Optimal. Leaf size=103 \[ -\frac{3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac{d \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac{3 c d-b e}{4 e^4 (d+e x)^4}-\frac{c}{3 e^4 (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.168545, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac{d \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac{3 c d-b e}{4 e^4 (d+e x)^4}-\frac{c}{3 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x + c*x^2))/(d + e*x)^7,x]
[Out]
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Rubi in Sympy [A] time = 27.047, size = 92, normalized size = 0.89 \[ - \frac{c}{3 e^{4} \left (d + e x\right )^{3}} + \frac{d \left (a e^{2} - b d e + c d^{2}\right )}{6 e^{4} \left (d + e x\right )^{6}} - \frac{b e - 3 c d}{4 e^{4} \left (d + e x\right )^{4}} - \frac{a e^{2} - 2 b d e + 3 c d^{2}}{5 e^{4} \left (d + e x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c*x**2+b*x+a)/(e*x+d)**7,x)
[Out]
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Mathematica [A] time = 0.0569547, size = 77, normalized size = 0.75 \[ -\frac{e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )+c \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x + c*x^2))/(d + e*x)^7,x]
[Out]
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Maple [A] time = 0.008, size = 93, normalized size = 0.9 \[ -{\frac{c}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{be-3\,cd}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{e}^{2}a-2\,bde+3\,c{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}+{\frac{d \left ({e}^{2}a-bde+c{d}^{2} \right ) }{6\,{e}^{4} \left ( ex+d \right ) ^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^2+b*x+a)/(e*x+d)^7,x)
[Out]
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Maxima [A] time = 0.702961, size = 185, normalized size = 1.8 \[ -\frac{20 \, c e^{3} x^{3} + c d^{3} + b d^{2} e + 2 \, a d e^{2} + 15 \,{\left (c d e^{2} + b e^{3}\right )} x^{2} + 6 \,{\left (c d^{2} e + b d e^{2} + 2 \, a e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(e*x + d)^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.256232, size = 185, normalized size = 1.8 \[ -\frac{20 \, c e^{3} x^{3} + c d^{3} + b d^{2} e + 2 \, a d e^{2} + 15 \,{\left (c d e^{2} + b e^{3}\right )} x^{2} + 6 \,{\left (c d^{2} e + b d e^{2} + 2 \, a e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(e*x + d)^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 24.9994, size = 148, normalized size = 1.44 \[ - \frac{2 a d e^{2} + b d^{2} e + c d^{3} + 20 c e^{3} x^{3} + x^{2} \left (15 b e^{3} + 15 c d e^{2}\right ) + x \left (12 a e^{3} + 6 b d e^{2} + 6 c d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**2+b*x+a)/(e*x+d)**7,x)
[Out]
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GIAC/XCAS [A] time = 0.261839, size = 105, normalized size = 1.02 \[ -\frac{{\left (20 \, c x^{3} e^{3} + 15 \, c d x^{2} e^{2} + 6 \, c d^{2} x e + c d^{3} + 15 \, b x^{2} e^{3} + 6 \, b d x e^{2} + b d^{2} e + 12 \, a x e^{3} + 2 \, a d e^{2}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(e*x + d)^7,x, algorithm="giac")
[Out]