Optimal. Leaf size=362 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{54 a^{10/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{27 a^{10/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{2/3} b e+a^{5/3} (-h)-2 a b^{2/3} f+14 b^{5/3} c\right )}{9 \sqrt{3} a^{10/3} b^{4/3}}+\frac{x \left (-2 b x (5 b c-2 a f)-3 b x^2 (3 b d-a g)+a (a h+5 b e)\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3} \]
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Rubi [A] time = 0.829961, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{54 a^{10/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{2/3} (a h+5 b e)+2 b^{2/3} (7 b c-a f)\right )}{27 a^{10/3} b^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-5 a^{2/3} b e+a^{5/3} (-h)-2 a b^{2/3} f+14 b^{5/3} c\right )}{9 \sqrt{3} a^{10/3} b^{4/3}}+\frac{x \left (-2 b x (5 b c-2 a f)-3 b x^2 (3 b d-a g)+a (a h+5 b e)\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1834
Rule 1871
Rule 1860
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^2 \left (a+b x^3\right )^3} \, dx &=\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^2 c-6 b^2 d x-b (5 b e+a h) x^2+4 b^2 \left (\frac{b c}{a}-f\right ) x^3+3 b^2 \left (\frac{b d}{a}-g\right ) x^4}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b^2}\\ &=\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{\int \frac{18 b^4 c+18 b^4 d x+2 b^3 (5 b e+a h) x^2-2 b^4 \left (\frac{5 b c}{a}-2 f\right ) x^3}{x^2 \left (a+b x^3\right )} \, dx}{18 a^2 b^4}\\ &=\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{\int \left (\frac{18 b^4 c}{a x^2}+\frac{18 b^4 d}{a x}+\frac{2 b^3 \left (a (5 b e+a h)-2 b (7 b c-a f) x-9 b^2 d x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^4}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}+\frac{\int \frac{a (5 b e+a h)-2 b (7 b c-a f) x-9 b^2 d x^2}{a+b x^3} \, dx}{9 a^3 b}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}+\frac{\int \frac{a (5 b e+a h)-2 b (7 b c-a f) x}{a+b x^3} \, dx}{9 a^3 b}-\frac{(b d) \int \frac{x^2}{a+b x^3} \, dx}{a^3}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}-\frac{d \log \left (a+b x^3\right )}{3 a^3}+\frac{\int \frac{\sqrt [3]{a} \left (-2 \sqrt [3]{a} b (7 b c-a f)+2 a \sqrt [3]{b} (5 b e+a h)\right )+\sqrt [3]{b} \left (-2 \sqrt [3]{a} b (7 b c-a f)-a \sqrt [3]{b} (5 b e+a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3} b^{4/3}}+\frac{\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3} b}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}+\frac{\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{4/3}}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{\left (14 b^{5/3} c-5 a^{2/3} b e-2 a b^{2/3} f-a^{5/3} h\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^3 b}-\frac{\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\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^{10/3} b^{4/3}}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{d \log (x)}{a^3}+\frac{\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{4/3}}-\frac{\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} b^{4/3}}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{\left (14 b^{5/3} c-5 a^{2/3} b e-2 a b^{2/3} f-a^{5/3} h\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^{10/3} b^{4/3}}\\ &=-\frac{c}{a^3 x}+\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac{x \left (a (5 b e+a h)-2 b (5 b c-2 a f) x-3 b (3 b d-a g) x^2\right )}{18 a^3 b \left (a+b x^3\right )}+\frac{\left (14 b^{5/3} c-5 a^{2/3} b e-2 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 )}{9 \sqrt{3} a^{10/3} b^{4/3}}+\frac{d \log (x)}{a^3}+\frac{\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{4/3}}-\frac{\left (2 b^{2/3} (7 b c-a f)+a^{2/3} (5 b e+a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} b^{4/3}}-\frac{d \log \left (a+b x^3\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.569985, size = 336, normalized size = 0.93 \[ -\frac{\frac{9 a^2 \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{b \left (a+b x^3\right )^2}-\frac{3 a \left (a^2 h x+a b (6 d+x (5 e+4 f x))-10 b^2 c x^2\right )}{b \left (a+b x^3\right )}+\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{2/3} b e+a^{5/3} h-2 a b^{2/3} f+14 b^{5/3} c\right )}{b^{4/3}}-\frac{2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^{2/3} b e+a^{5/3} h-2 a b^{2/3} f+14 b^{5/3} c\right )}{b^{4/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{2/3} b e+a^{5/3} h+2 a b^{2/3} f-14 b^{5/3} c\right )}{b^{4/3}}+18 a d \log \left (a+b x^3\right )+\frac{54 a c}{x}-54 a d \log (x)}{54 a^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 622, 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 [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 [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 = 1.09166, size = 556, normalized size = 1.54 \begin{align*} -\frac{d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{d \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c - 2 \, \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 )}{27 \, a^{4} b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} h + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b e - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b c + 2 \, \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 )}{54 \, a^{4} b^{2}} + \frac{6 \, a b^{2} d x^{4} - 4 \,{\left (7 \, b^{3} c - a b^{2} f\right )} x^{6} +{\left (a^{2} b h + 5 \, a b^{2} e\right )} x^{5} - 18 \, a^{2} b c - 7 \,{\left (7 \, a b^{2} c - a^{2} b f\right )} x^{3} - 2 \,{\left (a^{3} h - 4 \, a^{2} b e\right )} x^{2} + 3 \,{\left (3 \, a^{2} b d - a^{3} g\right )} x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} b x} + \frac{{\left (14 \, a^{3} b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{4} b^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{5} b^{2} h - 5 \, a^{4} b^{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} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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