Optimal. Leaf size=416 \[ -\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-e \left (b^3 d-a^3 e\right )-6 a b c d e+c^3 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-e \left (b^3 d-a^3 e\right )-6 a b c d e+c^3 d^2\right )}{3 d^{2/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2-6 a b c d e-3 a c^2 d^{4/3} e^{2/3}-3 b^2 c d^{4/3} e^{2/3}-b^3 d e+c^3 d^2\right )}{\sqrt{3} d^{2/3} e^{7/3}}-\frac{\log \left (d+e x^3\right ) \left (a^2 (-c) e-a b^2 e+b c^2 d\right )}{e^2}-\frac{x \left (-6 a b c e+b^3 (-e)+c^3 d\right )}{e^2}+\frac{3 c x^2 \left (a c+b^2\right )}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e} \]
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Rubi [A] time = 0.70029, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-e \left (b^3 d-a^3 e\right )-6 a b c d e+c^3 d^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (3 \sqrt [3]{d} e^{2/3} \left (a^2 (-b) e+a c^2 d+b^2 c d\right )-e \left (b^3 d-a^3 e\right )-6 a b c d e+c^3 d^2\right )}{3 d^{2/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2-6 a b c d e-3 a c^2 d^{4/3} e^{2/3}-3 b^2 c d^{4/3} e^{2/3}-b^3 d e+c^3 d^2\right )}{\sqrt{3} d^{2/3} e^{7/3}}-\frac{\log \left (d+e x^3\right ) \left (a^2 (-c) e-a b^2 e+b c^2 d\right )}{e^2}-\frac{x \left (-6 a b c e+b^3 (-e)+c^3 d\right )}{e^2}+\frac{3 c x^2 \left (a c+b^2\right )}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e} \]
Antiderivative was successfully verified.
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Rule 1887
Rule 1871
Rule 1860
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{d+e x^3} \, dx &=\int \left (-\frac{c^3 d-b^3 e-6 a b c e}{e^2}+\frac{3 c \left (b^2+a c\right ) x}{e}+\frac{3 b c^2 x^2}{e}+\frac{c^3 x^3}{e}+\frac{c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )-3 e \left (b^2 c d+a c^2 d-a^2 b e\right ) x-3 e \left (b c^2 d-a b^2 e-a^2 c e\right ) x^2}{e^2 \left (d+e x^3\right )}\right ) \, dx\\ &=-\frac{\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac{3 c \left (b^2+a c\right ) x^2}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e}+\frac{\int \frac{c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )-3 e \left (b^2 c d+a c^2 d-a^2 b e\right ) x-3 e \left (b c^2 d-a b^2 e-a^2 c e\right ) x^2}{d+e x^3} \, dx}{e^2}\\ &=-\frac{\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac{3 c \left (b^2+a c\right ) x^2}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e}+\frac{\int \frac{c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )-3 e \left (b^2 c d+a c^2 d-a^2 b e\right ) x}{d+e x^3} \, dx}{e^2}-\frac{\left (3 \left (b c^2 d-a b^2 e-a^2 c e\right )\right ) \int \frac{x^2}{d+e x^3} \, dx}{e}\\ &=-\frac{\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac{3 c \left (b^2+a c\right ) x^2}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e}-\frac{\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2}+\frac{\int \frac{\sqrt [3]{d} \left (-3 \sqrt [3]{d} e \left (b^2 c d+a c^2 d-a^2 b e\right )+2 \sqrt [3]{e} \left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )\right )\right )+\sqrt [3]{e} \left (-3 \sqrt [3]{d} e \left (b^2 c d+a c^2 d-a^2 b e\right )-\sqrt [3]{e} \left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )\right )\right ) x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{3 d^{2/3} e^{7/3}}+\frac{\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{2/3} e^2}\\ &=-\frac{\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac{3 c \left (b^2+a c\right ) x^2}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e}+\frac{\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac{\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2}+\frac{\left (c^3 d^2-3 b^2 c d^{4/3} e^{2/3}-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e+3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \int \frac{1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d} e^2}-\frac{\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \int \frac{-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{2/3} e^{7/3}}\\ &=-\frac{\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac{3 c \left (b^2+a c\right ) x^2}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e}+\frac{\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac{\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}-\frac{\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2}+\frac{\left (c^3 d^2-3 b^2 c d^{4/3} e^{2/3}-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e+3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} e^{7/3}}\\ &=-\frac{\left (c^3 d-b^3 e-6 a b c e\right ) x}{e^2}+\frac{3 c \left (b^2+a c\right ) x^2}{2 e}+\frac{b c^2 x^3}{e}+\frac{c^3 x^4}{4 e}-\frac{\left (c^3 d^2-3 b^2 c d^{4/3} e^{2/3}-3 a c^2 d^{4/3} e^{2/3}-b^3 d e-6 a b c d e+3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{7/3}}+\frac{\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac{\left (c^3 d^2-6 a b c d e-e \left (b^3 d-a^3 e\right )+3 \sqrt [3]{d} e^{2/3} \left (b^2 c d+a c^2 d-a^2 b e\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}-\frac{\left (b c^2 d-a b^2 e-a^2 c e\right ) \log \left (d+e x^3\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.451074, size = 439, normalized size = 1.06 \[ \frac{-\frac{2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (-3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2-6 a b c d e+3 a c^2 d^{4/3} e^{2/3}+3 b^2 c d^{4/3} e^{2/3}-b^3 d e+c^3 d^2\right )}{d^{2/3}}+\frac{4 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (-3 a^2 b \sqrt [3]{d} e^{5/3}+a^3 e^2-6 a b c d e+3 a c^2 d^{4/3} e^{2/3}+3 b^2 c d^{4/3} e^{2/3}-b^3 d e+c^3 d^2\right )}{d^{2/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (e \left (3 a^2 b \sqrt [3]{d} e^{2/3}+a^3 e-b^3 d\right )-3 c \left (2 a b d e+b^2 d^{4/3} e^{2/3}\right )-3 a c^2 d^{4/3} e^{2/3}+c^3 d^2\right )}{d^{2/3}}+12 \sqrt [3]{e} \log \left (d+e x^3\right ) \left (a^2 c e+a b^2 e-b c^2 d\right )+12 \sqrt [3]{e} x \left (6 a b c e+b^3 e-c^3 d\right )+18 c e^{4/3} x^2 \left (a c+b^2\right )+12 b c^2 e^{4/3} x^3+3 c^3 e^{4/3} x^4}{12 e^{7/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 837, normalized size = 2. \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 = 86.1001, size = 1314, normalized size = 3.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06767, size = 648, normalized size = 1.56 \begin{align*} -{\left (b c^{2} d - a b^{2} e - a^{2} c e\right )} e^{\left (-2\right )} \log \left ({\left | x^{3} e + d \right |}\right ) + \frac{\sqrt{3}{\left (\left (-d e^{2}\right )^{\frac{1}{3}} c^{3} d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b^{3} d e - 6 \, \left (-d e^{2}\right )^{\frac{1}{3}} a b c d e + 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} b^{2} c d + 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} a c^{2} d - 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} a^{2} b e + \left (-d e^{2}\right )^{\frac{1}{3}} a^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-3\right )}}{3 \, d} - \frac{{\left (c^{3} d^{2} e^{7} - 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} b^{2} c d e^{8} - 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} a c^{2} d e^{8} - b^{3} d e^{8} - 6 \, a b c d e^{8} + 3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} a^{2} b e^{9} + a^{3} e^{9}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} e^{\left (-9\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{3 \, d} + \frac{1}{4} \,{\left (c^{3} x^{4} e^{3} + 4 \, b c^{2} x^{3} e^{3} + 6 \, b^{2} c x^{2} e^{3} + 6 \, a c^{2} x^{2} e^{3} - 4 \, c^{3} d x e^{2} + 4 \, b^{3} x e^{3} + 24 \, a b c x e^{3}\right )} e^{\left (-4\right )} + \frac{{\left (\left (-d e^{2}\right )^{\frac{1}{3}} c^{3} d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b^{3} d e - 6 \, \left (-d e^{2}\right )^{\frac{1}{3}} a b c d e - 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} b^{2} c d - 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} a c^{2} d + 3 \, \left (-d e^{2}\right )^{\frac{2}{3}} a^{2} b e + \left (-d e^{2}\right )^{\frac{1}{3}} a^{3} e^{2}\right )} e^{\left (-3\right )} \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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