Optimal. Leaf size=145 \[ \frac{e^2 (a+b x)^2 (-4 a B e+A b e+3 b B d)}{2 b^5}-\frac{(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac{(b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{b^5}+\frac{3 e x (b d-a e) (-2 a B e+A b e+b B d)}{b^4}+\frac{B e^3 (a+b x)^3}{3 b^5} \]
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Rubi [A] time = 0.376969, antiderivative size = 145, 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 \[ \frac{e^2 (a+b x)^2 (-4 a B e+A b e+3 b B d)}{2 b^5}-\frac{(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac{(b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{b^5}+\frac{3 e x (b d-a e) (-2 a B e+A b e+b B d)}{b^4}+\frac{B e^3 (a+b x)^3}{3 b^5} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^3)/(a + b*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 51.1264, size = 143, normalized size = 0.99 \[ \frac{B e^{3} \left (a + b x\right )^{3}}{3 b^{5}} - \frac{3 e x \left (a e - b d\right ) \left (A b e - 2 B a e + B b d\right )}{b^{4}} + \frac{e^{2} \left (a + b x\right )^{2} \left (A b e - 4 B a e + 3 B b d\right )}{2 b^{5}} + \frac{\left (a e - b d\right )^{2} \left (3 A b e - 4 B a e + B b d\right ) \log{\left (a + b x \right )}}{b^{5}} + \frac{\left (A b - B a\right ) \left (a e - b d\right )^{3}}{b^{5} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**3/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.223717, size = 250, normalized size = 1.72 \[ \frac{-3 A b \left (-2 a^3 e^3+2 a^2 b e^2 (3 d+2 e x)+3 a b^2 e \left (-2 d^2-2 d e x+e^2 x^2\right )+b^3 \left (2 d^3-6 d e^2 x^2-e^3 x^3\right )\right )+B \left (-6 a^4 e^3+18 a^3 b e^2 (d+e x)+6 a^2 b^2 e \left (-3 d^2-6 d e x+2 e^2 x^2\right )+a b^3 \left (6 d^3+18 d^2 e x-27 d e^2 x^2-4 e^3 x^3\right )+b^4 e x^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (a+b x) (b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{6 b^5 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^3)/(a + b*x)^2,x]
[Out]
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Maple [B] time = 0.015, size = 376, normalized size = 2.6 \[{\frac{B{e}^{3}{x}^{3}}{3\,{b}^{2}}}+{\frac{{e}^{3}A{x}^{2}}{2\,{b}^{2}}}-{\frac{B{e}^{3}{x}^{2}a}{{b}^{3}}}+{\frac{3\,{e}^{2}B{x}^{2}d}{2\,{b}^{2}}}-2\,{\frac{aA{e}^{3}x}{{b}^{3}}}+3\,{\frac{{e}^{2}Adx}{{b}^{2}}}+3\,{\frac{B{a}^{2}{e}^{3}x}{{b}^{4}}}-6\,{\frac{{e}^{2}Badx}{{b}^{3}}}+3\,{\frac{eB{d}^{2}x}{{b}^{2}}}+3\,{\frac{\ln \left ( bx+a \right ) A{a}^{2}{e}^{3}}{{b}^{4}}}-6\,{\frac{\ln \left ( bx+a \right ) Aad{e}^{2}}{{b}^{3}}}+3\,{\frac{\ln \left ( bx+a \right ) A{d}^{2}e}{{b}^{2}}}-4\,{\frac{\ln \left ( bx+a \right ) B{a}^{3}{e}^{3}}{{b}^{5}}}+9\,{\frac{\ln \left ( bx+a \right ) B{a}^{2}d{e}^{2}}{{b}^{4}}}-6\,{\frac{\ln \left ( bx+a \right ) Ba{d}^{2}e}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ) B{d}^{3}}{{b}^{2}}}+{\frac{{a}^{3}A{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{{a}^{2}Ad{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+3\,{\frac{aA{d}^{2}e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{A{d}^{3}}{b \left ( bx+a \right ) }}-{\frac{B{a}^{4}{e}^{3}}{{b}^{5} \left ( bx+a \right ) }}+3\,{\frac{B{a}^{3}d{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{B{a}^{2}{d}^{2}e}{{b}^{3} \left ( bx+a \right ) }}+{\frac{Ba{d}^{3}}{{b}^{2} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^3/(b*x+a)^2,x)
[Out]
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Maxima [A] time = 1.36326, size = 369, normalized size = 2.54 \[ \frac{{\left (B a b^{3} - A b^{4}\right )} d^{3} - 3 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} -{\left (B a^{4} - A a^{3} b\right )} e^{3}}{b^{6} x + a b^{5}} + \frac{2 \, B b^{2} e^{3} x^{3} + 3 \,{\left (3 \, B b^{2} d e^{2} -{\left (2 \, B a b - A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B b^{2} d^{2} e - 3 \,{\left (2 \, B a b - A b^{2}\right )} d e^{2} +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} x}{6 \, b^{4}} + \frac{{\left (B b^{3} d^{3} - 3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e^{2} -{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^3/(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207824, size = 563, normalized size = 3.88 \[ \frac{2 \, B b^{4} e^{3} x^{4} + 6 \,{\left (B a b^{3} - A b^{4}\right )} d^{3} - 18 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 18 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - 6 \,{\left (B a^{4} - A a^{3} b\right )} e^{3} +{\left (9 \, B b^{4} d e^{2} -{\left (4 \, B a b^{3} - 3 \, A b^{4}\right )} e^{3}\right )} x^{3} + 3 \,{\left (6 \, B b^{4} d^{2} e - 3 \,{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d e^{2} +{\left (4 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B a b^{3} d^{2} e - 3 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d e^{2} +{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \,{\left (B a b^{3} d^{3} - 3 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{2} -{\left (4 \, B a^{4} - 3 \, A a^{3} b\right )} e^{3} +{\left (B b^{4} d^{3} - 3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} -{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{6} x + a b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^3/(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.2904, size = 250, normalized size = 1.72 \[ \frac{B e^{3} x^{3}}{3 b^{2}} - \frac{- A a^{3} b e^{3} + 3 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e + A b^{4} d^{3} + B a^{4} e^{3} - 3 B a^{3} b d e^{2} + 3 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3}}{a b^{5} + b^{6} x} - \frac{x^{2} \left (- A b e^{3} + 2 B a e^{3} - 3 B b d e^{2}\right )}{2 b^{3}} + \frac{x \left (- 2 A a b e^{3} + 3 A b^{2} d e^{2} + 3 B a^{2} e^{3} - 6 B a b d e^{2} + 3 B b^{2} d^{2} e\right )}{b^{4}} - \frac{\left (a e - b d\right )^{2} \left (- 3 A b e + 4 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**3/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.233693, size = 485, normalized size = 3.34 \[ \frac{{\left (b x + a\right )}^{3}{\left (2 \, B e^{3} + \frac{3 \,{\left (3 \, B b^{2} d e^{2} - 4 \, B a b e^{3} + A b^{2} e^{3}\right )}}{{\left (b x + a\right )} b} + \frac{18 \,{\left (B b^{4} d^{2} e - 3 \, B a b^{3} d e^{2} + A b^{4} d e^{2} + 2 \, B a^{2} b^{2} e^{3} - A a b^{3} e^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}}{6 \, b^{5}} - \frac{{\left (B b^{3} d^{3} - 6 \, B a b^{2} d^{2} e + 3 \, A b^{3} d^{2} e + 9 \, B a^{2} b d e^{2} - 6 \, A a b^{2} d e^{2} - 4 \, B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{5}} + \frac{\frac{B a b^{6} d^{3}}{b x + a} - \frac{A b^{7} d^{3}}{b x + a} - \frac{3 \, B a^{2} b^{5} d^{2} e}{b x + a} + \frac{3 \, A a b^{6} d^{2} e}{b x + a} + \frac{3 \, B a^{3} b^{4} d e^{2}}{b x + a} - \frac{3 \, A a^{2} b^{5} d e^{2}}{b x + a} - \frac{B a^{4} b^{3} e^{3}}{b x + a} + \frac{A a^{3} b^{4} e^{3}}{b x + a}}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^3/(b*x + a)^2,x, algorithm="giac")
[Out]