Optimal. Leaf size=331 \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 b^{10/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 b^{10/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} b^{10/3}}+\frac{x^2 (b c-a f)}{2 b^2}+\frac{x^3 (b d-a g)}{3 b^2}-\frac{a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}+\frac{x^4 (b e-a h)}{4 b^2}-\frac{a x (b e-a h)}{b^3}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b} \]
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Rubi [A] time = 1.06967, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1836, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{6 b^{10/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (b e-a h)+b^{2/3} (b c-a f)\right )}{3 b^{10/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} b^{10/3}}+\frac{x^2 (b c-a f)}{2 b^2}+\frac{x^3 (b d-a g)}{3 b^2}-\frac{a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}+\frac{x^4 (b e-a h)}{4 b^2}-\frac{a x (b e-a h)}{b^3}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b} \]
Antiderivative was successfully verified.
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Rule 1836
Rule 1887
Rule 1871
Rule 1860
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{x^4 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{a+b x^3} \, dx &=\frac{h x^7}{7 b}+\frac{\int \frac{x^4 \left (7 b c+7 b d x+7 (b e-a h) x^2+7 b f x^3+7 b g x^4\right )}{a+b x^3} \, dx}{7 b}\\ &=\frac{g x^6}{6 b}+\frac{h x^7}{7 b}+\frac{\int \frac{x^4 \left (42 b^2 c+42 b (b d-a g) x+42 b (b e-a h) x^2+42 b^2 f x^3\right )}{a+b x^3} \, dx}{42 b^2}\\ &=\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}+\frac{\int \frac{x^4 \left (210 b^2 (b c-a f)+210 b^2 (b d-a g) x+210 b^2 (b e-a h) x^2\right )}{a+b x^3} \, dx}{210 b^3}\\ &=\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}+\frac{\int \left (-210 a (b e-a h)+210 b (b c-a f) x+210 b (b d-a g) x^2+210 b (b e-a h) x^3+\frac{210 \left (a^2 (b e-a h)-a b (b c-a f) x-a b (b d-a g) x^2\right )}{a+b x^3}\right ) \, dx}{210 b^3}\\ &=-\frac{a (b e-a h) x}{b^3}+\frac{(b c-a f) x^2}{2 b^2}+\frac{(b d-a g) x^3}{3 b^2}+\frac{(b e-a h) x^4}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}+\frac{\int \frac{a^2 (b e-a h)-a b (b c-a f) x-a b (b d-a g) x^2}{a+b x^3} \, dx}{b^3}\\ &=-\frac{a (b e-a h) x}{b^3}+\frac{(b c-a f) x^2}{2 b^2}+\frac{(b d-a g) x^3}{3 b^2}+\frac{(b e-a h) x^4}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}+\frac{\int \frac{a^2 (b e-a h)-a b (b c-a f) x}{a+b x^3} \, dx}{b^3}-\frac{(a (b d-a g)) \int \frac{x^2}{a+b x^3} \, dx}{b^2}\\ &=-\frac{a (b e-a h) x}{b^3}+\frac{(b c-a f) x^2}{2 b^2}+\frac{(b d-a g) x^3}{3 b^2}+\frac{(b e-a h) x^4}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}-\frac{a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}+\frac{\int \frac{\sqrt [3]{a} \left (-a^{4/3} b (b c-a f)+2 a^2 \sqrt [3]{b} (b e-a h)\right )+\sqrt [3]{b} \left (-a^{4/3} b (b c-a f)-a^2 \sqrt [3]{b} (b e-a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^{10/3}}+\frac{\left (a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^3}\\ &=-\frac{a (b e-a h) x}{b^3}+\frac{(b c-a f) x^2}{2 b^2}+\frac{(b d-a g) x^3}{3 b^2}+\frac{(b e-a h) x^4}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}+\frac{a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac{\left (a \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^3}-\frac{\left (a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\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^{10/3}}\\ &=-\frac{a (b e-a h) x}{b^3}+\frac{(b c-a f) x^2}{2 b^2}+\frac{(b d-a g) x^3}{3 b^2}+\frac{(b e-a h) x^4}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}+\frac{a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}-\frac{a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}-\frac{\left (a^{2/3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\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^{10/3}}\\ &=-\frac{a (b e-a h) x}{b^3}+\frac{(b c-a f) x^2}{2 b^2}+\frac{(b d-a g) x^3}{3 b^2}+\frac{(b e-a h) x^4}{4 b^2}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b}+\frac{a^{2/3} \left (b^{5/3} c-a^{2/3} b e-a b^{2/3} f+a^{5/3} h\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{10/3}}+\frac{a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a^{2/3} \left (b^{2/3} (b c-a f)+a^{2/3} (b e-a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}-\frac{a (b d-a g) \log \left (a+b x^3\right )}{3 b^3}\\ \end{align*}
Mathematica [A] time = 0.438062, size = 334, normalized size = 1.01 \[ \frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b e+a^{5/3} h+a b^{2/3} f-b^{5/3} c\right )}{6 b^{10/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} b e+a^{5/3} (-h)-a b^{2/3} f+b^{5/3} c\right )}{3 b^{10/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^{2/3} b e+a^{5/3} h-a b^{2/3} f+b^{5/3} c\right )}{\sqrt{3} b^{10/3}}+\frac{x^2 (b c-a f)}{2 b^2}+\frac{x^3 (b d-a g)}{3 b^2}+\frac{a (a g-b d) \log \left (a+b x^3\right )}{3 b^3}+\frac{x^4 (b e-a h)}{4 b^2}+\frac{a x (a h-b e)}{b^3}+\frac{f x^5}{5 b}+\frac{g x^6}{6 b}+\frac{h x^7}{7 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 533, normalized size = 1.6 \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 [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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Sympy [B] time = 29.1659, size = 874, normalized size = 2.64 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{10} + t^{2} \left (- 27 a^{2} b^{7} g + 27 a b^{8} d\right ) + t \left (- 9 a^{4} b^{4} f h + 9 a^{4} b^{4} g^{2} + 9 a^{3} b^{5} c h - 18 a^{3} b^{5} d g + 9 a^{3} b^{5} e f - 9 a^{2} b^{6} c e + 9 a^{2} b^{6} d^{2}\right ) + a^{7} h^{3} - 3 a^{6} b e h^{2} + 3 a^{6} b f g h - a^{6} b g^{3} - 3 a^{5} b^{2} c g h - 3 a^{5} b^{2} d f h + 3 a^{5} b^{2} d g^{2} + 3 a^{5} b^{2} e^{2} h - 3 a^{5} b^{2} e f g + a^{5} b^{2} f^{3} + 3 a^{4} b^{3} c d h + 3 a^{4} b^{3} c e g - 3 a^{4} b^{3} c f^{2} - 3 a^{4} b^{3} d^{2} g + 3 a^{4} b^{3} d e f - a^{4} b^{3} e^{3} + 3 a^{3} b^{4} c^{2} f - 3 a^{3} b^{4} c d e + a^{3} b^{4} d^{3} - a^{2} b^{5} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 9 t^{2} a b^{7} f + 9 t^{2} b^{8} c - 3 t a^{4} b^{3} h^{2} + 6 t a^{3} b^{4} e h + 6 t a^{3} b^{4} f g - 6 t a^{2} b^{5} c g - 6 t a^{2} b^{5} d f - 3 t a^{2} b^{5} e^{2} + 6 t a b^{6} c d + a^{6} g h^{2} - a^{5} b d h^{2} - 2 a^{5} b e g h + 2 a^{5} b f^{2} h - a^{5} b f g^{2} - 4 a^{4} b^{2} c f h + a^{4} b^{2} c g^{2} + 2 a^{4} b^{2} d e h + 2 a^{4} b^{2} d f g + a^{4} b^{2} e^{2} g - 2 a^{4} b^{2} e f^{2} + 2 a^{3} b^{3} c^{2} h - 2 a^{3} b^{3} c d g + 4 a^{3} b^{3} c e f - a^{3} b^{3} d^{2} f - a^{3} b^{3} d e^{2} - 2 a^{2} b^{4} c^{2} e + a^{2} b^{4} c d^{2}}{a^{6} h^{3} - 3 a^{5} b e h^{2} + 3 a^{4} b^{2} e^{2} h - a^{4} b^{2} f^{3} + 3 a^{3} b^{3} c f^{2} - a^{3} b^{3} e^{3} - 3 a^{2} b^{4} c^{2} f + a b^{5} c^{3}} \right )} \right )\right )} + \frac{f x^{5}}{5 b} + \frac{g x^{6}}{6 b} + \frac{h x^{7}}{7 b} - \frac{x^{4} \left (a h - b e\right )}{4 b^{2}} - \frac{x^{3} \left (a g - b d\right )}{3 b^{2}} - \frac{x^{2} \left (a f - b c\right )}{2 b^{2}} + \frac{x \left (a^{2} h - a b e\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07807, size = 513, normalized size = 1.55 \begin{align*} -\frac{{\left (a b d - a^{2} g\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e - \left (-a b^{2}\right )^{\frac{2}{3}} b c + \left (-a b^{2}\right )^{\frac{2}{3}} a 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^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h - \left (-a b^{2}\right )^{\frac{1}{3}} a b e + \left (-a b^{2}\right )^{\frac{2}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} a 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^{4}} + \frac{60 \, b^{6} h x^{7} + 70 \, b^{6} g x^{6} + 84 \, b^{6} f x^{5} - 105 \, a b^{5} h x^{4} + 105 \, b^{6} x^{4} e + 140 \, b^{6} d x^{3} - 140 \, a b^{5} g x^{3} + 210 \, b^{6} c x^{2} - 210 \, a b^{5} f x^{2} + 420 \, a^{2} b^{4} h x - 420 \, a b^{5} x e}{420 \, b^{7}} + \frac{{\left (a b^{14} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{13} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} b^{12} h - a^{2} b^{13} 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^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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