Optimal. Leaf size=335 \[ \frac {x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac {x^2 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{3 \sqrt {3} b^{17/3}}+\frac {x^8 (b e-2 a f)}{8 b^3}+\frac {f x^{11}}{11 b^2} \]
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Rubi [A] time = 0.71, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1828, 1851, 1836, 1488, 292, 31, 634, 617, 204, 628} \[ \frac {x^2 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac {a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{3 \sqrt {3} b^{17/3}}+\frac {x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac {x^8 (b e-2 a f)}{8 b^3}+\frac {f x^{11}}{11 b^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1488
Rule 1828
Rule 1836
Rule 1851
Rubi steps
\begin {align*} \int \frac {x^7 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^7-3 a b^4 (b e-a f) x^{10}-3 a b^5 f x^{13}}{a+b x^3} \, dx}{3 a b^6}\\ &=\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {x \left (2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^6-3 a b^4 (b e-a f) x^9-3 a b^5 f x^{12}\right )}{a+b x^3} \, dx}{3 a b^6}\\ &=\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {x \left (22 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-33 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-33 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^6-33 a b^5 (b e-2 a f) x^9\right )}{a+b x^3} \, dx}{33 a b^7}\\ &=\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \frac {x \left (176 a^2 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-264 a b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-264 a b^5 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6\right )}{a+b x^3} \, dx}{264 a b^8}\\ &=\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\int \left (-264 a b^3 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x-264 a b^4 \left (b^2 d-2 a b e+3 a^2 f\right ) x^4-\frac {88 \left (-5 a^2 b^6 c+8 a^3 b^5 d-11 a^4 b^4 e+14 a^5 b^3 f\right ) x}{a+b x^3}\right ) \, dx}{264 a b^8}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac {\left (a \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{3 b^5}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^{16/3}}-\frac {\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 b^{16/3}}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac {\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\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}{18 b^{17/3}}-\frac {\left (a \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{16/3}}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}}-\frac {\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 b^{17/3}}\\ &=\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac {(b e-2 a f) x^8}{8 b^3}+\frac {f x^{11}}{11 b^2}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{17/3}}+\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac {a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 319, normalized size = 0.95 \[ \frac {792 b^{5/3} x^5 \left (3 a^2 f-2 a b e+b^2 d\right )+1980 b^{2/3} x^2 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )+\frac {1320 a b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}-440 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right )-440 \sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right )+220 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right )+495 b^{8/3} x^8 (b e-2 a f)+360 b^{11/3} f x^{11}}{3960 b^{17/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 455, normalized size = 1.36 \[ \frac {360 \, b^{4} f x^{14} + 45 \, {\left (11 \, b^{4} e - 14 \, a b^{3} f\right )} x^{11} + 99 \, {\left (8 \, b^{4} d - 11 \, a b^{3} e + 14 \, a^{2} b^{2} f\right )} x^{8} + 396 \, {\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{5} + 660 \, {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} x^{2} - 440 \, \sqrt {3} {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f + {\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + \sqrt {3} a}{3 \, a}\right ) + 220 \, {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f + {\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 440 \, {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f + {\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{3960 \, {\left (b^{6} x^{3} + a b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 442, normalized size = 1.32 \[ \frac {{\left (5 \, a b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 8 \, a^{2} b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a^{4} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 11 \, a^{3} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{5}} + \frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 11 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{7}} + \frac {a b^{3} c x^{2} - a^{2} b^{2} d x^{2} - a^{4} f x^{2} + a^{3} b x^{2} e}{3 \, {\left (b x^{3} + a\right )} b^{5}} - \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 11 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{7}} + \frac {40 \, b^{20} f x^{11} - 110 \, a b^{19} f x^{8} + 55 \, b^{20} x^{8} e + 88 \, b^{20} d x^{5} + 264 \, a^{2} b^{18} f x^{5} - 176 \, a b^{19} x^{5} e + 220 \, b^{20} c x^{2} - 440 \, a b^{19} d x^{2} - 880 \, a^{3} b^{17} f x^{2} + 660 \, a^{2} b^{18} x^{2} e}{440 \, b^{22}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 584, normalized size = 1.74 \[ \frac {f \,x^{11}}{11 b^{2}}-\frac {a f \,x^{8}}{4 b^{3}}+\frac {e \,x^{8}}{8 b^{2}}+\frac {3 a^{2} f \,x^{5}}{5 b^{4}}-\frac {2 a e \,x^{5}}{5 b^{3}}+\frac {d \,x^{5}}{5 b^{2}}-\frac {a^{4} f \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{5}}+\frac {a^{3} e \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{4}}-\frac {a^{2} d \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {a c \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {2 a^{3} f \,x^{2}}{b^{5}}+\frac {3 a^{2} e \,x^{2}}{2 b^{4}}-\frac {a d \,x^{2}}{b^{3}}+\frac {c \,x^{2}}{2 b^{2}}+\frac {14 \sqrt {3}\, a^{4} f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{6}}-\frac {14 a^{4} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{6}}+\frac {7 a^{4} f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{6}}-\frac {11 \sqrt {3}\, a^{3} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {11 a^{3} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}-\frac {11 a^{3} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {8 \sqrt {3}\, a^{2} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {8 a^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {4 a^{2} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {5 \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 )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {5 a c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {5 a c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 325, normalized size = 0.97 \[ \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{2}}{3 \, {\left (b^{6} x^{3} + a b^{5}\right )}} - \frac {\sqrt {3} {\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {40 \, b^{3} f x^{11} + 55 \, {\left (b^{3} e - 2 \, a b^{2} f\right )} x^{8} + 88 \, {\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{5} + 220 \, {\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x^{2}}{440 \, b^{5}} - \frac {{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.28, size = 362, normalized size = 1.08 \[ x^8\,\left (\frac {e}{8\,b^2}-\frac {a\,f}{4\,b^3}\right )-x^5\,\left (\frac {a^2\,f}{5\,b^4}-\frac {d}{5\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{5\,b}\right )+x^2\,\left (\frac {c}{2\,b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{2\,b^2}+\frac {a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )+\frac {f\,x^{11}}{11\,b^2}-\frac {x^2\,\left (\frac {f\,a^4}{3}-\frac {e\,a^3\,b}{3}+\frac {d\,a^2\,b^2}{3}-\frac {c\,a\,b^3}{3}\right )}{b^6\,x^3+a\,b^5}+\frac {a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-14\,f\,a^3+11\,e\,a^2\,b-8\,d\,a\,b^2+5\,c\,b^3\right )}{9\,b^{17/3}}-\frac {a^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+11\,e\,a^2\,b-8\,d\,a\,b^2+5\,c\,b^3\right )}{9\,b^{17/3}}+\frac {a^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+11\,e\,a^2\,b-8\,d\,a\,b^2+5\,c\,b^3\right )}{9\,b^{17/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 58.23, size = 539, normalized size = 1.61 \[ x^{8} \left (- \frac {a f}{4 b^{3}} + \frac {e}{8 b^{2}}\right ) + x^{5} \left (\frac {3 a^{2} f}{5 b^{4}} - \frac {2 a e}{5 b^{3}} + \frac {d}{5 b^{2}}\right ) + x^{2} \left (- \frac {2 a^{3} f}{b^{5}} + \frac {3 a^{2} e}{2 b^{4}} - \frac {a d}{b^{3}} + \frac {c}{2 b^{2}}\right ) + \frac {x^{2} \left (- a^{4} f + a^{3} b e - a^{2} b^{2} d + a b^{3} c\right )}{3 a b^{5} + 3 b^{6} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{17} + 2744 a^{11} f^{3} - 6468 a^{10} b e f^{2} + 4704 a^{9} b^{2} d f^{2} + 5082 a^{9} b^{2} e^{2} f - 2940 a^{8} b^{3} c f^{2} - 7392 a^{8} b^{3} d e f - 1331 a^{8} b^{3} e^{3} + 4620 a^{7} b^{4} c e f + 2688 a^{7} b^{4} d^{2} f + 2904 a^{7} b^{4} d e^{2} - 3360 a^{6} b^{5} c d f - 1815 a^{6} b^{5} c e^{2} - 2112 a^{6} b^{5} d^{2} e + 1050 a^{5} b^{6} c^{2} f + 2640 a^{5} b^{6} c d e + 512 a^{5} b^{6} d^{3} - 825 a^{4} b^{7} c^{2} e - 960 a^{4} b^{7} c d^{2} + 600 a^{3} b^{8} c^{2} d - 125 a^{2} b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {81 t^{2} b^{11}}{196 a^{7} f^{2} - 308 a^{6} b e f + 224 a^{5} b^{2} d f + 121 a^{5} b^{2} e^{2} - 140 a^{4} b^{3} c f - 176 a^{4} b^{3} d e + 110 a^{3} b^{4} c e + 64 a^{3} b^{4} d^{2} - 80 a^{2} b^{5} c d + 25 a b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{11}}{11 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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