Optimal. Leaf size=99 \[ -\frac{(A b-a B) (b d-a e)^2}{b^4 (a+b x)}+\frac{(b d-a e) \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^4}+\frac{e x (-2 a B e+A b e+2 b B d)}{b^3}+\frac{B e^2 x^2}{2 b^2} \]
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Rubi [A] time = 0.215418, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{(A b-a B) (b d-a e)^2}{b^4 (a+b x)}+\frac{(b d-a e) \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^4}+\frac{e x (-2 a B e+A b e+2 b B d)}{b^3}+\frac{B e^2 x^2}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^2)/(a + b*x)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B e^{2} \int x\, dx}{b^{2}} + \frac{\left (A b e - 2 B a e + 2 B b d\right ) \int e\, dx}{b^{3}} - \frac{\left (a e - b d\right ) \left (2 A b e - 3 B a e + B b d\right ) \log{\left (a + b x \right )}}{b^{4}} - \frac{\left (A b - B a\right ) \left (a e - b d\right )^{2}}{b^{4} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**2/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.131049, size = 153, normalized size = 1.55 \[ \frac{\log (a+b x) \left (3 a^2 B e^2-2 a A b e^2-4 a b B d e+2 A b^2 d e+b^2 B d^2\right )}{b^4}+\frac{a^3 B e^2-a^2 A b e^2-2 a^2 b B d e+2 a A b^2 d e+a b^2 B d^2-A b^3 d^2}{b^4 (a+b x)}+\frac{e x (-2 a B e+A b e+2 b B d)}{b^3}+\frac{B e^2 x^2}{2 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^2)/(a + b*x)^2,x]
[Out]
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Maple [B] time = 0.013, size = 223, normalized size = 2.3 \[{\frac{B{e}^{2}{x}^{2}}{2\,{b}^{2}}}+{\frac{{e}^{2}Ax}{{b}^{2}}}-2\,{\frac{Ba{e}^{2}x}{{b}^{3}}}+2\,{\frac{eBdx}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a \right ) Aa{e}^{2}}{{b}^{3}}}+2\,{\frac{\ln \left ( bx+a \right ) Ade}{{b}^{2}}}+3\,{\frac{\ln \left ( bx+a \right ) B{a}^{2}{e}^{2}}{{b}^{4}}}-4\,{\frac{\ln \left ( bx+a \right ) Bade}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ) B{d}^{2}}{{b}^{2}}}-{\frac{{a}^{2}A{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+2\,{\frac{aAde}{{b}^{2} \left ( bx+a \right ) }}-{\frac{A{d}^{2}}{b \left ( bx+a \right ) }}+{\frac{B{a}^{3}{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}-2\,{\frac{B{a}^{2}de}{{b}^{3} \left ( bx+a \right ) }}+{\frac{Ba{d}^{2}}{{b}^{2} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^2/(b*x+a)^2,x)
[Out]
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Maxima [A] time = 1.35616, size = 213, normalized size = 2.15 \[ \frac{{\left (B a b^{2} - A b^{3}\right )} d^{2} - 2 \,{\left (B a^{2} b - A a b^{2}\right )} d e +{\left (B a^{3} - A a^{2} b\right )} e^{2}}{b^{5} x + a b^{4}} + \frac{B b e^{2} x^{2} + 2 \,{\left (2 \, B b d e -{\left (2 \, B a - A b\right )} e^{2}\right )} x}{2 \, b^{3}} + \frac{{\left (B b^{2} d^{2} - 2 \,{\left (2 \, B a b - A b^{2}\right )} d e +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21542, size = 336, normalized size = 3.39 \[ \frac{B b^{3} e^{2} x^{3} + 2 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} - 4 \,{\left (B a^{2} b - A a b^{2}\right )} d e + 2 \,{\left (B a^{3} - A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (2 \, B a b^{2} d e -{\left (2 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x + 2 \,{\left (B a b^{2} d^{2} - 2 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} d e +{\left (3 \, B a^{3} - 2 \, A a^{2} b\right )} e^{2} +{\left (B b^{3} d^{2} - 2 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d e +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.76636, size = 148, normalized size = 1.49 \[ \frac{B e^{2} x^{2}}{2 b^{2}} + \frac{- A a^{2} b e^{2} + 2 A a b^{2} d e - A b^{3} d^{2} + B a^{3} e^{2} - 2 B a^{2} b d e + B a b^{2} d^{2}}{a b^{4} + b^{5} x} - \frac{x \left (- A b e^{2} + 2 B a e^{2} - 2 B b d e\right )}{b^{3}} + \frac{\left (a e - b d\right ) \left (- 2 A b e + 3 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**2/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.223935, size = 306, normalized size = 3.09 \[ \frac{{\left (b x + a\right )}^{2}{\left (B e^{2} + \frac{2 \,{\left (2 \, B b^{2} d e - 3 \, B a b e^{2} + A b^{2} e^{2}\right )}}{{\left (b x + a\right )} b}\right )}}{2 \, b^{4}} - \frac{{\left (B b^{2} d^{2} - 4 \, B a b d e + 2 \, A b^{2} d e + 3 \, B a^{2} e^{2} - 2 \, A a b e^{2}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{\frac{B a b^{4} d^{2}}{b x + a} - \frac{A b^{5} d^{2}}{b x + a} - \frac{2 \, B a^{2} b^{3} d e}{b x + a} + \frac{2 \, A a b^{4} d e}{b x + a} + \frac{B a^{3} b^{2} e^{2}}{b x + a} - \frac{A a^{2} b^{3} e^{2}}{b x + a}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a)^2,x, algorithm="giac")
[Out]