Optimal. Leaf size=381 \[ \frac {3 b c-a d}{7 a^4 x^7}-\frac {c}{10 a^3 x^{10}}-\frac {a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{9 \sqrt {3} a^{19/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac {b x^2 \left (-5 a^3 f+8 a^2 b e-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac {b x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.71, antiderivative size = 381, 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} \[ \frac {b x^2 \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{9 a^6 \left (a+b x^3\right )}+\frac {b x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{54 a^{19/3}}+\frac {3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{a^6 x}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{27 a^{19/3}}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (35 a^2 b e-14 a^3 f-65 a b^2 d+104 b^3 c\right )}{9 \sqrt {3} a^{19/3}}-\frac {a^2 e-3 a b d+6 b^2 c}{4 a^5 x^4}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {c}{10 a^3 x^{10}} \]
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^{11} \left (a+b x^3\right )^3} \, dx &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \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 {4 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}}{x^{11} \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}+\frac {\int \frac {18 b^7 c-18 b^7 \left (\frac {2 b c}{a}-d\right ) x^3+18 b^7 \left (\frac {3 b^2 c}{a^2}-\frac {2 b d}{a}+e\right ) x^6-18 b^7 \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 {2 b^8 \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^{12}}{a^4}}{x^{11} \left (a+b x^3\right )} \, dx}{18 a^2 b^7}\\ &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^7 c}{a x^{11}}+\frac {18 b^7 (-3 b c+a d)}{a^2 x^8}+\frac {18 b^7 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^5}+\frac {18 b^7 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^2}-\frac {2 b^8 \left (-104 b^3 c+65 a b^2 d-35 a^2 b e+14 a^3 f\right ) x}{a^4 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^7}\\ &=-\frac {c}{10 a^3 x^{10}}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}+\frac {\left (b \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{9 a^6}\\ &=-\frac {c}{10 a^3 x^{10}}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac {\left (b^{2/3} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{19/3}}+\frac {\left (b^{2/3} \left (104 b^3 c-65 a b^2 d+35 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}{27 a^{19/3}}\\ &=-\frac {c}{10 a^3 x^{10}}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{19/3}}+\frac {\left (\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 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}{54 a^{19/3}}+\frac {\left (b^{2/3} \left (104 b^3 c-65 a b^2 d+35 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}{18 a^6}\\ &=-\frac {c}{10 a^3 x^{10}}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{19/3}}+\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 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 )}{54 a^{19/3}}+\frac {\left (\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 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 )}{9 a^{19/3}}\\ &=-\frac {c}{10 a^3 x^{10}}+\frac {3 b c-a d}{7 a^4 x^7}-\frac {6 b^2 c-3 a b d+a^2 e}{4 a^5 x^4}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) x^2}{9 a^6 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 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 )}{9 \sqrt {3} a^{19/3}}-\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{19/3}}+\frac {\sqrt [3]{b} \left (104 b^3 c-65 a b^2 d+35 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 )}{54 a^{19/3}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 366, normalized size = 0.96 \[ \frac {-\frac {540 a^{7/3} (a d-3 b c)}{x^7}-\frac {378 a^{10/3} c}{x^{10}}-\frac {945 a^{4/3} \left (a^2 e-3 a b d+6 b^2 c\right )}{x^4}-\frac {420 \sqrt [3]{a} b x^2 \left (5 a^3 f-8 a^2 b e+11 a b^2 d-14 b^3 c\right )}{a+b x^3}-\frac {3780 \sqrt [3]{a} \left (a^3 f-3 a^2 b e+6 a b^2 d-10 b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (14 a^3 f-35 a^2 b e+65 a b^2 d-104 b^3 c\right )-140 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )-\frac {630 a^{4/3} b x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+35 a^2 b e-65 a b^2 d+104 b^3 c\right )}{3780 a^{19/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 621, normalized size = 1.63 \[ \frac {420 \, {\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{15} + 735 \, {\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{12} + 270 \, {\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{9} - 27 \, {\left (104 \, a^{3} b^{2} c - 65 \, a^{4} b d + 35 \, a^{5} e\right )} x^{6} - 378 \, a^{5} c + 108 \, {\left (8 \, a^{4} b c - 5 \, a^{5} d\right )} x^{3} + 140 \, \sqrt {3} {\left ({\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{16} + 2 \, {\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{13} + {\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{10}\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 ) + 70 \, {\left ({\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{16} + 2 \, {\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{13} + {\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{10}\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 ) - 140 \, {\left ({\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{16} + 2 \, {\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{13} + {\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{10}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{3780 \, {\left (a^{6} b^{2} x^{16} + 2 \, a^{7} b x^{13} + a^{8} x^{10}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 486, normalized size = 1.28 \[ -\frac {{\left (104 \, b^{4} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 65 \, a b^{3} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 14 \, a^{3} b f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 35 \, a^{2} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7}} - \frac {\sqrt {3} {\left (104 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{7} b} + \frac {{\left (104 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 65 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 35 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{7} b} + \frac {28 \, b^{5} c x^{5} - 22 \, a b^{4} d x^{5} - 10 \, a^{3} b^{2} f x^{5} + 16 \, a^{2} b^{3} x^{5} e + 31 \, a b^{4} c x^{2} - 25 \, a^{2} b^{3} d x^{2} - 13 \, a^{4} b f x^{2} + 19 \, a^{3} b^{2} x^{2} e}{18 \, {\left (b x^{3} + a\right )}^{2} a^{6}} + \frac {1400 \, b^{3} c x^{9} - 840 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 420 \, a^{2} b x^{9} e - 210 \, a b^{2} c x^{6} + 105 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 60 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{6} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 659, normalized size = 1.73 \[ -\frac {5 b^{2} f \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {8 b^{3} e \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{4}}-\frac {11 b^{4} d \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{5}}+\frac {14 b^{5} c \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{6}}-\frac {13 b f \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {19 b^{2} e \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{3}}-\frac {25 b^{3} d \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{4}}+\frac {31 b^{4} c \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{5}}-\frac {14 \sqrt {3}\, 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^{3}}+\frac {14 f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 f \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^{3}}+\frac {35 \sqrt {3}\, b 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^{4}}-\frac {35 b e \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 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^{4}}-\frac {65 \sqrt {3}\, b^{2} 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^{5}}+\frac {65 b^{2} d \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} d \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} 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^{6}}-\frac {104 b^{3} c \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} 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^{6}}-\frac {f}{a^{3} x}+\frac {3 b e}{a^{4} x}-\frac {6 b^{2} d}{a^{5} x}+\frac {10 b^{3} c}{a^{6} x}-\frac {e}{4 a^{3} x^{4}}+\frac {3 b d}{4 a^{4} x^{4}}-\frac {3 b^{2} c}{2 a^{5} x^{4}}-\frac {d}{7 a^{3} x^{7}}+\frac {3 b c}{7 a^{4} x^{7}}-\frac {c}{10 a^{3} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.20, size = 376, normalized size = 0.99 \[ \frac {140 \, {\left (104 \, b^{5} c - 65 \, a b^{4} d + 35 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{15} + 245 \, {\left (104 \, a b^{4} c - 65 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{12} + 90 \, {\left (104 \, a^{2} b^{3} c - 65 \, a^{3} b^{2} d + 35 \, a^{4} b e - 14 \, a^{5} f\right )} x^{9} - 9 \, {\left (104 \, a^{3} b^{2} c - 65 \, a^{4} b d + 35 \, a^{5} e\right )} x^{6} - 126 \, a^{5} c + 36 \, {\left (8 \, a^{4} b c - 5 \, a^{5} d\right )} x^{3}}{1260 \, {\left (a^{6} b^{2} x^{16} + 2 \, a^{7} b x^{13} + a^{8} x^{10}\right )}} + \frac {\sqrt {3} {\left (104 \, b^{3} c - 65 \, a b^{2} d + 35 \, a^{2} b e - 14 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (104 \, b^{3} c - 65 \, a b^{2} d + 35 \, a^{2} b e - 14 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (104 \, b^{3} c - 65 \, a b^{2} d + 35 \, a^{2} b e - 14 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.28, size = 359, normalized size = 0.94 \[ -\frac {\frac {c}{10\,a}-\frac {x^9\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{14\,a^4}+\frac {x^3\,\left (5\,a\,d-8\,b\,c\right )}{35\,a^2}+\frac {x^6\,\left (35\,e\,a^2-65\,d\,a\,b+104\,c\,b^2\right )}{140\,a^3}-\frac {7\,b\,x^{12}\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{36\,a^5}-\frac {b^2\,x^{15}\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{9\,a^6}}{a^2\,x^{10}+2\,a\,b\,x^{13}+b^2\,x^{16}}-\frac {b^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{27\,a^{19/3}}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{27\,a^{19/3}}-\frac {b^{1/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-14\,f\,a^3+35\,e\,a^2\,b-65\,d\,a\,b^2+104\,c\,b^3\right )}{27\,a^{19/3}} \]
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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