Optimal. Leaf size=127 \[ \frac{1}{6} e x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 x^4 (3 b e+2 c d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8 \]
[Out]
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Rubi [A] time = 0.306682, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{1}{6} e x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 x^4 (3 b e+2 c d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8 \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 34.7703, size = 124, normalized size = 0.98 \[ \frac{b^{2} d^{3} x^{3}}{3} + \frac{b d^{2} x^{4} \left (3 b e + 2 c d\right )}{4} + \frac{c^{2} e^{3} x^{8}}{8} + \frac{c e^{2} x^{7} \left (2 b e + 3 c d\right )}{7} + \frac{d x^{5} \left (3 b^{2} e^{2} + 6 b c d e + c^{2} d^{2}\right )}{5} + \frac{e x^{6} \left (b^{2} e^{2} + 6 b c d e + 3 c^{2} d^{2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.0317404, size = 127, normalized size = 1. \[ \frac{1}{6} e x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^3 x^3+\frac{1}{4} b d^2 x^4 (3 b e+2 c d)+\frac{1}{7} c e^2 x^7 (2 b e+3 c d)+\frac{1}{8} c^2 e^3 x^8 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 128, normalized size = 1. \[{\frac{{c}^{2}{e}^{3}{x}^{8}}{8}}+{\frac{ \left ( 2\,{e}^{3}bc+3\,d{e}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{3}{b}^{2}+6\,d{e}^{2}bc+3\,{d}^{2}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,d{e}^{2}{b}^{2}+6\,{d}^{2}ebc+{c}^{2}{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{2}e{b}^{2}+2\,{d}^{3}bc \right ){x}^{4}}{4}}+{\frac{{b}^{2}{d}^{3}{x}^{3}}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.690927, size = 171, normalized size = 1.35 \[ \frac{1}{8} \, c^{2} e^{3} x^{8} + \frac{1}{3} \, b^{2} d^{3} x^{3} + \frac{1}{7} \,{\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + b^{2} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{3} + 6 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, b c d^{3} + 3 \, b^{2} d^{2} e\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.19507, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e^{3} c^{2} + \frac{3}{7} x^{7} e^{2} d c^{2} + \frac{2}{7} x^{7} e^{3} c b + \frac{1}{2} x^{6} e d^{2} c^{2} + x^{6} e^{2} d c b + \frac{1}{6} x^{6} e^{3} b^{2} + \frac{1}{5} x^{5} d^{3} c^{2} + \frac{6}{5} x^{5} e d^{2} c b + \frac{3}{5} x^{5} e^{2} d b^{2} + \frac{1}{2} x^{4} d^{3} c b + \frac{3}{4} x^{4} e d^{2} b^{2} + \frac{1}{3} x^{3} d^{3} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.172782, size = 138, normalized size = 1.09 \[ \frac{b^{2} d^{3} x^{3}}{3} + \frac{c^{2} e^{3} x^{8}}{8} + x^{7} \left (\frac{2 b c e^{3}}{7} + \frac{3 c^{2} d e^{2}}{7}\right ) + x^{6} \left (\frac{b^{2} e^{3}}{6} + b c d e^{2} + \frac{c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac{3 b^{2} d e^{2}}{5} + \frac{6 b c d^{2} e}{5} + \frac{c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{3 b^{2} d^{2} e}{4} + \frac{b c d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.206011, size = 177, normalized size = 1.39 \[ \frac{1}{8} \, c^{2} x^{8} e^{3} + \frac{3}{7} \, c^{2} d x^{7} e^{2} + \frac{1}{2} \, c^{2} d^{2} x^{6} e + \frac{1}{5} \, c^{2} d^{3} x^{5} + \frac{2}{7} \, b c x^{7} e^{3} + b c d x^{6} e^{2} + \frac{6}{5} \, b c d^{2} x^{5} e + \frac{1}{2} \, b c d^{3} x^{4} + \frac{1}{6} \, b^{2} x^{6} e^{3} + \frac{3}{5} \, b^{2} d x^{5} e^{2} + \frac{3}{4} \, b^{2} d^{2} x^{4} e + \frac{1}{3} \, b^{2} d^{3} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^3,x, algorithm="giac")
[Out]