Optimal. Leaf size=272 \[ \frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )}{6 d^{2/3} e^{5/3}}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )}{3 d^{2/3} e^{5/3}}+\frac{\left (2 a c+b^2\right ) \log \left (d+e x^3\right )}{3 e}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (-a e \left (a \sqrt [3]{e}+2 b \sqrt [3]{d}\right )+2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{\sqrt{3} d^{2/3} e^{5/3}}+\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e} \]
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Rubi [A] time = 0.889187, antiderivative size = 270, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^2 (-e)-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}+2 b c d\right )}{6 d^{2/3} e^{4/3}}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )}{3 d^{2/3} e^{5/3}}+\frac{\left (2 a c+b^2\right ) \log \left (d+e x^3\right )}{3 e}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (-a e \left (a \sqrt [3]{e}+2 b \sqrt [3]{d}\right )+2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{\sqrt{3} d^{2/3} e^{5/3}}+\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/(d + e*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 b c x}{e} + \frac{c^{2} \int x\, dx}{e} + \frac{\left (2 a c + b^{2}\right ) \log{\left (d + e x^{3} \right )}}{3 e} + \frac{\left (a^{2} e^{\frac{4}{3}} - 2 a b \sqrt [3]{d} e - 2 b c d \sqrt [3]{e} + c^{2} d^{\frac{4}{3}}\right ) \log{\left (\sqrt [3]{d} + \sqrt [3]{e} x \right )}}{3 d^{\frac{2}{3}} e^{\frac{5}{3}}} - \frac{\left (a^{2} e^{\frac{4}{3}} - 2 a b \sqrt [3]{d} e - 2 b c d \sqrt [3]{e} + c^{2} d^{\frac{4}{3}}\right ) \log{\left (d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{2}{3}} e^{\frac{5}{3}}} - \frac{\sqrt{3} \left (a^{2} e^{\frac{4}{3}} + 2 a b \sqrt [3]{d} e - 2 b c d \sqrt [3]{e} - c^{2} d^{\frac{4}{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} - \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{3 d^{\frac{2}{3}} e^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(e*x**3+d),x)
[Out]
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Mathematica [A] time = 0.799703, size = 269, normalized size = 0.99 \[ \frac{2 e^{2/3} \left (2 a c+b^2\right ) \log \left (d+e x^3\right )-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a e \left (a \sqrt [3]{e}-2 b \sqrt [3]{d}\right )-2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{d^{2/3}}+\frac{2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a e \left (a \sqrt [3]{e}-2 b \sqrt [3]{d}\right )-2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{d^{2/3}}+\frac{2 \sqrt{3} \left (c d^{2/3}-a e^{2/3}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (a e^{2/3}+2 b \sqrt [3]{d} \sqrt [3]{e}+c d^{2/3}\right )}{d^{2/3}}+12 b c e^{2/3} x+3 c^2 e^{2/3} x^2}{6 e^{5/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/(d + e*x^3),x]
[Out]
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Maple [B] time = 0.007, size = 444, normalized size = 1.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(e*x^3+d),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x^3 + d),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x^3 + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 15.6229, size = 546, normalized size = 2.01 \[ \frac{2 b c x}{e} + \frac{c^{2} x^{2}}{2 e} + \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{5} + t^{2} \left (- 54 a c d^{2} e^{4} - 27 b^{2} d^{2} e^{4}\right ) + t \left (18 a^{3} b d e^{4} + 27 a^{2} c^{2} d^{2} e^{3} + 9 b^{4} d^{2} e^{3} + 18 b c^{3} d^{3} e^{2}\right ) - a^{6} e^{4} - 6 a^{4} b c d e^{3} + 2 a^{3} b^{3} d e^{3} - 2 a^{3} c^{3} d^{2} e^{2} - 9 a^{2} b^{2} c^{2} d^{2} e^{2} + 6 a b^{4} c d^{2} e^{2} - 6 a b c^{4} d^{3} e - b^{6} d^{2} e^{2} + 2 b^{3} c^{3} d^{3} e - c^{6} d^{4}, \left ( t \mapsto t \log{\left (x + \frac{18 t^{2} a b d^{2} e^{4} - 9 t^{2} c^{2} d^{3} e^{3} + 3 t a^{4} d e^{4} - 36 t a^{2} b c d^{2} e^{3} - 12 t a b^{3} d^{2} e^{3} + 12 t a c^{3} d^{3} e^{2} + 18 t b^{2} c^{2} d^{3} e^{2} - 2 a^{5} c d e^{3} + 7 a^{4} b^{2} d e^{3} + 8 a^{3} b c^{2} d^{2} e^{2} - 4 a^{2} b^{3} c d^{2} e^{2} - 2 a^{2} c^{4} d^{3} e + 2 a b^{5} d^{2} e^{2} + 4 a b^{2} c^{3} d^{3} e - 5 b^{4} c^{2} d^{3} e - 4 b c^{5} d^{4}}{a^{6} e^{4} - 6 a^{4} b c d e^{3} + 8 a^{3} b^{3} d e^{3} + 6 a b c^{4} d^{3} e - 8 b^{3} c^{3} d^{3} e - c^{6} d^{4}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(e*x**3+d),x)
[Out]
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GIAC/XCAS [A] time = 0.218029, size = 383, normalized size = 1.41 \[ \frac{1}{3} \,{\left (b^{2} + 2 \, a c\right )} e^{\left (-1\right )}{\rm ln}\left ({\left | x^{3} e + d \right |}\right ) - \frac{\sqrt{3}{\left (2 \, \left (-d e^{2}\right )^{\frac{1}{3}} b c d e - \left (-d e^{2}\right )^{\frac{2}{3}} c^{2} d + 2 \, \left (-d e^{2}\right )^{\frac{2}{3}} a b e - \left (-d e^{2}\right )^{\frac{1}{3}} a^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-3\right )}}{3 \, d} + \frac{{\left (\left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} c^{2} d e^{4} + 2 \, b c d e^{4} - 2 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} a b e^{5} - a^{2} e^{5}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} e^{\left (-5\right )}{\rm ln}\left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{3 \, d} + \frac{1}{2} \,{\left (c^{2} x^{2} e + 4 \, b c x e\right )} e^{\left (-2\right )} - \frac{{\left (2 \, \left (-d e^{2}\right )^{\frac{1}{3}} b c d e + \left (-d e^{2}\right )^{\frac{2}{3}} c^{2} d - 2 \, \left (-d e^{2}\right )^{\frac{2}{3}} a b e - \left (-d e^{2}\right )^{\frac{1}{3}} a^{2} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x^3 + d),x, algorithm="giac")
[Out]