Optimal. Leaf size=91 \[ \frac{(A b-a B) (b d-a e)^2 \log (a+b x)}{b^4}+\frac{e x (A b-a B) (b d-a e)}{b^3}+\frac{(d+e x)^2 (A b-a B)}{2 b^2}+\frac{B (d+e x)^3}{3 b e} \]
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Rubi [A] time = 0.124051, antiderivative size = 91, 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 \[ \frac{(A b-a B) (b d-a e)^2 \log (a+b x)}{b^4}+\frac{e x (A b-a B) (b d-a e)}{b^3}+\frac{(d+e x)^2 (A b-a B)}{2 b^2}+\frac{B (d+e x)^3}{3 b e} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^2)/(a + b*x),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B \left (d + e x\right )^{3}}{3 b e} + \frac{\left (d + e x\right )^{2} \left (A b - B a\right )}{2 b^{2}} - \frac{\left (A b - B a\right ) \left (a e - b d\right ) \int e\, dx}{b^{3}} + \frac{\left (A b - B a\right ) \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**2/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0942936, size = 102, normalized size = 1.12 \[ \frac{b x \left (6 a^2 B e^2-3 a b e (2 A e+4 B d+B e x)+b^2 \left (3 A e (4 d+e x)+2 B \left (3 d^2+3 d e x+e^2 x^2\right )\right )\right )+6 (A b-a B) (b d-a e)^2 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^2)/(a + b*x),x]
[Out]
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Maple [B] time = 0.004, size = 197, normalized size = 2.2 \[{\frac{B{x}^{3}{e}^{2}}{3\,b}}+{\frac{A{x}^{2}{e}^{2}}{2\,b}}-{\frac{B{x}^{2}a{e}^{2}}{2\,{b}^{2}}}+{\frac{B{x}^{2}de}{b}}-{\frac{aA{e}^{2}x}{{b}^{2}}}+2\,{\frac{Adex}{b}}+{\frac{B{a}^{2}{e}^{2}x}{{b}^{3}}}-2\,{\frac{Badex}{{b}^{2}}}+{\frac{B{d}^{2}x}{b}}+{\frac{\ln \left ( bx+a \right ) A{a}^{2}{e}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx+a \right ) Aade}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) A{d}^{2}}{b}}-{\frac{\ln \left ( bx+a \right ) B{a}^{3}{e}^{2}}{{b}^{4}}}+2\,{\frac{\ln \left ( bx+a \right ) B{a}^{2}de}{{b}^{3}}}-{\frac{\ln \left ( bx+a \right ) Ba{d}^{2}}{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^2/(b*x+a),x)
[Out]
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Maxima [A] time = 1.35531, size = 209, normalized size = 2.3 \[ \frac{2 \, B b^{2} e^{2} x^{3} + 3 \,{\left (2 \, B b^{2} d e -{\left (B a b - A b^{2}\right )} e^{2}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} - 2 \,{\left (B a b - A b^{2}\right )} d e +{\left (B a^{2} - A a b\right )} e^{2}\right )} x}{6 \, b^{3}} - \frac{{\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} - 2 \,{\left (B a^{2} b - A a b^{2}\right )} d e +{\left (B a^{3} - A a^{2} b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a),x, algorithm="maxima")
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Fricas [A] time = 0.214637, size = 213, normalized size = 2.34 \[ \frac{2 \, B b^{3} e^{2} x^{3} + 3 \,{\left (2 \, B b^{3} d e -{\left (B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 6 \,{\left (B b^{3} d^{2} - 2 \,{\left (B a b^{2} - A b^{3}\right )} d e +{\left (B a^{2} b - A a b^{2}\right )} e^{2}\right )} x - 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} - 2 \,{\left (B a^{2} b - A a b^{2}\right )} d e +{\left (B a^{3} - A a^{2} b\right )} e^{2}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.50195, size = 117, normalized size = 1.29 \[ \frac{B e^{2} x^{3}}{3 b} - \frac{x^{2} \left (- A b e^{2} + B a e^{2} - 2 B b d e\right )}{2 b^{2}} + \frac{x \left (- A a b e^{2} + 2 A b^{2} d e + B a^{2} e^{2} - 2 B a b d e + B b^{2} d^{2}\right )}{b^{3}} - \frac{\left (- A b + B a\right ) \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**2/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.216391, size = 221, normalized size = 2.43 \[ \frac{2 \, B b^{2} x^{3} e^{2} + 6 \, B b^{2} d x^{2} e + 6 \, B b^{2} d^{2} x - 3 \, B a b x^{2} e^{2} + 3 \, A b^{2} x^{2} e^{2} - 12 \, B a b d x e + 12 \, A b^{2} d x e + 6 \, B a^{2} x e^{2} - 6 \, A a b x e^{2}}{6 \, b^{3}} - \frac{{\left (B a b^{2} d^{2} - A b^{3} d^{2} - 2 \, B a^{2} b d e + 2 \, A a b^{2} d e + B a^{3} e^{2} - A a^{2} b e^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a),x, algorithm="giac")
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