Optimal. Leaf size=110 \[ a^4 A \log (x)+a^4 B x+2 a^3 A c x^2+\frac{4}{3} a^3 B c x^3+\frac{3}{2} a^2 A c^2 x^4+\frac{6}{5} a^2 B c^2 x^5+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]
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Rubi [A] time = 0.105935, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a^4 A \log (x)+a^4 B x+2 a^3 A c x^2+\frac{4}{3} a^3 B c x^3+\frac{3}{2} a^2 A c^2 x^4+\frac{6}{5} a^2 B c^2 x^5+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^4)/x,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ A a^{4} \log{\left (x \right )} + 4 A a^{3} c \int x\, dx + \frac{3 A a^{2} c^{2} x^{4}}{2} + \frac{2 A a c^{3} x^{6}}{3} + \frac{A c^{4} x^{8}}{8} + \frac{4 B a^{3} c x^{3}}{3} + \frac{6 B a^{2} c^{2} x^{5}}{5} + \frac{4 B a c^{3} x^{7}}{7} + \frac{B c^{4} x^{9}}{9} + a^{4} \int B\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**4/x,x)
[Out]
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Mathematica [A] time = 0.0128156, size = 110, normalized size = 1. \[ a^4 A \log (x)+a^4 B x+2 a^3 A c x^2+\frac{4}{3} a^3 B c x^3+\frac{3}{2} a^2 A c^2 x^4+\frac{6}{5} a^2 B c^2 x^5+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^4)/x,x]
[Out]
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Maple [A] time = 0.004, size = 97, normalized size = 0.9 \[{a}^{4}Bx+2\,{a}^{3}Ac{x}^{2}+{\frac{4\,{a}^{3}Bc{x}^{3}}{3}}+{\frac{3\,{a}^{2}A{c}^{2}{x}^{4}}{2}}+{\frac{6\,{a}^{2}B{c}^{2}{x}^{5}}{5}}+{\frac{2\,aA{c}^{3}{x}^{6}}{3}}+{\frac{4\,aB{c}^{3}{x}^{7}}{7}}+{\frac{A{c}^{4}{x}^{8}}{8}}+{\frac{B{c}^{4}{x}^{9}}{9}}+{a}^{4}A\ln \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^4/x,x)
[Out]
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Maxima [A] time = 0.685199, size = 130, normalized size = 1.18 \[ \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^4*(B*x + A)/x,x, algorithm="maxima")
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Fricas [A] time = 0.267559, size = 130, normalized size = 1.18 \[ \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^4*(B*x + A)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.41325, size = 117, normalized size = 1.06 \[ A a^{4} \log{\left (x \right )} + 2 A a^{3} c x^{2} + \frac{3 A a^{2} c^{2} x^{4}}{2} + \frac{2 A a c^{3} x^{6}}{3} + \frac{A c^{4} x^{8}}{8} + B a^{4} x + \frac{4 B a^{3} c x^{3}}{3} + \frac{6 B a^{2} c^{2} x^{5}}{5} + \frac{4 B a c^{3} x^{7}}{7} + \frac{B c^{4} x^{9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**4/x,x)
[Out]
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GIAC/XCAS [A] time = 0.278248, size = 131, normalized size = 1.19 \[ \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4}{\rm ln}\left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^4*(B*x + A)/x,x, algorithm="giac")
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