Optimal. Leaf size=66 \[ a^4 A \log (x)+4 a^3 A b x+3 a^2 A b^2 x^2+\frac{4}{3} a A b^3 x^3+\frac{B (a+b x)^5}{5 b}+\frac{1}{4} A b^4 x^4 \]
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Rubi [A] time = 0.0611273, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ a^4 A \log (x)+4 a^3 A b x+3 a^2 A b^2 x^2+\frac{4}{3} a A b^3 x^3+\frac{B (a+b x)^5}{5 b}+\frac{1}{4} A b^4 x^4 \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ A a^{4} \log{\left (x \right )} + 4 A a^{3} b x + 6 A a^{2} b^{2} \int x\, dx + \frac{4 A a b^{3} x^{3}}{3} + \frac{A b^{4} x^{4}}{4} + \frac{B \left (a + b x\right )^{5}}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x,x)
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Mathematica [A] time = 0.0425827, size = 83, normalized size = 1.26 \[ a^4 A \log (x)+a^4 B x+2 a^3 b x (2 A+B x)+a^2 b^2 x^2 (3 A+2 B x)+\frac{1}{3} a b^3 x^3 (4 A+3 B x)+\frac{1}{20} b^4 x^4 (5 A+4 B x) \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x,x]
[Out]
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Maple [A] time = 0.005, size = 94, normalized size = 1.4 \[{\frac{{b}^{4}B{x}^{5}}{5}}+{\frac{A{b}^{4}{x}^{4}}{4}}+B{x}^{4}a{b}^{3}+{\frac{4\,aA{b}^{3}{x}^{3}}{3}}+2\,B{x}^{3}{a}^{2}{b}^{2}+3\,{a}^{2}A{b}^{2}{x}^{2}+2\,B{x}^{2}{a}^{3}b+4\,{a}^{3}Abx+{a}^{4}Bx+{a}^{4}A\ln \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x,x)
[Out]
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Maxima [A] time = 0.678168, size = 126, normalized size = 1.91 \[ \frac{1}{5} \, B b^{4} x^{5} + A a^{4} \log \left (x\right ) + \frac{1}{4} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.301912, size = 126, normalized size = 1.91 \[ \frac{1}{5} \, B b^{4} x^{5} + A a^{4} \log \left (x\right ) + \frac{1}{4} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.4388, size = 95, normalized size = 1.44 \[ A a^{4} \log{\left (x \right )} + \frac{B b^{4} x^{5}}{5} + x^{4} \left (\frac{A b^{4}}{4} + B a b^{3}\right ) + x^{3} \left (\frac{4 A a b^{3}}{3} + 2 B a^{2} b^{2}\right ) + x^{2} \left (3 A a^{2} b^{2} + 2 B a^{3} b\right ) + x \left (4 A a^{3} b + B a^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x,x)
[Out]
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GIAC/XCAS [A] time = 0.272813, size = 127, normalized size = 1.92 \[ \frac{1}{5} \, B b^{4} x^{5} + B a b^{3} x^{4} + \frac{1}{4} \, A b^{4} x^{4} + 2 \, B a^{2} b^{2} x^{3} + \frac{4}{3} \, A a b^{3} x^{3} + 2 \, B a^{3} b x^{2} + 3 \, A a^{2} b^{2} x^{2} + B a^{4} x + 4 \, A a^{3} b x + A a^{4}{\rm ln}\left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x,x, algorithm="giac")
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