Optimal. Leaf size=189 \[ -\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x (c+d x)}{3 a \left (a+b x^3\right )} \]
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Rubi [A] time = 0.258876, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}+\frac{\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac{\left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{2/3}}+\frac{x (c+d x)}{3 a \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3)^3,x]
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Rubi in Sympy [A] time = 50.3101, size = 172, normalized size = 0.91 \[ \frac{x \left (c + d x\right )}{3 a \left (a + b x^{3}\right )} + \frac{\left (\frac{\sqrt [3]{a} d}{2} - \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\left (\sqrt [3]{a} d - 2 \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (\sqrt [3]{a} d + 2 \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*d*x**4+b*c*x**3+a*d*x+a*c)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.32869, size = 180, normalized size = 0.95 \[ \frac{\frac{\left (a^{2/3} d-2 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{2 \left (2 \sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac{2 \sqrt{3} \sqrt [3]{a} \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 a x (c+d x)}{a+b x^3}}{18 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3)^3,x]
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Maple [A] time = 0.004, size = 238, normalized size = 1.3 \[{\frac{cx}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{2\,c}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,c\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d{x}^{2}}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{d}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*d*x^4+b*c*x^3+a*d*x+a*c)/(b*x^3+a)^3,x)
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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*d*x^4 + b*c*x^3 + a*d*x + a*c)/(b*x^3 + a)^3,x, algorithm="maxima")
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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*d*x^4 + b*c*x^3 + a*d*x + a*c)/(b*x^3 + a)^3,x, algorithm="fricas")
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Sympy [A] time = 1.88176, size = 105, normalized size = 0.56 \[ \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{2} + 54 t a^{2} b c d + a d^{3} - 8 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a^{4} b d + 36 t a^{2} b c^{2} + 4 a c d^{2}}{a d^{3} + 8 b c^{3}} \right )} \right )\right )} + \frac{c x + d x^{2}}{3 a^{2} + 3 a b x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x**4+b*c*x**3+a*d*x+a*c)/(b*x**3+a)**3,x)
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GIAC/XCAS [A] time = 0.22452, size = 252, normalized size = 1.33 \[ -\frac{{\left (d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{d x^{2} + c x}{3 \,{\left (b x^{3} + a\right )} a} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^4 + b*c*x^3 + a*d*x + a*c)/(b*x^3 + a)^3,x, algorithm="giac")
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