Optimal. Leaf size=89 \[ -\frac{a^4 A}{3 x^3}-\frac{a^3 (a B+4 A b)}{2 x^2}-\frac{2 a^2 b (2 a B+3 A b)}{x}+b^3 x (4 a B+A b)+2 a b^2 \log (x) (3 a B+2 A b)+\frac{1}{2} b^4 B x^2 \]
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Rubi [A] time = 0.137364, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{a^4 A}{3 x^3}-\frac{a^3 (a B+4 A b)}{2 x^2}-\frac{2 a^2 b (2 a B+3 A b)}{x}+b^3 x (4 a B+A b)+2 a b^2 \log (x) (3 a B+2 A b)+\frac{1}{2} b^4 B x^2 \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{4}}{3 x^{3}} + B b^{4} \int x\, dx - \frac{a^{3} \left (4 A b + B a\right )}{2 x^{2}} - \frac{2 a^{2} b \left (3 A b + 2 B a\right )}{x} + 2 a b^{2} \left (2 A b + 3 B a\right ) \log{\left (x \right )} + b^{3} x \left (A b + 4 B a\right ) \]
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**4,x)
[Out]
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Mathematica [A] time = 0.0627554, size = 86, normalized size = 0.97 \[ -\frac{a^4 (2 A+3 B x)}{6 x^3}-\frac{2 a^3 b (A+2 B x)}{x^2}-\frac{6 a^2 A b^2}{x}+2 a b^2 \log (x) (3 a B+2 A b)+4 a b^3 B x+\frac{1}{2} b^4 x (2 A+B x) \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^4,x]
[Out]
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Maple [A] time = 0.011, size = 95, normalized size = 1.1 \[{\frac{{b}^{4}B{x}^{2}}{2}}+Ax{b}^{4}+4\,Bxa{b}^{3}+4\,A\ln \left ( x \right ) a{b}^{3}+6\,B\ln \left ( x \right ){a}^{2}{b}^{2}-{\frac{A{a}^{4}}{3\,{x}^{3}}}-2\,{\frac{A{a}^{3}b}{{x}^{2}}}-{\frac{B{a}^{4}}{2\,{x}^{2}}}-6\,{\frac{A{a}^{2}{b}^{2}}{x}}-4\,{\frac{B{a}^{3}b}{x}} \]
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^4,x)
[Out]
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Maxima [A] time = 0.678093, size = 130, normalized size = 1.46 \[ \frac{1}{2} \, B b^{4} x^{2} +{\left (4 \, B a b^{3} + A b^{4}\right )} x + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \log \left (x\right ) - \frac{2 \, A a^{4} + 12 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{6 \, x^{3}} \]
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^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.299699, size = 136, normalized size = 1.53 \[ \frac{3 \, B b^{4} x^{5} - 2 \, A a^{4} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 12 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} \log \left (x\right ) - 12 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{6 \, x^{3}} \]
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^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.40605, size = 95, normalized size = 1.07 \[ \frac{B b^{4} x^{2}}{2} + 2 a b^{2} \left (2 A b + 3 B a\right ) \log{\left (x \right )} + x \left (A b^{4} + 4 B a b^{3}\right ) - \frac{2 A a^{4} + x^{2} \left (36 A a^{2} b^{2} + 24 B a^{3} b\right ) + x \left (12 A a^{3} b + 3 B a^{4}\right )}{6 x^{3}} \]
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**4,x)
[Out]
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GIAC/XCAS [A] time = 0.269347, size = 130, normalized size = 1.46 \[ \frac{1}{2} \, B b^{4} x^{2} + 4 \, B a b^{3} x + A b^{4} x + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A a^{4} + 12 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{6 \, x^{3}} \]
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^4,x, algorithm="giac")
[Out]