Optimal. Leaf size=62 \[ \frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{d (c d-b e)}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2} \]
[Out]
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Rubi [A] time = 0.104668, antiderivative size = 62, 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 \[ \frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{d (c d-b e)}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 15.6049, size = 53, normalized size = 0.85 \[ - \frac{c}{2 e^{3} \left (d + e x\right )^{2}} + \frac{d \left (b e - c d\right )}{4 e^{3} \left (d + e x\right )^{4}} - \frac{b e - 2 c d}{3 e^{3} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.0286682, size = 43, normalized size = 0.69 \[ -\frac{b e (d+4 e x)+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)/(d + e*x)^5,x]
[Out]
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Maple [A] time = 0.008, size = 56, normalized size = 0.9 \[ -{\frac{c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{be-2\,cd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{d \left ( be-cd \right ) }{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)/(e*x+d)^5,x)
[Out]
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Maxima [A] time = 0.712231, size = 108, normalized size = 1.74 \[ -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216018, size = 108, normalized size = 1.74 \[ -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.80525, size = 85, normalized size = 1.37 \[ - \frac{b d e + c d^{2} + 6 c e^{2} x^{2} + x \left (4 b e^{2} + 4 c d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.207775, size = 101, normalized size = 1.63 \[ -\frac{1}{12} \,{\left (\frac{6 \, c e}{{\left (x e + d\right )}^{2}} - \frac{8 \, c d e}{{\left (x e + d\right )}^{3}} + \frac{3 \, c d^{2} e}{{\left (x e + d\right )}^{4}} + \frac{4 \, b e^{2}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b d e^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^5,x, algorithm="giac")
[Out]