Optimal. Leaf size=424 \[ \frac {3 b c-a d}{10 a^4 x^{10}}-\frac {c}{13 a^3 x^{13}}-\frac {a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{54 a^{22/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{9 \sqrt {3} a^{22/3}}-\frac {b^2 x^2 \left (-8 a^3 f+11 a^2 b e-14 a b^2 d+17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}-\frac {b \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{4 a^6 x^4}-\frac {b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.85, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1829, 1834, 292, 31, 634, 617, 204, 628} \[ -\frac {b^2 x^2 \left (11 a^2 b e-8 a^3 f-14 a b^2 d+17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}-\frac {b^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2}+\frac {3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{4 a^6 x^4}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{54 a^{22/3}}-\frac {b \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (65 a^2 b e-35 a^3 f-104 a b^2 d+152 b^3 c\right )}{9 \sqrt {3} a^{22/3}}-\frac {a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {c}{13 a^3 x^{13}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1829
Rule 1834
Rubi steps
\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^3} \, dx &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^3 c+6 b^3 \left (\frac {b c}{a}-d\right ) x^3-\frac {6 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac {6 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac {6 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}+\frac {4 b^5 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{15}}{a^5}}{x^{14} \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {\int \frac {18 b^8 c-18 b^8 \left (\frac {2 b c}{a}-d\right ) x^3+18 b^8 \left (\frac {3 b^2 c}{a^2}-\frac {2 b d}{a}+e\right ) x^6-18 b^8 \left (\frac {4 b^3 c}{a^3}-\frac {3 b^2 d}{a^2}+\frac {2 b e}{a}-f\right ) x^9+\frac {18 b^9 \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) x^{12}}{a^4}-\frac {2 b^{10} \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^{15}}{a^5}}{x^{14} \left (a+b x^3\right )} \, dx}{18 a^2 b^8}\\ &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^8 c}{a x^{14}}+\frac {18 b^8 (-3 b c+a d)}{a^2 x^{11}}+\frac {18 b^8 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^8}+\frac {18 b^8 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^5}-\frac {18 b^9 \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^5 x^2}+\frac {2 b^{10} \left (-152 b^3 c+104 a b^2 d-65 a^2 b e+35 a^3 f\right ) x}{a^5 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^8}\\ &=-\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}-\frac {\left (b^2 \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{9 a^7}\\ &=-\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{22/3}}-\frac {\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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}{27 a^{22/3}}\\ &=-\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac {\left (b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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}{54 a^{22/3}}-\frac {\left (b^{5/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^7}\\ &=-\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{22/3}}-\frac {\left (b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 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 )}{9 a^{22/3}}\\ &=-\frac {c}{13 a^3 x^{13}}+\frac {3 b c-a d}{10 a^4 x^{10}}-\frac {6 b^2 c-3 a b d+a^2 e}{7 a^5 x^7}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{4 a^6 x^4}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right )}{a^7 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^6 \left (a+b x^3\right )^2}-\frac {b^2 \left (17 b^3 c-14 a b^2 d+11 a^2 b e-8 a^3 f\right ) x^2}{9 a^7 \left (a+b x^3\right )}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{22/3}}+\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{22/3}}-\frac {b^{4/3} \left (152 b^3 c-104 a b^2 d+65 a^2 b e-35 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{22/3}}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 419, normalized size = 0.99 \[ \frac {3 b c-a d}{10 a^4 x^{10}}-\frac {c}{13 a^3 x^{13}}-\frac {a^2 e-3 a b d+6 b^2 c}{7 a^5 x^7}+\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^3 f-65 a^2 b e+104 a b^2 d-152 b^3 c\right )}{54 a^{22/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{27 a^{22/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-35 a^3 f+65 a^2 b e-104 a b^2 d+152 b^3 c\right )}{9 \sqrt {3} a^{22/3}}+\frac {b^2 x^2 \left (8 a^3 f-11 a^2 b e+14 a b^2 d-17 b^3 c\right )}{9 a^7 \left (a+b x^3\right )}+\frac {b \left (3 a^3 f-6 a^2 b e+10 a b^2 d-15 b^3 c\right )}{a^7 x}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{4 a^6 x^4}+\frac {b^2 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 a^6 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 686, normalized size = 1.62 \[ -\frac {5460 \, {\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{18} + 9555 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{15} + 3510 \, {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{12} - 351 \, {\left (152 \, a^{3} b^{3} c - 104 \, a^{4} b^{2} d + 65 \, a^{5} b e - 35 \, a^{6} f\right )} x^{9} + 3780 \, a^{6} c + 108 \, {\left (152 \, a^{4} b^{2} c - 104 \, a^{5} b d + 65 \, a^{6} e\right )} x^{6} - 378 \, {\left (19 \, a^{5} b c - 13 \, a^{6} d\right )} x^{3} + 1820 \, \sqrt {3} {\left ({\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{19} + 2 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{16} + {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 910 \, {\left ({\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{19} + 2 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{16} + {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 1820 \, {\left ({\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{19} + 2 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{16} + {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{49140 \, {\left (a^{7} b^{2} x^{19} + 2 \, a^{8} b x^{16} + a^{9} x^{13}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 531, normalized size = 1.25 \[ \frac {\sqrt {3} {\left (152 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 104 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 35 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{8}} + \frac {{\left (152 \, b^{5} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 104 \, a b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 35 \, a^{3} b^{2} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 65 \, a^{2} b^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{8}} - \frac {{\left (152 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 104 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 35 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 65 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{8}} - \frac {34 \, b^{6} c x^{5} - 28 \, a b^{5} d x^{5} - 16 \, a^{3} b^{3} f x^{5} + 22 \, a^{2} b^{4} x^{5} e + 37 \, a b^{5} c x^{2} - 31 \, a^{2} b^{4} d x^{2} - 19 \, a^{4} b^{2} f x^{2} + 25 \, a^{3} b^{3} x^{2} e}{18 \, {\left (b x^{3} + a\right )}^{2} a^{7}} - \frac {27300 \, b^{4} c x^{12} - 18200 \, a b^{3} d x^{12} - 5460 \, a^{3} b f x^{12} + 10920 \, a^{2} b^{2} x^{12} e - 4550 \, a b^{3} c x^{9} + 2730 \, a^{2} b^{2} d x^{9} + 455 \, a^{4} f x^{9} - 1365 \, a^{3} b x^{9} e + 1560 \, a^{2} b^{2} c x^{6} - 780 \, a^{3} b d x^{6} + 260 \, a^{4} x^{6} e - 546 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{7} x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 716, normalized size = 1.69 \[ \frac {8 b^{3} f \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{4}}-\frac {11 b^{4} e \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{5}}+\frac {14 b^{5} d \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{6}}-\frac {17 b^{6} c \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{7}}+\frac {19 b^{2} f \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{3}}-\frac {25 b^{3} e \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{4}}+\frac {31 b^{4} d \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{5}}-\frac {37 b^{5} c \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{6}}+\frac {35 \sqrt {3}\, b f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {35 b f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {35 b f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {65 \sqrt {3}\, b^{2} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}+\frac {65 b^{2} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}-\frac {65 b^{2} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}+\frac {104 \sqrt {3}\, b^{3} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{6}}-\frac {104 b^{3} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{6}}+\frac {52 b^{3} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{6}}-\frac {152 \sqrt {3}\, b^{4} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{7}}+\frac {152 b^{4} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{7}}-\frac {76 b^{4} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{7}}+\frac {3 b f}{a^{4} x}-\frac {6 b^{2} e}{a^{5} x}+\frac {10 b^{3} d}{a^{6} x}-\frac {15 b^{4} c}{a^{7} x}-\frac {f}{4 a^{3} x^{4}}+\frac {3 b e}{4 a^{4} x^{4}}-\frac {3 b^{2} d}{2 a^{5} x^{4}}+\frac {5 b^{3} c}{2 a^{6} x^{4}}-\frac {e}{7 a^{3} x^{7}}+\frac {3 b d}{7 a^{4} x^{7}}-\frac {6 b^{2} c}{7 a^{5} x^{7}}-\frac {d}{10 a^{3} x^{10}}+\frac {3 b c}{10 a^{4} x^{10}}-\frac {c}{13 a^{3} x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 427, normalized size = 1.01 \[ -\frac {1820 \, {\left (152 \, b^{6} c - 104 \, a b^{5} d + 65 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )} x^{18} + 3185 \, {\left (152 \, a b^{5} c - 104 \, a^{2} b^{4} d + 65 \, a^{3} b^{3} e - 35 \, a^{4} b^{2} f\right )} x^{15} + 1170 \, {\left (152 \, a^{2} b^{4} c - 104 \, a^{3} b^{3} d + 65 \, a^{4} b^{2} e - 35 \, a^{5} b f\right )} x^{12} - 117 \, {\left (152 \, a^{3} b^{3} c - 104 \, a^{4} b^{2} d + 65 \, a^{5} b e - 35 \, a^{6} f\right )} x^{9} + 1260 \, a^{6} c + 36 \, {\left (152 \, a^{4} b^{2} c - 104 \, a^{5} b d + 65 \, a^{6} e\right )} x^{6} - 126 \, {\left (19 \, a^{5} b c - 13 \, a^{6} d\right )} x^{3}}{16380 \, {\left (a^{7} b^{2} x^{19} + 2 \, a^{8} b x^{16} + a^{9} x^{13}\right )}} - \frac {\sqrt {3} {\left (152 \, b^{4} c - 104 \, a b^{3} d + 65 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (152 \, b^{4} c - 104 \, a b^{3} d + 65 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (152 \, b^{4} c - 104 \, a b^{3} d + 65 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.30, size = 397, normalized size = 0.94 \[ \frac {b^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{27\,a^{22/3}}-\frac {\frac {c}{13\,a}-\frac {x^9\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{140\,a^4}+\frac {x^3\,\left (13\,a\,d-19\,b\,c\right )}{130\,a^2}+\frac {x^6\,\left (65\,e\,a^2-104\,d\,a\,b+152\,c\,b^2\right )}{455\,a^3}+\frac {b\,x^{12}\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{14\,a^5}+\frac {7\,b^2\,x^{15}\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{36\,a^6}+\frac {b^3\,x^{18}\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{9\,a^7}}{a^2\,x^{13}+2\,a\,b\,x^{16}+b^2\,x^{19}}-\frac {b^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{27\,a^{22/3}}+\frac {b^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-35\,f\,a^3+65\,e\,a^2\,b-104\,d\,a\,b^2+152\,c\,b^3\right )}{27\,a^{22/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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