Optimal. Leaf size=201 \[ \frac{(5 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{8/3}}-\frac{x^2 (5 a B+A b)}{18 a b^2 \left (a+b x^3\right )}+\frac{x^5 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.30396, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{(5 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{8/3}}-\frac{x^2 (5 a B+A b)}{18 a b^2 \left (a+b x^3\right )}+\frac{x^5 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Rubi in Sympy [A] time = 40.8409, size = 184, normalized size = 0.92 \[ \frac{x^{5} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{x^{2} \left (A b + 5 B a\right )}{18 a b^{2} \left (a + b x^{3}\right )} - \frac{\left (A b + 5 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{4}{3}} b^{\frac{8}{3}}} + \frac{\left (A b + 5 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{4}{3}} b^{\frac{8}{3}}} - \frac{\sqrt{3} \left (A b + 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{4}{3}} b^{\frac{8}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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Mathematica [A] time = 0.338836, size = 181, normalized size = 0.9 \[ \frac{\frac{(5 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}-\frac{2 (5 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}-\frac{2 \sqrt{3} (5 a B+A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{4/3}}+\frac{6 b^{2/3} x^2 (A b-4 a B)}{a \left (a+b x^3\right )}-\frac{9 b^{2/3} x^2 (A b-a B)}{\left (a+b x^3\right )^2}}{54 b^{8/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(A + B*x^3))/(a + b*x^3)^3,x]
[Out]
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Maple [A] time = 0.014, size = 241, normalized size = 1.2 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{ \left ( Ab-4\,Ba \right ){x}^{5}}{9\,ab}}-{\frac{ \left ( Ab+5\,Ba \right ){x}^{2}}{18\,{b}^{2}}} \right ) }-{\frac{A}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,B}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{A}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,B}{54\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}A}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{3}B}{27\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(B*x^3+A)/(b*x^3+a)^3,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^3 + A)*x^4/(b*x^3 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238096, size = 433, normalized size = 2.15 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \,{\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \,{\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \,{\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (2 \,{\left (4 \, B a b - A b^{2}\right )} x^{5} +{\left (5 \, B a^{2} + A a b\right )} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^4/(b*x^3 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.02935, size = 153, normalized size = 0.76 \[ - \frac{x^{5} \left (- 2 A b^{2} + 8 B a b\right ) + x^{2} \left (A a b + 5 B a^{2}\right )}{18 a^{3} b^{2} + 36 a^{2} b^{3} x^{3} + 18 a b^{4} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{4} b^{8} + A^{3} b^{3} + 15 A^{2} B a b^{2} + 75 A B^{2} a^{2} b + 125 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a^{3} b^{5}}{A^{2} b^{2} + 10 A B a b + 25 B^{2} a^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(B*x**3+A)/(b*x**3+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.224988, size = 300, normalized size = 1.49 \[ -\frac{{\left (5 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} + A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b^{2}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a + \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{4}} - \frac{8 \, B a b x^{5} - 2 \, A b^{2} x^{5} + 5 \, B a^{2} x^{2} + A a b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a + \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^4/(b*x^3 + a)^3,x, algorithm="giac")
[Out]