Optimal. Leaf size=279 \[ \frac {x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^4}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{14/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} b^{14/3}}+\frac {x^8 (b e-a f)}{8 b^2}+\frac {f x^{11}}{11 b} \]
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Rubi [A] time = 0.27, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1836, 1488, 292, 31, 634, 617, 204, 628} \[ \frac {x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 b^4}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{14/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} b^{14/3}}+\frac {x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac {x^8 (b e-a f)}{8 b^2}+\frac {f x^{11}}{11 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1488
Rule 1836
Rubi steps
\begin {align*} \int \frac {x^4 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac {f x^{11}}{11 b}+\frac {\int \frac {x^4 \left (11 b c+11 b d x^3+11 (b e-a f) x^6\right )}{a+b x^3} \, dx}{11 b}\\ &=\frac {f x^{11}}{11 b}+\frac {\int \left (\frac {11 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^3}+\frac {11 \left (b^2 d-a b e+a^2 f\right ) x^4}{b^2}+\frac {11 (b e-a f) x^7}{b}+\frac {11 \left (-a b^3 c+a^2 b^2 d-a^3 b e+a^4 f\right ) x}{b^3 \left (a+b x^3\right )}\right ) \, dx}{11 b}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}-\frac {\left (a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{b^4}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}+\frac {\left (a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{13/3}}-\frac {\left (a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-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}{3 b^{13/3}}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}+\frac {a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{14/3}}-\frac {\left (a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-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}{6 b^{14/3}}-\frac {\left (a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^{13/3}}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}+\frac {a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{14/3}}-\frac {a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{14/3}}-\frac {\left (a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-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 )}{b^{14/3}}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}+\frac {a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{14/3}}+\frac {a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{14/3}}-\frac {a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{14/3}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 266, normalized size = 0.95 \[ \frac {264 b^{5/3} x^5 \left (a^2 f-a b e+b^2 d\right )+660 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )-440 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-440 \sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+220 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+165 b^{8/3} x^8 (b e-a f)+120 b^{11/3} f x^{11}}{1320 b^{14/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 281, normalized size = 1.01 \[ \frac {120 \, b^{3} f x^{11} + 165 \, {\left (b^{3} e - a b^{2} f\right )} x^{8} + 264 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{5} + 660 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2} - 440 \, \sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + \sqrt {3} a}{3 \, a}\right ) + 220 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 440 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{1320 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 386, normalized size = 1.38 \[ \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{6}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{6}} + \frac {{\left (a b^{10} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{9} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} b^{7} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} b^{8} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{11}} + \frac {40 \, b^{10} f x^{11} - 55 \, a b^{9} f x^{8} + 55 \, b^{10} x^{8} e + 88 \, b^{10} d x^{5} + 88 \, a^{2} b^{8} f x^{5} - 88 \, a b^{9} x^{5} e + 220 \, b^{10} c x^{2} - 220 \, a b^{9} d x^{2} - 220 \, a^{3} b^{7} f x^{2} + 220 \, a^{2} b^{8} x^{2} e}{440 \, b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 502, normalized size = 1.80 \[ \frac {f \,x^{11}}{11 b}-\frac {a f \,x^{8}}{8 b^{2}}+\frac {e \,x^{8}}{8 b}+\frac {a^{2} f \,x^{5}}{5 b^{3}}-\frac {a e \,x^{5}}{5 b^{2}}+\frac {d \,x^{5}}{5 b}-\frac {a^{3} f \,x^{2}}{2 b^{4}}+\frac {a^{2} e \,x^{2}}{2 b^{3}}-\frac {a d \,x^{2}}{2 b^{2}}+\frac {c \,x^{2}}{2 b}+\frac {\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 )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}-\frac {a^{4} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {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 )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}-\frac {\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 )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {a^{3} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {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 )}{6 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {\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 )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {a^{2} d \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} d \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 {\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}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 269, normalized size = 0.96 \[ -\frac {\sqrt {3} {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {40 \, b^{3} f x^{11} + 55 \, {\left (b^{3} e - a b^{2} f\right )} x^{8} + 88 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{5} + 220 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2}}{440 \, b^{4}} - \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.15, size = 267, normalized size = 0.96 \[ x^8\,\left (\frac {e}{8\,b}-\frac {a\,f}{8\,b^2}\right )+x^5\,\left (\frac {d}{5\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{5\,b}\right )+x^2\,\left (\frac {c}{2\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{2\,b}\right )+\frac {f\,x^{11}}{11\,b}+\frac {a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{14/3}}-\frac {a^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{14/3}}+\frac {a^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{14/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.48, size = 469, normalized size = 1.68 \[ x^{8} \left (- \frac {a f}{8 b^{2}} + \frac {e}{8 b}\right ) + x^{5} \left (\frac {a^{2} f}{5 b^{3}} - \frac {a e}{5 b^{2}} + \frac {d}{5 b}\right ) + x^{2} \left (- \frac {a^{3} f}{2 b^{4}} + \frac {a^{2} e}{2 b^{3}} - \frac {a d}{2 b^{2}} + \frac {c}{2 b}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{14} + a^{11} f^{3} - 3 a^{10} b e f^{2} + 3 a^{9} b^{2} d f^{2} + 3 a^{9} b^{2} e^{2} f - 3 a^{8} b^{3} c f^{2} - 6 a^{8} b^{3} d e f - a^{8} b^{3} e^{3} + 6 a^{7} b^{4} c e f + 3 a^{7} b^{4} d^{2} f + 3 a^{7} b^{4} d e^{2} - 6 a^{6} b^{5} c d f - 3 a^{6} b^{5} c e^{2} - 3 a^{6} b^{5} d^{2} e + 3 a^{5} b^{6} c^{2} f + 6 a^{5} b^{6} c d e + a^{5} b^{6} d^{3} - 3 a^{4} b^{7} c^{2} e - 3 a^{4} b^{7} c d^{2} + 3 a^{3} b^{8} c^{2} d - a^{2} b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {9 t^{2} b^{9}}{a^{7} f^{2} - 2 a^{6} b e f + 2 a^{5} b^{2} d f + a^{5} b^{2} e^{2} - 2 a^{4} b^{3} c f - 2 a^{4} b^{3} d e + 2 a^{3} b^{4} c e + a^{3} b^{4} d^{2} - 2 a^{2} b^{5} c d + a b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{11}}{11 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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