Optimal. Leaf size=375 \[ -\frac{b^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{4 a^5 x^4}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{18 a^{19/3}}-\frac{b \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{3 \sqrt{3} a^{19/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{c}{13 a^2 x^{13}} \]
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Rubi [A] time = 0.533546, antiderivative size = 375, normalized size of antiderivative = 1., 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 = {1829, 1834, 292, 31, 634, 617, 204, 628} \[ -\frac{b^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{4 a^5 x^4}-\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{18 a^{19/3}}-\frac{b \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{3 \sqrt{3} a^{19/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{c}{13 a^2 x^{13}} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1834
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}-\frac{\int \frac{-3 b^3 c+3 b^3 \left (\frac{b c}{a}-d\right ) x^3-\frac{3 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac{3 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac{3 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}+\frac{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 )} \, dx}{3 a b^3}\\ &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^{14}}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^{11}}-\frac{3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^8}-\frac{3 b^3 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^5}+\frac{3 b^4 \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^5 x^2}-\frac{b^5 \left (-16 b^3 c+13 a b^2 d-10 a^2 b e+7 a^3 f\right ) x}{a^5 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{13 a^2 x^{13}}+\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}-\frac{\left (b^2 \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac{x}{a+b x^3} \, dx}{3 a^6}\\ &=-\frac{c}{13 a^2 x^{13}}+\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac{\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{19/3}}-\frac{\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 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 a^{19/3}}\\ &=-\frac{c}{13 a^2 x^{13}}+\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac{b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac{\left (b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 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 a^{19/3}}-\frac{\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 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 a^6}\\ &=-\frac{c}{13 a^2 x^{13}}+\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac{b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac{b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{19/3}}-\frac{\left (b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 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 a^{19/3}}\\ &=-\frac{c}{13 a^2 x^{13}}+\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac{b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 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} a^{19/3}}+\frac{b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac{b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{19/3}}\\ \end{align*}
Mathematica [A] time = 0.360017, size = 370, normalized size = 0.99 \[ \frac{b^2 x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{4 a^5 x^4}+\frac{b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-10 a^2 b e+7 a^3 f+13 a b^2 d-16 b^3 c\right )}{18 a^{19/3}}+\frac{b \left (-3 a^2 b e+2 a^3 f+4 a b^2 d-5 b^3 c\right )}{a^6 x}+\frac{b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac{b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{3 \sqrt{3} a^{19/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac{2 b c-a d}{10 a^3 x^{10}}-\frac{c}{13 a^2 x^{13}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 631, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36125, size = 1188, normalized size = 3.17 \begin{align*} -\frac{5460 \,{\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{15} + 4095 \,{\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{12} - 585 \,{\left (16 \, a^{2} b^{3} c - 13 \, a^{3} b^{2} d + 10 \, a^{4} b e - 7 \, a^{5} f\right )} x^{9} + 234 \,{\left (16 \, a^{3} b^{2} c - 13 \, a^{4} b d + 10 \, a^{5} e\right )} x^{6} + 1260 \, a^{5} c - 126 \,{\left (16 \, a^{4} b c - 13 \, a^{5} d\right )} x^{3} + 1820 \, \sqrt{3}{\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} +{\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} 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 (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} +{\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} 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 (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} +{\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} 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 )}{16380 \,{\left (a^{6} b x^{16} + a^{7} x^{13}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08059, size = 651, normalized size = 1.74 \begin{align*} \frac{\sqrt{3}{\left (16 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 10 \, \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 \, a^{7}} + \frac{{\left (16 \, b^{5} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 13 \, a b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 7 \, a^{3} b^{2} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 10 \, 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 )}{9 \, a^{7}} - \frac{{\left (16 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 10 \, \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 \, a^{7}} - \frac{b^{5} c x^{2} - a b^{4} d x^{2} - a^{3} b^{2} f x^{2} + a^{2} b^{3} x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{6}} - \frac{9100 \, b^{4} c x^{12} - 7280 \, a b^{3} d x^{12} - 3640 \, a^{3} b f x^{12} + 5460 \, a^{2} b^{2} x^{12} e - 1820 \, a b^{3} c x^{9} + 1365 \, a^{2} b^{2} d x^{9} + 455 \, a^{4} f x^{9} - 910 \, a^{3} b x^{9} e + 780 \, a^{2} b^{2} c x^{6} - 520 \, a^{3} b d x^{6} + 260 \, a^{4} x^{6} e - 364 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{6} x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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