Optimal. Leaf size=55 \[ \frac{1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \]
[Out]
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Rubi [A] time = 0.083849, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{1}{12} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3+\frac{1}{4} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^3 \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 20.4882, size = 49, normalized size = 0.89 \[ \frac{d^{3} \left (b + 2 c x\right )^{2} \left (a + b x + c x^{2}\right )^{3}}{4} + \frac{d^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{3}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**3*(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.045571, size = 97, normalized size = 1.76 \[ \frac{1}{3} d^3 x (b+c x) \left (3 a^2 \left (b^2+2 b c x+2 c^2 x^2\right )+a x \left (3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right )+x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^2,x]
[Out]
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Maple [B] time = 0.001, size = 237, normalized size = 4.3 \[{c}^{5}{d}^{3}{x}^{8}+4\,b{d}^{3}{c}^{4}{x}^{7}+{\frac{ \left ( 30\,{b}^{2}{d}^{3}{c}^{3}+8\,{c}^{3}{d}^{3} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 13\,{b}^{3}{c}^{2}{d}^{3}+12\,b{d}^{3}{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +16\,{c}^{3}{d}^{3}ab \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{4}{d}^{3}c+6\,{b}^{2}{d}^{3}c \left ( 2\,ac+{b}^{2} \right ) +24\,{b}^{2}{d}^{3}{c}^{2}a+8\,{c}^{3}{d}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( 2\,ac+{b}^{2} \right ) +12\,{b}^{3}{d}^{3}ca+12\,b{d}^{3}{c}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{b}^{2}{d}^{3}c{a}^{2}+2\,{b}^{4}{d}^{3}a \right ){x}^{2}}{2}}+{b}^{3}{d}^{3}{a}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^3*(c*x^2+b*x+a)^2,x)
[Out]
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Maxima [A] time = 0.681441, size = 217, normalized size = 3.95 \[ c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac{1}{3} \,{\left (19 \, b^{2} c^{3} + 8 \, a c^{4}\right )} d^{3} x^{6} + a^{2} b^{3} d^{3} x +{\left (5 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d^{3} x^{5} +{\left (2 \, b^{4} c + 9 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (b^{5} + 14 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} d^{3} x^{3} +{\left (a b^{4} + 3 \, a^{2} b^{2} c\right )} d^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20621, size = 1, normalized size = 0.02 \[ x^{8} d^{3} c^{5} + 4 x^{7} d^{3} c^{4} b + \frac{19}{3} x^{6} d^{3} c^{3} b^{2} + \frac{8}{3} x^{6} d^{3} c^{4} a + 5 x^{5} d^{3} c^{2} b^{3} + 8 x^{5} d^{3} c^{3} b a + 2 x^{4} d^{3} c b^{4} + 9 x^{4} d^{3} c^{2} b^{2} a + 2 x^{4} d^{3} c^{3} a^{2} + \frac{1}{3} x^{3} d^{3} b^{5} + \frac{14}{3} x^{3} d^{3} c b^{3} a + 4 x^{3} d^{3} c^{2} b a^{2} + x^{2} d^{3} b^{4} a + 3 x^{2} d^{3} c b^{2} a^{2} + x d^{3} b^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.200634, size = 194, normalized size = 3.53 \[ a^{2} b^{3} d^{3} x + 4 b c^{4} d^{3} x^{7} + c^{5} d^{3} x^{8} + x^{6} \left (\frac{8 a c^{4} d^{3}}{3} + \frac{19 b^{2} c^{3} d^{3}}{3}\right ) + x^{5} \left (8 a b c^{3} d^{3} + 5 b^{3} c^{2} d^{3}\right ) + x^{4} \left (2 a^{2} c^{3} d^{3} + 9 a b^{2} c^{2} d^{3} + 2 b^{4} c d^{3}\right ) + x^{3} \left (4 a^{2} b c^{2} d^{3} + \frac{14 a b^{3} c d^{3}}{3} + \frac{b^{5} d^{3}}{3}\right ) + x^{2} \left (3 a^{2} b^{2} c d^{3} + a b^{4} d^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**3*(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.211994, size = 261, normalized size = 4.75 \[ c^{5} d^{3} x^{8} + 4 \, b c^{4} d^{3} x^{7} + \frac{19}{3} \, b^{2} c^{3} d^{3} x^{6} + \frac{8}{3} \, a c^{4} d^{3} x^{6} + 5 \, b^{3} c^{2} d^{3} x^{5} + 8 \, a b c^{3} d^{3} x^{5} + 2 \, b^{4} c d^{3} x^{4} + 9 \, a b^{2} c^{2} d^{3} x^{4} + 2 \, a^{2} c^{3} d^{3} x^{4} + \frac{1}{3} \, b^{5} d^{3} x^{3} + \frac{14}{3} \, a b^{3} c d^{3} x^{3} + 4 \, a^{2} b c^{2} d^{3} x^{3} + a b^{4} d^{3} x^{2} + 3 \, a^{2} b^{2} c d^{3} x^{2} + a^{2} b^{3} d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^2,x, algorithm="giac")
[Out]