Optimal. Leaf size=156 \[ \frac{(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac{c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5} \]
[Out]
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Rubi [A] time = 0.379796, 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 \[ \frac{(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac{c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 a d \left (a e + b d\right ) \int x\, dx + \frac{c^{2} e^{2} x^{7}}{7} + \frac{c e x^{6} \left (b e + c d\right )}{3} + d^{2} \int a^{2}\, dx + x^{5} \left (\frac{2 a c e^{2}}{5} + \frac{b^{2} e^{2}}{5} + \frac{4 b c d e}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac{a b e^{2}}{2} + a c d e + \frac{b^{2} d e}{2} + \frac{b c d^{2}}{2}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{4 a b d e}{3} + \frac{2 a c d^{2}}{3} + \frac{b^{2} d^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0843891, size = 153, normalized size = 0.98 \[ \frac{1}{3} x^3 \left (a^2 e^2+4 a b d e+2 a c d^2+b^2 d^2\right )+a^2 d^2 x+\frac{1}{5} x^5 \left (2 a c e^2+b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{2} x^4 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+a d x^2 (a e+b d)+\frac{1}{3} c e x^6 (b e+c d)+\frac{1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(a + b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0., size = 155, normalized size = 1. \[{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{2}bc+2\,{c}^{2}de \right ){x}^{6}}{6}}+{\frac{ \left ({c}^{2}{d}^{2}+4\,bcde+{e}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,c{d}^{2}b+2\,de \left ( 2\,ac+{b}^{2} \right ) +2\,{e}^{2}ab \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,abde+{a}^{2}{e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{2}+2\,{d}^{2}ab \right ){x}^{2}}{2}}+{a}^{2}{d}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(c*x^2+b*x+a)^2,x)
[Out]
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Maxima [A] time = 0.815428, size = 197, normalized size = 1.26 \[ \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \,{\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{2} + 4 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{1}{2} \,{\left (b c d^{2} + a b e^{2} +{\left (b^{2} + 2 \, a c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (4 \, a b d e + a^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{3} +{\left (a b d^{2} + a^{2} d e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.18302, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{3} x^{6} e^{2} c b + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{4}{5} x^{5} e d c b + \frac{1}{5} x^{5} e^{2} b^{2} + \frac{2}{5} x^{5} e^{2} c a + \frac{1}{2} x^{4} d^{2} c b + \frac{1}{2} x^{4} e d b^{2} + x^{4} e d c a + \frac{1}{2} x^{4} e^{2} b a + \frac{1}{3} x^{3} d^{2} b^{2} + \frac{2}{3} x^{3} d^{2} c a + \frac{4}{3} x^{3} e d b a + \frac{1}{3} x^{3} e^{2} a^{2} + x^{2} d^{2} b a + x^{2} e d a^{2} + x d^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.187853, size = 173, normalized size = 1.11 \[ a^{2} d^{2} x + \frac{c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac{b c e^{2}}{3} + \frac{c^{2} d e}{3}\right ) + x^{5} \left (\frac{2 a c e^{2}}{5} + \frac{b^{2} e^{2}}{5} + \frac{4 b c d e}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac{a b e^{2}}{2} + a c d e + \frac{b^{2} d e}{2} + \frac{b c d^{2}}{2}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{4 a b d e}{3} + \frac{2 a c d^{2}}{3} + \frac{b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.203025, size = 240, normalized size = 1.54 \[ \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{1}{3} \, b c x^{6} e^{2} + \frac{4}{5} \, b c d x^{5} e + \frac{1}{2} \, b c d^{2} x^{4} + \frac{1}{5} \, b^{2} x^{5} e^{2} + \frac{2}{5} \, a c x^{5} e^{2} + \frac{1}{2} \, b^{2} d x^{4} e + a c d x^{4} e + \frac{1}{3} \, b^{2} d^{2} x^{3} + \frac{2}{3} \, a c d^{2} x^{3} + \frac{1}{2} \, a b x^{4} e^{2} + \frac{4}{3} \, a b d x^{3} e + a b d^{2} x^{2} + \frac{1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d)^2,x, algorithm="giac")
[Out]