Optimal. Leaf size=145 \[ -\frac{b^2 x (-3 a B e-A b e+3 b B d)}{e^4}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (d+e x)}-\frac{(b d-a e)^3 (B d-A e)}{2 e^5 (d+e x)^2}+\frac{3 b (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)}{e^5}+\frac{b^3 B x^2}{2 e^3} \]
[Out]
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Rubi [A] time = 0.367595, 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 \[ -\frac{b^2 x (-3 a B e-A b e+3 b B d)}{e^4}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{e^5 (d+e x)}-\frac{(b d-a e)^3 (B d-A e)}{2 e^5 (d+e x)^2}+\frac{3 b (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)}{e^5}+\frac{b^3 B x^2}{2 e^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^3*(A + B*x))/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B b^{3} \int x\, dx}{e^{3}} + \frac{3 b \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{e^{5}} + \frac{\left (A b e + 3 B a e - 3 B b d\right ) \int b^{2}\, dx}{e^{4}} - \frac{\left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{e^{5} \left (d + e x\right )} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{3}}{2 e^{5} \left (d + e x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3*(B*x+A)/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.195833, size = 238, normalized size = 1.64 \[ \frac{-a^3 e^3 (A e+B (d+2 e x))-3 a^2 b e^2 (A e (d+2 e x)-B d (3 d+4 e x))+3 a b^2 e \left (A d e (3 d+4 e x)+B \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )\right )+6 b (d+e x)^2 (b d-a e) \log (d+e x) (-a B e-A b e+2 b B d)+b^3 \left (A e \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+B \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )\right )}{2 e^5 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^3,x]
[Out]
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Maple [B] time = 0.013, size = 404, normalized size = 2.8 \[{\frac{{b}^{3}B{x}^{2}}{2\,{e}^{3}}}+{\frac{{b}^{3}Ax}{{e}^{3}}}+3\,{\frac{a{b}^{2}Bx}{{e}^{3}}}-3\,{\frac{{b}^{3}Bdx}{{e}^{4}}}+3\,{\frac{{b}^{2}\ln \left ( ex+d \right ) Aa}{{e}^{3}}}-3\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Ad}{{e}^{4}}}+3\,{\frac{b\ln \left ( ex+d \right ) B{a}^{2}}{{e}^{3}}}-9\,{\frac{{b}^{2}\ln \left ( ex+d \right ) Bda}{{e}^{4}}}+6\,{\frac{{b}^{3}\ln \left ( ex+d \right ) B{d}^{2}}{{e}^{5}}}-3\,{\frac{A{a}^{2}b}{{e}^{2} \left ( ex+d \right ) }}+6\,{\frac{a{b}^{2}Ad}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{{b}^{3}A{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{B{a}^{3}}{{e}^{2} \left ( ex+d \right ) }}+6\,{\frac{{a}^{2}bBd}{{e}^{3} \left ( ex+d \right ) }}-9\,{\frac{{b}^{2}a{d}^{2}B}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{b}^{3}B{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{3}A}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{3\,Ad{a}^{2}b}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{3\,A{d}^{2}a{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{3}A{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{Bd{a}^{3}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{3\,B{d}^{2}{a}^{2}b}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{3\,{b}^{2}a{d}^{3}B}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}B{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3*(B*x+A)/(e*x+d)^3,x)
[Out]
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Maxima [A] time = 1.37558, size = 370, normalized size = 2.55 \[ \frac{7 \, B b^{3} d^{4} - A a^{3} e^{4} - 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 2 \,{\left (4 \, B b^{3} d^{3} e - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac{B b^{3} e x^{2} - 2 \,{\left (3 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} x}{2 \, e^{4}} + \frac{3 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^3/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212612, size = 567, normalized size = 3.91 \[ \frac{B b^{3} e^{4} x^{4} + 7 \, B b^{3} d^{4} - A a^{3} e^{4} - 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 2 \,{\left (2 \, B b^{3} d e^{3} -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} -{\left (11 \, B b^{3} d^{2} e^{2} - 4 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3}\right )} x^{2} + 2 \,{\left (B b^{3} d^{3} e - 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 6 \,{\left (2 \, B b^{3} d^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e +{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} +{\left (2 \, B b^{3} d^{2} e^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} +{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 2 \,{\left (2 \, B b^{3} d^{3} e -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} +{\left (B a^{2} b + A a b^{2}\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^3/(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 21.7589, size = 296, normalized size = 2.04 \[ \frac{B b^{3} x^{2}}{2 e^{3}} + \frac{3 b \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{A a^{3} e^{4} + 3 A a^{2} b d e^{3} - 9 A a b^{2} d^{2} e^{2} + 5 A b^{3} d^{3} e + B a^{3} d e^{3} - 9 B a^{2} b d^{2} e^{2} + 15 B a b^{2} d^{3} e - 7 B b^{3} d^{4} + x \left (6 A a^{2} b e^{4} - 12 A a b^{2} d e^{3} + 6 A b^{3} d^{2} e^{2} + 2 B a^{3} e^{4} - 12 B a^{2} b d e^{3} + 18 B a b^{2} d^{2} e^{2} - 8 B b^{3} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} + \frac{x \left (A b^{3} e + 3 B a b^{2} e - 3 B b^{3} d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**3*(B*x+A)/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.227196, size = 369, normalized size = 2.54 \[ 3 \,{\left (2 \, B b^{3} d^{2} - 3 \, B a b^{2} d e - A b^{3} d e + B a^{2} b e^{2} + A a b^{2} e^{2}\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B b^{3} x^{2} e^{3} - 6 \, B b^{3} d x e^{2} + 6 \, B a b^{2} x e^{3} + 2 \, A b^{3} x e^{3}\right )} e^{\left (-6\right )} + \frac{{\left (7 \, B b^{3} d^{4} - 15 \, B a b^{2} d^{3} e - 5 \, A b^{3} d^{3} e + 9 \, B a^{2} b d^{2} e^{2} + 9 \, A a b^{2} d^{2} e^{2} - B a^{3} d e^{3} - 3 \, A a^{2} b d e^{3} - A a^{3} e^{4} + 2 \,{\left (4 \, B b^{3} d^{3} e - 9 \, B a b^{2} d^{2} e^{2} - 3 \, A b^{3} d^{2} e^{2} + 6 \, B a^{2} b d e^{3} + 6 \, A a b^{2} d e^{3} - B a^{3} e^{4} - 3 \, A a^{2} b e^{4}\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^3/(e*x + d)^3,x, algorithm="giac")
[Out]