Optimal. Leaf size=156 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{5 e^5 (d+e x)^5}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{3 e^5 (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.32752, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{5 e^5 (d+e x)^5}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{3 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/(d + e*x)^8,x]
[Out]
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Rubi in Sympy [A] time = 49.0007, size = 146, normalized size = 0.94 \[ - \frac{c^{2}}{3 e^{5} \left (d + e x\right )^{3}} - \frac{c \left (b e - 2 c d\right )}{2 e^{5} \left (d + e x\right )^{4}} - \frac{2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{5 e^{5} \left (d + e x\right )^{5}} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{5} \left (d + e x\right )^{6}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2}}{7 e^{5} \left (d + e x\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(e*x+d)**8,x)
[Out]
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Mathematica [A] time = 0.138564, size = 161, normalized size = 1.03 \[ -\frac{2 e^2 \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )+c e \left (4 a e \left (d^2+7 d e x+21 e^2 x^2\right )+3 b \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{210 e^5 (d+e x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/(d + e*x)^8,x]
[Out]
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Maple [A] time = 0.009, size = 195, normalized size = 1.3 \[ -{\frac{{a}^{2}{e}^{4}-2\,d{e}^{3}ab+2\,ac{d}^{2}{e}^{2}+{b}^{2}{d}^{2}{e}^{2}-2\,{d}^{3}ebc+{c}^{2}{d}^{4}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{2\,ab{e}^{3}-4\,ad{e}^{2}c-2\,{b}^{2}d{e}^{2}+6\,bc{d}^{2}e-4\,{c}^{2}{d}^{3}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{c \left ( be-2\,cd \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(e*x+d)^8,x)
[Out]
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Maxima [A] time = 0.827731, size = 331, normalized size = 2.12 \[ -\frac{70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 10 \, a b d e^{3} + 30 \, a^{2} e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 35 \,{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \,{\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 7 \,{\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 10 \, a b e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{210 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203772, size = 331, normalized size = 2.12 \[ -\frac{70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 10 \, a b d e^{3} + 30 \, a^{2} e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 35 \,{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \,{\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 7 \,{\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 10 \, a b e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{210 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^8,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(e*x+d)**8,x)
[Out]
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GIAC/XCAS [A] time = 0.20432, size = 243, normalized size = 1.56 \[ -\frac{{\left (70 \, c^{2} x^{4} e^{4} + 70 \, c^{2} d x^{3} e^{3} + 42 \, c^{2} d^{2} x^{2} e^{2} + 14 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 105 \, b c x^{3} e^{4} + 63 \, b c d x^{2} e^{3} + 21 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 42 \, b^{2} x^{2} e^{4} + 84 \, a c x^{2} e^{4} + 14 \, b^{2} d x e^{3} + 28 \, a c d x e^{3} + 2 \, b^{2} d^{2} e^{2} + 4 \, a c d^{2} e^{2} + 70 \, a b x e^{4} + 10 \, a b d e^{3} + 30 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{210 \,{\left (x e + d\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^8,x, algorithm="giac")
[Out]