Optimal. Leaf size=55 \[ \frac{1}{20} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4+\frac{1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4 \]
[Out]
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Rubi [A] time = 0.10119, 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 \[ \frac{1}{20} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4+\frac{1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4 \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 19.6558, size = 49, normalized size = 0.89 \[ \frac{d^{3} \left (b + 2 c x\right )^{2} \left (a + b x + c x^{2}\right )^{4}}{5} + \frac{d^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{4}}{20} \]
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)**3,x)
[Out]
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Mathematica [B] time = 0.0672048, size = 132, normalized size = 2.4 \[ \frac{1}{20} d^3 x (b+c x) \left (20 a^3 \left (b^2+2 b c x+2 c^2 x^2\right )+10 a^2 x \left (3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right )+20 a x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )+x^3 (b+c x)^3 \left (5 b^2+16 b c x+16 c^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^3,x]
[Out]
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Maple [B] time = 0.003, size = 534, normalized size = 9.7 \[{\frac{4\,{c}^{6}{d}^{3}{x}^{10}}{5}}+4\,b{d}^{3}{c}^{5}{x}^{9}+{\frac{ \left ( 42\,{b}^{2}{d}^{3}{c}^{4}+8\,{c}^{3}{d}^{3} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 19\,{b}^{3}{d}^{3}{c}^{3}+12\,b{d}^{3}{c}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +8\,{c}^{3}{d}^{3} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{4}{d}^{3}{c}^{2}+6\,{b}^{2}{d}^{3}c \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +12\,b{d}^{3}{c}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +8\,{c}^{3}{d}^{3} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +6\,{b}^{2}{d}^{3}c \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +12\,b{d}^{3}{c}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +24\,{c}^{3}{d}^{3}{a}^{2}b \right ){x}^{5}}{5}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +6\,{b}^{2}{d}^{3}c \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +36\,{b}^{2}{d}^{3}{c}^{2}{a}^{2}+8\,{c}^{3}{d}^{3}{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{3}{d}^{3} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +18\,{b}^{3}{d}^{3}c{a}^{2}+12\,b{d}^{3}{c}^{2}{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{b}^{2}{d}^{3}c{a}^{3}+3\,{b}^{4}{d}^{3}{a}^{2} \right ){x}^{2}}{2}}+{b}^{3}{d}^{3}{a}^{3}x \]
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)^3,x)
[Out]
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Maxima [A] time = 0.701136, size = 328, normalized size = 5.96 \[ \frac{4}{5} \, c^{6} d^{3} x^{10} + 4 \, b c^{5} d^{3} x^{9} + \frac{3}{4} \,{\left (11 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{3} x^{8} + 3 \,{\left (3 \, b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x^{7} + a^{3} b^{3} d^{3} x + \frac{1}{2} \,{\left (11 \, b^{4} c^{2} + 38 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{3} x^{6} + \frac{3}{5} \,{\left (3 \, b^{5} c + 25 \, a b^{3} c^{2} + 20 \, a^{2} b c^{3}\right )} d^{3} x^{5} + \frac{1}{4} \,{\left (b^{6} + 24 \, a b^{4} c + 54 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d^{3} x^{4} +{\left (a b^{5} + 7 \, a^{2} b^{3} c + 4 \, a^{3} b c^{2}\right )} d^{3} x^{3} + \frac{3}{2} \,{\left (a^{2} b^{4} + 2 \, a^{3} b^{2} c\right )} d^{3} x^{2} \]
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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.193382, size = 1, normalized size = 0.02 \[ \frac{4}{5} x^{10} d^{3} c^{6} + 4 x^{9} d^{3} c^{5} b + \frac{33}{4} x^{8} d^{3} c^{4} b^{2} + 3 x^{8} d^{3} c^{5} a + 9 x^{7} d^{3} c^{3} b^{3} + 12 x^{7} d^{3} c^{4} b a + \frac{11}{2} x^{6} d^{3} c^{2} b^{4} + 19 x^{6} d^{3} c^{3} b^{2} a + 4 x^{6} d^{3} c^{4} a^{2} + \frac{9}{5} x^{5} d^{3} c b^{5} + 15 x^{5} d^{3} c^{2} b^{3} a + 12 x^{5} d^{3} c^{3} b a^{2} + \frac{1}{4} x^{4} d^{3} b^{6} + 6 x^{4} d^{3} c b^{4} a + \frac{27}{2} x^{4} d^{3} c^{2} b^{2} a^{2} + 2 x^{4} d^{3} c^{3} a^{3} + x^{3} d^{3} b^{5} a + 7 x^{3} d^{3} c b^{3} a^{2} + 4 x^{3} d^{3} c^{2} b a^{3} + \frac{3}{2} x^{2} d^{3} b^{4} a^{2} + 3 x^{2} d^{3} c b^{2} a^{3} + x d^{3} b^{3} a^{3} \]
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)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.257579, size = 299, normalized size = 5.44 \[ a^{3} b^{3} d^{3} x + 4 b c^{5} d^{3} x^{9} + \frac{4 c^{6} d^{3} x^{10}}{5} + x^{8} \left (3 a c^{5} d^{3} + \frac{33 b^{2} c^{4} d^{3}}{4}\right ) + x^{7} \left (12 a b c^{4} d^{3} + 9 b^{3} c^{3} d^{3}\right ) + x^{6} \left (4 a^{2} c^{4} d^{3} + 19 a b^{2} c^{3} d^{3} + \frac{11 b^{4} c^{2} d^{3}}{2}\right ) + x^{5} \left (12 a^{2} b c^{3} d^{3} + 15 a b^{3} c^{2} d^{3} + \frac{9 b^{5} c d^{3}}{5}\right ) + x^{4} \left (2 a^{3} c^{3} d^{3} + \frac{27 a^{2} b^{2} c^{2} d^{3}}{2} + 6 a b^{4} c d^{3} + \frac{b^{6} d^{3}}{4}\right ) + x^{3} \left (4 a^{3} b c^{2} d^{3} + 7 a^{2} b^{3} c d^{3} + a b^{5} d^{3}\right ) + x^{2} \left (3 a^{3} b^{2} c d^{3} + \frac{3 a^{2} b^{4} d^{3}}{2}\right ) \]
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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.214251, size = 402, normalized size = 7.31 \[ \frac{4}{5} \, c^{6} d^{3} x^{10} + 4 \, b c^{5} d^{3} x^{9} + \frac{33}{4} \, b^{2} c^{4} d^{3} x^{8} + 3 \, a c^{5} d^{3} x^{8} + 9 \, b^{3} c^{3} d^{3} x^{7} + 12 \, a b c^{4} d^{3} x^{7} + \frac{11}{2} \, b^{4} c^{2} d^{3} x^{6} + 19 \, a b^{2} c^{3} d^{3} x^{6} + 4 \, a^{2} c^{4} d^{3} x^{6} + \frac{9}{5} \, b^{5} c d^{3} x^{5} + 15 \, a b^{3} c^{2} d^{3} x^{5} + 12 \, a^{2} b c^{3} d^{3} x^{5} + \frac{1}{4} \, b^{6} d^{3} x^{4} + 6 \, a b^{4} c d^{3} x^{4} + \frac{27}{2} \, a^{2} b^{2} c^{2} d^{3} x^{4} + 2 \, a^{3} c^{3} d^{3} x^{4} + a b^{5} d^{3} x^{3} + 7 \, a^{2} b^{3} c d^{3} x^{3} + 4 \, a^{3} b c^{2} d^{3} x^{3} + \frac{3}{2} \, a^{2} b^{4} d^{3} x^{2} + 3 \, a^{3} b^{2} c d^{3} x^{2} + a^{3} b^{3} d^{3} x \]
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)^3,x, algorithm="giac")
[Out]