Optimal. Leaf size=186 \[ \frac{a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{2/3}}-\frac{a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}-\frac{a \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{2/3}}+\frac{b^2 \log \left (c+d x^3\right )}{3 d} \]
[Out]
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Rubi [A] time = 0.331619, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{2/3}}-\frac{a \left (2 b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{2/3}}-\frac{a \left (a \sqrt [3]{d}+2 b \sqrt [3]{c}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{2/3}}+\frac{b^2 \log \left (c+d x^3\right )}{3 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2/(c + d*x^3),x]
[Out]
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Rubi in Sympy [A] time = 42.178, size = 175, normalized size = 0.94 \[ \frac{a \left (a \sqrt [3]{d} - 2 b \sqrt [3]{c}\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} d^{\frac{2}{3}}} - \frac{a \left (a \sqrt [3]{d} - 2 b \sqrt [3]{c}\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} d^{\frac{2}{3}}} - \frac{\sqrt{3} a \left (a \sqrt [3]{d} + 2 b \sqrt [3]{c}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} d^{\frac{2}{3}}} + \frac{b^{2} \log{\left (c + d x^{3} \right )}}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2/(d*x**3+c),x)
[Out]
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Mathematica [A] time = 0.165063, size = 200, normalized size = 1.08 \[ -\frac{\left (a^2 \sqrt [3]{c} \sqrt [3]{d}-2 a b c^{2/3}\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c d^{2/3}}+\frac{\left (a^2 \sqrt [3]{c} \sqrt [3]{d}-2 a b c^{2/3}\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c d^{2/3}}+\frac{\left (a^2 \sqrt [3]{c} \sqrt [3]{d}+2 a b c^{2/3}\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} x-\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c d^{2/3}}+\frac{b^2 \log \left (c+d x^3\right )}{3 d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2/(c + d*x^3),x]
[Out]
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Maple [A] time = 0.004, size = 211, normalized size = 1.1 \[{\frac{{a}^{2}}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{6\,d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}\sqrt{3}}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,ab}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{ab}{3\,d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{2\,\sqrt{3}ab}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{b}^{2}\ln \left ( d{x}^{3}+c \right ) }{3\,d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2/(d*x^3+c),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((b*x + a)^2/(d*x^3 + c),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((b*x + a)^2/(d*x^3 + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.98675, size = 156, normalized size = 0.84 \[ \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{3} - 27 t^{2} b^{2} c^{2} d^{2} + t \left (18 a^{3} b c d^{2} + 9 b^{4} c^{2} d\right ) - a^{6} d^{2} + 2 a^{3} b^{3} c d - b^{6} c^{2}, \left ( t \mapsto t \log{\left (x + \frac{18 t^{2} b c^{2} d^{2} + 3 t a^{3} c d^{2} - 12 t b^{3} c^{2} d + 7 a^{3} b^{2} c d + 2 b^{5} c^{2}}{a^{5} d^{2} + 8 a^{2} b^{3} c d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2/(d*x**3+c),x)
[Out]
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GIAC/XCAS [A] time = 0.21605, size = 262, normalized size = 1.41 \[ \frac{b^{2}{\rm ln}\left ({\left | d x^{3} + c \right |}\right )}{3 \, d} - \frac{{\left (2 \, a b d \left (-\frac{c}{d}\right )^{\frac{1}{3}} + a^{2} d\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d} + \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d - 2 \, \left (-c d^{2}\right )^{\frac{2}{3}} a b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \, c d^{2}} + \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} a^{2} c d^{3} + 2 \, \left (-c d^{2}\right )^{\frac{2}{3}} a b c d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \, c^{2} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(d*x^3 + c),x, algorithm="giac")
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