Optimal. Leaf size=60 \[ \frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{d (c d-b e)}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)} \]
[Out]
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Rubi [A] time = 0.103923, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{d (c d-b e)}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 15.5663, size = 49, normalized size = 0.82 \[ - \frac{c}{e^{3} \left (d + e x\right )} + \frac{d \left (b e - c d\right )}{3 e^{3} \left (d + e x\right )^{3}} - \frac{b e - 2 c d}{2 e^{3} \left (d + e x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.0275592, size = 44, normalized size = 0.73 \[ -\frac{b e (d+3 e x)+2 c \left (d^2+3 d e x+3 e^2 x^2\right )}{6 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)/(d + e*x)^4,x]
[Out]
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Maple [A] time = 0.009, size = 56, normalized size = 0.9 \[ -{\frac{be-2\,cd}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{d \left ( be-cd \right ) }{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{c}{{e}^{3} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)/(e*x+d)^4,x)
[Out]
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Maxima [A] time = 0.697397, size = 96, normalized size = 1.6 \[ -\frac{6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 3 \,{\left (2 \, c d e + b e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212469, size = 96, normalized size = 1.6 \[ -\frac{6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 3 \,{\left (2 \, c d e + b e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.4236, size = 75, normalized size = 1.25 \[ - \frac{b d e + 2 c d^{2} + 6 c e^{2} x^{2} + x \left (3 b e^{2} + 6 c d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.205952, size = 61, normalized size = 1.02 \[ -\frac{{\left (6 \, c x^{2} e^{2} + 6 \, c d x e + 2 \, c d^{2} + 3 \, b x e^{2} + b d e\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^4,x, algorithm="giac")
[Out]