Optimal. Leaf size=77 \[ \frac{-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac{(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac{b B}{3 e^3 (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.128985, antiderivative size = 77, 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 \[ \frac{-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac{(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac{b B}{3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(A + B*x))/(d + e*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 20.2022, size = 70, normalized size = 0.91 \[ - \frac{B b}{3 e^{3} \left (d + e x\right )^{3}} - \frac{A b e + B a e - 2 B b d}{4 e^{3} \left (d + e x\right )^{4}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )}{5 e^{3} \left (d + e x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(B*x+A)/(e*x+d)**6,x)
[Out]
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Mathematica [A] time = 0.0550409, size = 65, normalized size = 0.84 \[ -\frac{3 a e (4 A e+B (d+5 e x))+b \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{60 e^3 (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(A + B*x))/(d + e*x)^6,x]
[Out]
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Maple [A] time = 0.009, size = 79, normalized size = 1. \[ -{\frac{Bb}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Abe+Bae-2\,Bbd}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{aA{e}^{2}-Abde-Bade+bB{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(B*x+A)/(e*x+d)^6,x)
[Out]
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Maxima [A] time = 1.33772, size = 158, normalized size = 2.05 \[ -\frac{20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \,{\left (B a + A b\right )} d e + 5 \,{\left (2 \, B b d e + 3 \,{\left (B a + A b\right )} e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(e*x + d)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213581, size = 158, normalized size = 2.05 \[ -\frac{20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \,{\left (B a + A b\right )} d e + 5 \,{\left (2 \, B b d e + 3 \,{\left (B a + A b\right )} e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(e*x + d)^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.1612, size = 134, normalized size = 1.74 \[ - \frac{12 A a e^{2} + 3 A b d e + 3 B a d e + 2 B b d^{2} + 20 B b e^{2} x^{2} + x \left (15 A b e^{2} + 15 B a e^{2} + 10 B b d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(B*x+A)/(e*x+d)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.219513, size = 96, normalized size = 1.25 \[ -\frac{{\left (20 \, B b x^{2} e^{2} + 10 \, B b d x e + 2 \, B b d^{2} + 15 \, B a x e^{2} + 15 \, A b x e^{2} + 3 \, B a d e + 3 \, A b d e + 12 \, A a e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(e*x + d)^6,x, algorithm="giac")
[Out]