Optimal. Leaf size=173 \[ -\frac{(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{7/3}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^4}{4 b} \]
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Rubi [A] time = 0.265867, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{7/3}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^4}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^2/(a + b*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{d^{2} x^{4}}{4 b} - \frac{\left (a d - 2 b c\right ) \int d\, dx}{b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\left (a d - b c\right )^{2} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**2/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.185366, size = 167, normalized size = 0.97 \[ \frac{-2 (b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+3 a^{2/3} b^{4/3} d^2 x^4-12 a^{2/3} \sqrt [3]{b} d x (a d-2 b c)+4 (b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+4 \sqrt{3} (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{12 a^{2/3} b^{7/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3)^2/(a + b*x^3),x]
[Out]
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Maple [B] time = 0.005, size = 334, normalized size = 1.9 \[{\frac{{d}^{2}{x}^{4}}{4\,b}}-{\frac{a{d}^{2}x}{{b}^{2}}}+2\,{\frac{dxc}{b}}+{\frac{{a}^{2}{d}^{2}}{3\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,acd}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{c}^{2}}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}{d}^{2}}{6\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{acd}{3\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{c}^{2}}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{a}^{2}{d}^{2}}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,\sqrt{3}cad}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{c}^{2}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^2/(b*x^3+a),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((d*x^3 + c)^2/(b*x^3 + a),x, algorithm="maxima")
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Fricas [A] time = 0.212444, size = 259, normalized size = 1.5 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 4 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 12 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (b d^{2} x^{4} + 4 \,{\left (2 \, b c d - a d^{2}\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{36 \, \left (a^{2} b\right )^{\frac{1}{3}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^2/(b*x^3 + a),x, algorithm="fricas")
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Sympy [A] time = 2.99486, size = 156, normalized size = 0.9 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{7} - a^{6} d^{6} + 6 a^{5} b c d^{5} - 15 a^{4} b^{2} c^{2} d^{4} + 20 a^{3} b^{3} c^{3} d^{3} - 15 a^{2} b^{4} c^{4} d^{2} + 6 a b^{5} c^{5} d - b^{6} c^{6}, \left ( t \mapsto t \log{\left (\frac{3 t a b^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )} \right )\right )} + \frac{d^{2} x^{4}}{4 b} - \frac{x \left (a d^{2} - 2 b c d\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+c)**2/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.218411, size = 336, normalized size = 1.94 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b c d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b c d + \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} - \frac{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{4}} + \frac{b^{3} d^{2} x^{4} + 8 \, b^{3} c d x - 4 \, a b^{2} d^{2} x}{4 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^2/(b*x^3 + a),x, algorithm="giac")
[Out]