Optimal. Leaf size=161 \[ a^3 d x+\frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8 \]
[Out]
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Rubi [A] time = 0.394989, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a^3 d x+\frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8 \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ a^{3} \int d\, dx + a^{2} \left (a e + 3 b d\right ) \int x\, dx + a x^{3} \left (a b e + a c d + b^{2} d\right ) + \frac{c^{3} e x^{8}}{8} + \frac{c^{2} x^{7} \left (3 b e + c d\right )}{7} + \frac{c x^{6} \left (a c e + b^{2} e + b c d\right )}{2} + x^{5} \left (\frac{6 a b c e}{5} + \frac{3 a c^{2} d}{5} + \frac{b^{3} e}{5} + \frac{3 b^{2} c d}{5}\right ) + x^{4} \left (\frac{3 a^{2} c e}{4} + \frac{3 a b^{2} e}{4} + \frac{3 a b c d}{2} + \frac{b^{3} d}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.0736927, size = 161, normalized size = 1. \[ a^3 d x+\frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(a + b*x + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.001, size = 223, normalized size = 1.4 \[{\frac{{c}^{3}e{x}^{8}}{8}}+{\frac{ \left ( 3\,eb{c}^{2}+d{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{c}^{2}bd+e \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +e \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( d \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +e \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +3\,{a}^{2}be \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{3}e+3\,{a}^{2}bd \right ){x}^{2}}{2}}+{a}^{3}dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+b*x+a)^3,x)
[Out]
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Maxima [A] time = 0.803088, size = 220, normalized size = 1.37 \[ \frac{1}{8} \, c^{3} e x^{8} + \frac{1}{7} \,{\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d +{\left (b^{2} c + a c^{2}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left (3 \,{\left (b^{2} c + a c^{2}\right )} d +{\left (b^{3} + 6 \, a b c\right )} e\right )} x^{5} + a^{3} d x + \frac{1}{4} \,{\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \,{\left (a b^{2} + a^{2} c\right )} e\right )} x^{4} +{\left (a^{2} b e +{\left (a b^{2} + a^{2} c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.197419, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e c^{3} + \frac{1}{7} x^{7} d c^{3} + \frac{3}{7} x^{7} e c^{2} b + \frac{1}{2} x^{6} d c^{2} b + \frac{1}{2} x^{6} e c b^{2} + \frac{1}{2} x^{6} e c^{2} a + \frac{3}{5} x^{5} d c b^{2} + \frac{1}{5} x^{5} e b^{3} + \frac{3}{5} x^{5} d c^{2} a + \frac{6}{5} x^{5} e c b a + \frac{1}{4} x^{4} d b^{3} + \frac{3}{2} x^{4} d c b a + \frac{3}{4} x^{4} e b^{2} a + \frac{3}{4} x^{4} e c a^{2} + x^{3} d b^{2} a + x^{3} d c a^{2} + x^{3} e b a^{2} + \frac{3}{2} x^{2} d b a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.205821, size = 190, normalized size = 1.18 \[ a^{3} d x + \frac{c^{3} e x^{8}}{8} + x^{7} \left (\frac{3 b c^{2} e}{7} + \frac{c^{3} d}{7}\right ) + x^{6} \left (\frac{a c^{2} e}{2} + \frac{b^{2} c e}{2} + \frac{b c^{2} d}{2}\right ) + x^{5} \left (\frac{6 a b c e}{5} + \frac{3 a c^{2} d}{5} + \frac{b^{3} e}{5} + \frac{3 b^{2} c d}{5}\right ) + x^{4} \left (\frac{3 a^{2} c e}{4} + \frac{3 a b^{2} e}{4} + \frac{3 a b c d}{2} + \frac{b^{3} d}{4}\right ) + x^{3} \left (a^{2} b e + a^{2} c d + a b^{2} d\right ) + x^{2} \left (\frac{a^{3} e}{2} + \frac{3 a^{2} b d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.204104, size = 266, normalized size = 1.65 \[ \frac{1}{8} \, c^{3} x^{8} e + \frac{1}{7} \, c^{3} d x^{7} + \frac{3}{7} \, b c^{2} x^{7} e + \frac{1}{2} \, b c^{2} d x^{6} + \frac{1}{2} \, b^{2} c x^{6} e + \frac{1}{2} \, a c^{2} x^{6} e + \frac{3}{5} \, b^{2} c d x^{5} + \frac{3}{5} \, a c^{2} d x^{5} + \frac{1}{5} \, b^{3} x^{5} e + \frac{6}{5} \, a b c x^{5} e + \frac{1}{4} \, b^{3} d x^{4} + \frac{3}{2} \, a b c d x^{4} + \frac{3}{4} \, a b^{2} x^{4} e + \frac{3}{4} \, a^{2} c x^{4} e + a b^{2} d x^{3} + a^{2} c d x^{3} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(e*x + d),x, algorithm="giac")
[Out]