \,x^{2}+2 a h x -b e x}{b^{3}}+\frac {\frac {\left (\frac {1}{3} a b f -\frac {1}{3} b^{2} c \right ) x^{2}+\left (-\frac {1}{3} a^{2} h +\frac {1}{3} a b e \right ) x -\frac {a \left (a g -b d \right )}{3}}{b \,x^{3}+a}+\frac {\left (7 a^{2} h -4 a b e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {\left (-5 a b f +2 b^{2} c \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {\left (-6 a b g +3 b^{2} d \right ) \ln \left (b \,x^{3}+a \right )}{9 b}}{b^{3}}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 371, normalized size = 1.10 \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 (b^{4} x^{3} + a b^{3}\right )}} + \frac {\sqrt {3} {\left (2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 5 \, a b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + 7 \, a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, 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 b^{3}} + \frac {3 \, b h x^{4} + 4 \, b g x^{3} + 6 \, b f x^{2} - 12 \, {\left (2 \, a h - b e\right )} x}{12 \, b^{3}} + \frac {{\left (6 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 12 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, a^{2} h + 4 \, 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 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (3 \, b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 6 \, a b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + 7 \, a^{2} h - 4 \, a b e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{4} \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 = 2.10, size = 16147, normalized size = 47.91 \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 = 0.56, size = 357, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {3} {\left (7 \, a^{2} h - 4 \, a b e - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c + 5 \, \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}} b^{2}} - \frac {{\left (7 \, a^{2} h - 4 \, a b e + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - 5 \, \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}} b^{2}} + \frac {{\left (b d - 2 \, a g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \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 )} b^{3}} - \frac {{\left (2 \, b^{6} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b^{5} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 7 \, a^{2} b^{4} h - 4 \, a b^{5} 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 b^{7}} + \frac {3 \, b^{6} h x^{4} + 4 \, b^{6} g x^{3} + 6 \, b^{6} f x^{2} - 24 \, a b^{5} h x + 12 \, b^{6} x e}{12 \, b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.11, size = 1241, normalized size = 3.68 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (729\,a\,b^{10}\,z^3-729\,a\,b^8\,d\,z^2+1458\,a^2\,b^7\,g\,z^2-216\,a\,b^6\,c\,e\,z-945\,a^3\,b^4\,f\,h\,z-972\,a^2\,b^5\,d\,g\,z+540\,a^2\,b^5\,e\,f\,z+378\,a^2\,b^5\,c\,h\,z+243\,a\,b^6\,d^2\,z+972\,a^3\,b^4\,g^2\,z-630\,a^4\,b\,f\,g\,h+72\,a\,b^4\,c\,d\,e+360\,a^3\,b^2\,e\,f\,g+315\,a^3\,b^2\,d\,f\,h+252\,a^3\,b^2\,c\,g\,h-180\,a^2\,b^3\,d\,e\,f-144\,a^2\,b^3\,c\,e\,g-126\,a^2\,b^3\,c\,d\,h+588\,a^4\,b\,e\,h^2-60\,a\,b^4\,c^2\,f-336\,a^3\,b^2\,e^2\,h-324\,a^3\,b^2\,d\,g^2+162\,a^2\,b^3\,d^2\,g+150\,a^2\,b^3\,c\,f^2-125\,a^3\,b^2\,f^3+64\,a^2\,b^3\,e^3+216\,a^4\,b\,g^3-27\,a\,b^4\,d^3-343\,a^5\,h^3+8\,b^5\,c^3,z,k\right )\,\left (\frac {108\,a^2\,b^3\,g-54\,a\,b^4\,d}{9\,b^4}+\frac {x\,\left (63\,a^2\,b^3\,h-36\,a\,b^4\,e\right )}{9\,b^4}+\mathrm {root}\left (729\,a\,b^{10}\,z^3-729\,a\,b^8\,d\,z^2+1458\,a^2\,b^7\,g\,z^2-216\,a\,b^6\,c\,e\,z-945\,a^3\,b^4\,f\,h\,z-972\,a^2\,b^5\,d\,g\,z+540\,a^2\,b^5\,e\,f\,z+378\,a^2\,b^5\,c\,h\,z+243\,a\,b^6\,d^2\,z+972\,a^3\,b^4\,g^2\,z-630\,a^4\,b\,f\,g\,h+72\,a\,b^4\,c\,d\,e+360\,a^3\,b^2\,e\,f\,g+315\,a^3\,b^2\,d\,f\,h+252\,a^3\,b^2\,c\,g\,h-180\,a^2\,b^3\,d\,e\,f-144\,a^2\,b^3\,c\,e\,g-126\,a^2\,b^3\,c\,d\,h+588\,a^4\,b\,e\,h^2-60\,a\,b^4\,c^2\,f-336\,a^3\,b^2\,e^2\,h-324\,a^3\,b^2\,d\,g^2+162\,a^2\,b^3\,d^2\,g+150\,a^2\,b^3\,c\,f^2-125\,a^3\,b^2\,f^3+64\,a^2\,b^3\,e^3+216\,a^4\,b\,g^3-27\,a\,b^4\,d^3-343\,a^5\,h^3+8\,b^5\,c^3,z,k\right )\,a\,b^2\,9\right )+\frac {36\,a^3\,g^2+9\,a\,b^2\,d^2-35\,a^3\,f\,h-8\,a\,b^2\,c\,e+14\,a^2\,b\,c\,h-36\,a^2\,b\,d\,g+20\,a^2\,b\,e\,f}{9\,b^4}+\frac {x\,\left (4\,b^3\,c^2+25\,a^2\,b\,f^2+42\,a^3\,g\,h-20\,a\,b^2\,c\,f+12\,a\,b^2\,d\,e-21\,a^2\,b\,d\,h-24\,a^2\,b\,e\,g\right )}{9\,b^4}\right )\,\mathrm {root}\left (729\,a\,b^{10}\,z^3-729\,a\,b^8\,d\,z^2+1458\,a^2\,b^7\,g\,z^2-216\,a\,b^6\,c\,e\,z-945\,a^3\,b^4\,f\,h\,z-972\,a^2\,b^5\,d\,g\,z+540\,a^2\,b^5\,e\,f\,z+378\,a^2\,b^5\,c\,h\,z+243\,a\,b^6\,d^2\,z+972\,a^3\,b^4\,g^2\,z-630\,a^4\,b\,f\,g\,h+72\,a\,b^4\,c\,d\,e+360\,a^3\,b^2\,e\,f\,g+315\,a^3\,b^2\,d\,f\,h+252\,a^3\,b^2\,c\,g\,h-180\,a^2\,b^3\,d\,e\,f-144\,a^2\,b^3\,c\,e\,g-126\,a^2\,b^3\,c\,d\,h+588\,a^4\,b\,e\,h^2-60\,a\,b^4\,c^2\,f-336\,a^3\,b^2\,e^2\,h-324\,a^3\,b^2\,d\,g^2+162\,a^2\,b^3\,d^2\,g+150\,a^2\,b^3\,c\,f^2-125\,a^3\,b^2\,f^3+64\,a^2\,b^3\,e^3+216\,a^4\,b\,g^3-27\,a\,b^4\,d^3-343\,a^5\,h^3+8\,b^5\,c^3,z,k\right )\right )+x\,\left (\frac {e}{b^2}-\frac {2\,a\,h}{b^3}\right )-\frac {x\,\left (\frac {a^2\,h}{3}-\frac {a\,b\,e}{3}\right )+\frac {a^2\,g}{3}+x^2\,\left (\frac {b^2\,c}{3}-\frac {a\,b\,f}{3}\right )-\frac {a\,b\,d}{3}}{b^4\,x^3+a\,b^3}+\frac {f\,x^2}{2\,b^2}+\frac {g\,x^3}{3\,b^2}+\frac {h\,x^4}{4\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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[Out]
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