Optimal. Leaf size=59 \[ \frac{(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac{B x (b d-a e)}{b^2}+\frac{e (A+B x)^2}{2 b B} \]
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Rubi [A] time = 0.0948833, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac{B x (b d-a e)}{b^2}+\frac{e (A+B x)^2}{2 b B} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x))/(a + b*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a e - b d\right ) \int B\, dx}{b^{2}} - \frac{\left (A b - B a\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{3}} + \frac{e \left (A + B x\right )^{2}}{2 B b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0385519, size = 56, normalized size = 0.95 \[ \frac{b x (b (2 A e+2 B d+B e x)-2 a B e)+2 (A b-a B) (b d-a e) \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x))/(a + b*x),x]
[Out]
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Maple [A] time = 0.004, size = 90, normalized size = 1.5 \[{\frac{B{x}^{2}e}{2\,b}}+{\frac{Aex}{b}}-{\frac{Baex}{{b}^{2}}}+{\frac{Bdx}{b}}-{\frac{\ln \left ( bx+a \right ) Aae}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) Ad}{b}}+{\frac{\ln \left ( bx+a \right ) B{a}^{2}e}{{b}^{3}}}-{\frac{\ln \left ( bx+a \right ) Bad}{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)/(b*x+a),x)
[Out]
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Maxima [A] time = 1.35463, size = 97, normalized size = 1.64 \[ \frac{B b e x^{2} + 2 \,{\left (B b d -{\left (B a - A b\right )} e\right )} x}{2 \, b^{2}} - \frac{{\left ({\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222031, size = 101, normalized size = 1.71 \[ \frac{B b^{2} e x^{2} + 2 \,{\left (B b^{2} d -{\left (B a b - A b^{2}\right )} e\right )} x - 2 \,{\left ({\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.80958, size = 53, normalized size = 0.9 \[ \frac{B e x^{2}}{2 b} - \frac{x \left (- A b e + B a e - B b d\right )}{b^{2}} + \frac{\left (- A b + B a\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.214571, size = 100, normalized size = 1.69 \[ \frac{B b x^{2} e + 2 \, B b d x - 2 \, B a x e + 2 \, A b x e}{2 \, b^{2}} - \frac{{\left (B a b d - A b^{2} d - B a^{2} e + A a b e\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(b*x + a),x, algorithm="giac")
[Out]