Optimal. Leaf size=101 \[ -\frac{3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac{d \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{3 c d-b e}{2 e^4 (d+e x)^2}-\frac{c}{e^4 (d+e x)} \]
[Out]
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Rubi [A] time = 0.180288, antiderivative size = 101, 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 \[ -\frac{3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac{d \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{3 c d-b e}{2 e^4 (d+e x)^2}-\frac{c}{e^4 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x + c*x^2))/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 26.6824, size = 88, normalized size = 0.87 \[ - \frac{c}{e^{4} \left (d + e x\right )} + \frac{d \left (a e^{2} - b d e + c d^{2}\right )}{4 e^{4} \left (d + e x\right )^{4}} - \frac{b e - 3 c d}{2 e^{4} \left (d + e x\right )^{2}} - \frac{a e^{2} - 2 b d e + 3 c d^{2}}{3 e^{4} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c*x**2+b*x+a)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.0576718, size = 77, normalized size = 0.76 \[ -\frac{e \left (a e (d+4 e x)+b \left (d^2+4 d e x+6 e^2 x^2\right )\right )+3 c \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x + c*x^2))/(d + e*x)^5,x]
[Out]
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Maple [A] time = 0.008, size = 93, normalized size = 0.9 \[ -{\frac{{e}^{2}a-2\,bde+3\,c{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{c}{{e}^{4} \left ( ex+d \right ) }}+{\frac{d \left ({e}^{2}a-bde+c{d}^{2} \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{be-3\,cd}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^2+b*x+a)/(e*x+d)^5,x)
[Out]
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Maxima [A] time = 0.698463, size = 157, normalized size = 1.55 \[ -\frac{12 \, c e^{3} x^{3} + 3 \, c d^{3} + b d^{2} e + a d e^{2} + 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{2} + 4 \,{\left (3 \, c d^{2} e + b d e^{2} + a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257816, size = 157, normalized size = 1.55 \[ -\frac{12 \, c e^{3} x^{3} + 3 \, c d^{3} + b d^{2} e + a d e^{2} + 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{2} + 4 \,{\left (3 \, c d^{2} e + b d e^{2} + a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.2714, size = 124, normalized size = 1.23 \[ - \frac{a d e^{2} + b d^{2} e + 3 c d^{3} + 12 c e^{3} x^{3} + x^{2} \left (6 b e^{3} + 18 c d e^{2}\right ) + x \left (4 a e^{3} + 4 b d e^{2} + 12 c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**2+b*x+a)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.261039, size = 181, normalized size = 1.79 \[ -\frac{1}{12} \,{\left (\frac{12 \, c e^{5}}{x e + d} - \frac{18 \, c d e^{5}}{{\left (x e + d\right )}^{2}} + \frac{12 \, c d^{2} e^{5}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c d^{3} e^{5}}{{\left (x e + d\right )}^{4}} + \frac{6 \, b e^{6}}{{\left (x e + d\right )}^{2}} - \frac{8 \, b d e^{6}}{{\left (x e + d\right )}^{3}} + \frac{3 \, b d^{2} e^{6}}{{\left (x e + d\right )}^{4}} + \frac{4 \, a e^{7}}{{\left (x e + d\right )}^{3}} - \frac{3 \, a d e^{7}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-9\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*x/(e*x + d)^5,x, algorithm="giac")
[Out]