Optimal. Leaf size=171 \[ \frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.343308, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x^3))/(a + b*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 33.8584, size = 156, normalized size = 0.91 \[ \frac{x^{2} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{\left (A b + 2 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{4}{3}} b^{\frac{5}{3}}} + \frac{\left (A b + 2 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (A b + 2 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x**3+A)/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.17991, size = 146, normalized size = 0.85 \[ \frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{6 \sqrt [3]{a} b^{2/3} x^2 (a B-A b)}{a+b x^3}-2 (2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} (2 a B+A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^{4/3} b^{5/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x^3))/(a + b*x^3)^2,x]
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Maple [A] time = 0.012, size = 223, normalized size = 1.3 \[{\frac{ \left ( Ab-Ba \right ){x}^{2}}{3\,ab \left ( b{x}^{3}+a \right ) }}-{\frac{A}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,B}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{A}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{9\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}A}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,\sqrt{3}B}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x^3+A)/(b*x^3+a)^2,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*x^3 + A)*x/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238198, size = 293, normalized size = 1.71 \[ -\frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}}{\left (B a - A b\right )} x^{2} + \sqrt{3}{\left ({\left (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{2} + A a b\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{2} + A a b\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{2} + A a b\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right )\right )}}{54 \,{\left (a b^{2} x^{3} + a^{2} b\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.02384, size = 117, normalized size = 0.68 \[ - \frac{x^{2} \left (- A b + B a\right )}{3 a^{2} b + 3 a b^{2} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{4} b^{5} + A^{3} b^{3} + 6 A^{2} B a b^{2} + 12 A B^{2} a^{2} b + 8 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{3} b^{3}}{A^{2} b^{2} + 4 A B a b + 4 B^{2} a^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x**3+A)/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.22073, size = 273, normalized size = 1.6 \[ -\frac{{\left (2 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} + A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b} - \frac{B a x^{2} - A b x^{2}}{3 \,{\left (b x^{3} + a\right )} a b} - \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a + \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a + \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x/(b*x^3 + a)^2,x, algorithm="giac")
[Out]