Optimal. Leaf size=331 \[ -\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 b^{10/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 b^{10/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt {3} b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac {a x (b e-a h)}{b^3}+\frac {x^2 (b c-a f)}{2 b^2}+\frac {x^3 (b d-a g)}{3 b^2}+\frac {x^4 (b e-a h)}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b} \]
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Rubi [A] time = 1.07, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1836, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 b^{10/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 b^{10/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt {3} b^{10/3}}+\frac {x^2 (b c-a f)}{2 b^2}+\frac {x^3 (b d-a g)}{3 b^2}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}+\frac {x^4 (b e-a h)}{4 b^2}-\frac {a x (b e-a h)}{b^3}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 260
Rule 617
Rule 628
Rule 634
Rule 1836
Rule 1860
Rule 1871
Rule 1887
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 )}{a+b x^3} \, dx &=\frac {h x^7}{7 b}+\frac {\int \frac {x^4 \left (7 b c+7 b d x+7 (b e-a h) x^2+7 b f x^3+7 b g x^4\right )}{a+b x^3} \, dx}{7 b}\\ &=\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \frac {x^4 \left (42 b^2 c+42 b (b d-a g) x+42 b (b e-a h) x^2+42 b^2 f x^3\right )}{a+b x^3} \, dx}{42 b^2}\\ &=\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \frac {x^4 \left (210 b^2 (b c-a f)+210 b^2 (b d-a g) x+210 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{210 b^3}\\ &=\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \left (-210 a (b e-a h)+210 b (b c-a f) x+210 b (b d-a g) x^2+210 b (b e-a h) x^3+\frac {210 \left (a^2 (b e-a h)-a b (b c-a f) x-a b (b d-a g) x^2\right )}{a+b x^3}\right ) \, dx}{210 b^3}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \frac {a^2 (b e-a h)-a b (b c-a f) x-a b (b d-a g) x^2}{a+b x^3} \, dx}{b^3}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {\int \frac {a^2 (b e-a h)-a b (b c-a f) x}{a+b x^3} \, dx}{b^3}-\frac {(a (b d-a g)) \int \frac {x^2}{a+b x^3} \, dx}{b^2}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}+\frac {\int \frac {\sqrt [3]{a} \left (-a^{4/3} b (b c-a f)+2 a^2 \sqrt [3]{b} (b e-a h)\right )+\sqrt [3]{b} \left (-a^{4/3} b (b c-a f)-a^2 \sqrt [3]{b} (b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^{10/3}}+\frac {\left (a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^3}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac {\left (a \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^3}-\frac {\left (a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right )\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 b^{10/3}}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac {\left (a^{2/3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{10/3}}\\ &=-\frac {a (b e-a h) x}{b^3}+\frac {(b c-a f) x^2}{2 b^2}+\frac {(b d-a g) x^3}{3 b^2}+\frac {(b e-a h) x^4}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b}+\frac {a^{2/3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{10/3}}+\frac {a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}-\frac {a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 334, normalized size = 1.01 \[ \frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right )}{6 b^{10/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right )}{3 b^{10/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt {3} b^{10/3}}+\frac {a (a g-b d) \log \left (a+b x^3\right )}{3 b^3}+\frac {a x (a h-b e)}{b^3}+\frac {x^2 (b c-a f)}{2 b^2}+\frac {x^3 (b d-a g)}{3 b^2}+\frac {x^4 (b e-a h)}{4 b^2}+\frac {f x^5}{5 b}+\frac {g x^6}{6 b}+\frac {h x^7}{7 b} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 380, normalized size = 1.15 \[ -\frac {{\left (a b d - a^{2} g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac {1}{3}} a b e - \left (-a b^{2}\right )^{\frac {2}{3}} b c + \left (-a b^{2}\right )^{\frac {2}{3}} a f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac {1}{3}} a b e + \left (-a b^{2}\right )^{\frac {2}{3}} b c - \left (-a b^{2}\right )^{\frac {2}{3}} a f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4}} + \frac {60 \, b^{6} h x^{7} + 70 \, b^{6} g x^{6} + 84 \, b^{6} f x^{5} - 105 \, a b^{5} h x^{4} + 105 \, b^{6} x^{4} e + 140 \, b^{6} d x^{3} - 140 \, a b^{5} g x^{3} + 210 \, b^{6} c x^{2} - 210 \, a b^{5} f x^{2} + 420 \, a^{2} b^{4} h x - 420 \, a b^{5} x e}{420 \, b^{7}} + \frac {{\left (a b^{14} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{13} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} b^{12} h - a^{2} b^{13} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 533, normalized size = 1.61 \[ \frac {h \,x^{7}}{7 b}+\frac {g \,x^{6}}{6 b}+\frac {f \,x^{5}}{5 b}-\frac {a h \,x^{4}}{4 b^{2}}+\frac {e \,x^{4}}{4 b}-\frac {a g \,x^{3}}{3 b^{2}}+\frac {d \,x^{3}}{3 b}-\frac {a f \,x^{2}}{2 b^{2}}+\frac {c \,x^{2}}{2 b}-\frac {\sqrt {3}\, a^{3} h \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {a^{3} h \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {a^{3} h \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {\sqrt {3}\, a^{2} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {a^{2} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {a^{2} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {\sqrt {3}\, a^{2} f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {a^{2} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {a^{2} f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {a^{2} g \ln \left (b \,x^{3}+a \right )}{3 b^{3}}+\frac {a^{2} h x}{b^{3}}-\frac {\sqrt {3}\, a c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {a c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {a c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {a d \ln \left (b \,x^{3}+a \right )}{3 b^{2}}-\frac {a e x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 378, normalized size = 1.14 \[ -\frac {\sqrt {3} {\left (a b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} h \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{3}} + \frac {60 \, b^{2} h x^{7} + 70 \, b^{2} g x^{6} + 84 \, b^{2} f x^{5} + 105 \, {\left (b^{2} e - a b h\right )} x^{4} + 140 \, {\left (b^{2} d - a b g\right )} x^{3} + 210 \, {\left (b^{2} c - a b f\right )} x^{2} - 420 \, {\left (a b e - a^{2} h\right )} x}{420 \, b^{3}} - \frac {{\left (2 \, a b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b g \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b f \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b e - a^{3} h\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a b^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} b g \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b f \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b e + a^{3} h\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.09, size = 1271, normalized size = 3.84 \[ x^2\,\left (\frac {c}{2\,b}-\frac {a\,f}{2\,b^2}\right )+x^3\,\left (\frac {d}{3\,b}-\frac {a\,g}{3\,b^2}\right )+x^4\,\left (\frac {e}{4\,b}-\frac {a\,h}{4\,b^2}\right )+\left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,b^{10}\,z^3+27\,a\,b^8\,d\,z^2-27\,a^2\,b^7\,g\,z^2-9\,a^4\,b^4\,f\,h\,z-18\,a^3\,b^5\,d\,g\,z+9\,a^3\,b^5\,e\,f\,z+9\,a^3\,b^5\,c\,h\,z-9\,a^2\,b^6\,c\,e\,z+9\,a^4\,b^4\,g^2\,z+9\,a^2\,b^6\,d^2\,z+3\,a^6\,b\,f\,g\,h-3\,a^5\,b^2\,e\,f\,g-3\,a^5\,b^2\,d\,f\,h-3\,a^5\,b^2\,c\,g\,h+3\,a^4\,b^3\,d\,e\,f+3\,a^4\,b^3\,c\,e\,g+3\,a^4\,b^3\,c\,d\,h-3\,a^3\,b^4\,c\,d\,e-3\,a^6\,b\,e\,h^2+3\,a^5\,b^2\,e^2\,h+3\,a^5\,b^2\,d\,g^2-3\,a^4\,b^3\,d^2\,g-3\,a^4\,b^3\,c\,f^2+3\,a^3\,b^4\,c^2\,f+a^5\,b^2\,f^3+a^3\,b^4\,d^3+a^7\,h^3-a^4\,b^3\,e^3-a^2\,b^5\,c^3-a^6\,b\,g^3,z,k\right )\,\left (\frac {6\,a^2\,b^4\,d-6\,a^3\,b^3\,g}{b^4}+\frac {x\,\left (3\,a^2\,b^4\,e-3\,a^3\,b^3\,h\right )}{b^4}+\mathrm {root}\left (27\,b^{10}\,z^3+27\,a\,b^8\,d\,z^2-27\,a^2\,b^7\,g\,z^2-9\,a^4\,b^4\,f\,h\,z-18\,a^3\,b^5\,d\,g\,z+9\,a^3\,b^5\,e\,f\,z+9\,a^3\,b^5\,c\,h\,z-9\,a^2\,b^6\,c\,e\,z+9\,a^4\,b^4\,g^2\,z+9\,a^2\,b^6\,d^2\,z+3\,a^6\,b\,f\,g\,h-3\,a^5\,b^2\,e\,f\,g-3\,a^5\,b^2\,d\,f\,h-3\,a^5\,b^2\,c\,g\,h+3\,a^4\,b^3\,d\,e\,f+3\,a^4\,b^3\,c\,e\,g+3\,a^4\,b^3\,c\,d\,h-3\,a^3\,b^4\,c\,d\,e-3\,a^6\,b\,e\,h^2+3\,a^5\,b^2\,e^2\,h+3\,a^5\,b^2\,d\,g^2-3\,a^4\,b^3\,d^2\,g-3\,a^4\,b^3\,c\,f^2+3\,a^3\,b^4\,c^2\,f+a^5\,b^2\,f^3+a^3\,b^4\,d^3+a^7\,h^3-a^4\,b^3\,e^3-a^2\,b^5\,c^3-a^6\,b\,g^3,z,k\right )\,a\,b^2\,9\right )+\frac {a^5\,g^2+a^3\,b^2\,d^2-a^5\,f\,h+a^4\,b\,c\,h-2\,a^4\,b\,d\,g+a^4\,b\,e\,f-a^3\,b^2\,c\,e}{b^4}+\frac {x\,\left (a^4\,b\,f^2+a^2\,b^3\,c^2+a^5\,g\,h-a^4\,b\,d\,h-a^4\,b\,e\,g-2\,a^3\,b^2\,c\,f+a^3\,b^2\,d\,e\right )}{b^4}\right )\,\mathrm {root}\left (27\,b^{10}\,z^3+27\,a\,b^8\,d\,z^2-27\,a^2\,b^7\,g\,z^2-9\,a^4\,b^4\,f\,h\,z-18\,a^3\,b^5\,d\,g\,z+9\,a^3\,b^5\,e\,f\,z+9\,a^3\,b^5\,c\,h\,z-9\,a^2\,b^6\,c\,e\,z+9\,a^4\,b^4\,g^2\,z+9\,a^2\,b^6\,d^2\,z+3\,a^6\,b\,f\,g\,h-3\,a^5\,b^2\,e\,f\,g-3\,a^5\,b^2\,d\,f\,h-3\,a^5\,b^2\,c\,g\,h+3\,a^4\,b^3\,d\,e\,f+3\,a^4\,b^3\,c\,e\,g+3\,a^4\,b^3\,c\,d\,h-3\,a^3\,b^4\,c\,d\,e-3\,a^6\,b\,e\,h^2+3\,a^5\,b^2\,e^2\,h+3\,a^5\,b^2\,d\,g^2-3\,a^4\,b^3\,d^2\,g-3\,a^4\,b^3\,c\,f^2+3\,a^3\,b^4\,c^2\,f+a^5\,b^2\,f^3+a^3\,b^4\,d^3+a^7\,h^3-a^4\,b^3\,e^3-a^2\,b^5\,c^3-a^6\,b\,g^3,z,k\right )\right )+\frac {f\,x^5}{5\,b}+\frac {g\,x^6}{6\,b}+\frac {h\,x^7}{7\,b}-\frac {a\,x\,\left (\frac {e}{b}-\frac {a\,h}{b^2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 60.52, size = 881, normalized size = 2.66 \[ x^{4} \left (- \frac {a h}{4 b^{2}} + \frac {e}{4 b}\right ) + x^{3} \left (- \frac {a g}{3 b^{2}} + \frac {d}{3 b}\right ) + x^{2} \left (- \frac {a f}{2 b^{2}} + \frac {c}{2 b}\right ) + x \left (\frac {a^{2} h}{b^{3}} - \frac {a e}{b^{2}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{10} + t^{2} \left (- 27 a^{2} b^{7} g + 27 a b^{8} d\right ) + t \left (- 9 a^{4} b^{4} f h + 9 a^{4} b^{4} g^{2} + 9 a^{3} b^{5} c h - 18 a^{3} b^{5} d g + 9 a^{3} b^{5} e f - 9 a^{2} b^{6} c e + 9 a^{2} b^{6} d^{2}\right ) + a^{7} h^{3} - 3 a^{6} b e h^{2} + 3 a^{6} b f g h - a^{6} b g^{3} - 3 a^{5} b^{2} c g h - 3 a^{5} b^{2} d f h + 3 a^{5} b^{2} d g^{2} + 3 a^{5} b^{2} e^{2} h - 3 a^{5} b^{2} e f g + a^{5} b^{2} f^{3} + 3 a^{4} b^{3} c d h + 3 a^{4} b^{3} c e g - 3 a^{4} b^{3} c f^{2} - 3 a^{4} b^{3} d^{2} g + 3 a^{4} b^{3} d e f - a^{4} b^{3} e^{3} + 3 a^{3} b^{4} c^{2} f - 3 a^{3} b^{4} c d e + a^{3} b^{4} d^{3} - a^{2} b^{5} c^{3}, \left (t \mapsto t \log {\left (x + \frac {- 9 t^{2} a b^{7} f + 9 t^{2} b^{8} c - 3 t a^{4} b^{3} h^{2} + 6 t a^{3} b^{4} e h + 6 t a^{3} b^{4} f g - 6 t a^{2} b^{5} c g - 6 t a^{2} b^{5} d f - 3 t a^{2} b^{5} e^{2} + 6 t a b^{6} c d + a^{6} g h^{2} - a^{5} b d h^{2} - 2 a^{5} b e g h + 2 a^{5} b f^{2} h - a^{5} b f g^{2} - 4 a^{4} b^{2} c f h + a^{4} b^{2} c g^{2} + 2 a^{4} b^{2} d e h + 2 a^{4} b^{2} d f g + a^{4} b^{2} e^{2} g - 2 a^{4} b^{2} e f^{2} + 2 a^{3} b^{3} c^{2} h - 2 a^{3} b^{3} c d g + 4 a^{3} b^{3} c e f - a^{3} b^{3} d^{2} f - a^{3} b^{3} d e^{2} - 2 a^{2} b^{4} c^{2} e + a^{2} b^{4} c d^{2}}{a^{6} h^{3} - 3 a^{5} b e h^{2} + 3 a^{4} b^{2} e^{2} h - a^{4} b^{2} f^{3} + 3 a^{3} b^{3} c f^{2} - a^{3} b^{3} e^{3} - 3 a^{2} b^{4} c^{2} f + a b^{5} c^{3}} \right )} \right )\right )} + \frac {f x^{5}}{5 b} + \frac {g x^{6}}{6 b} + \frac {h x^{7}}{7 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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