Optimal. Leaf size=92 \[ \frac{3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac{d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac{(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac{d^3 (a+b x)^8}{8 b^4} \]
[Out]
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Rubi [A] time = 0.334437, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac{d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac{(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac{d^3 (a+b x)^8}{8 b^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 47.9887, size = 80, normalized size = 0.87 \[ \frac{d^{3} \left (a + b x\right )^{8}}{8 b^{4}} - \frac{3 d^{2} \left (a + b x\right )^{7} \left (a d - b c\right )}{7 b^{4}} + \frac{d \left (a + b x\right )^{6} \left (a d - b c\right )^{2}}{2 b^{4}} - \frac{\left (a + b x\right )^{5} \left (a d - b c\right )^{3}}{5 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
[Out]
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Mathematica [B] time = 0.0541907, size = 217, normalized size = 2.36 \[ a^4 c^3 x+\frac{1}{2} a^3 c^2 x^2 (3 a d+4 b c)+\frac{1}{2} b^2 d x^6 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c x^3 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac{1}{5} b x^5 \left (4 a^3 d^3+18 a^2 b c d^2+12 a b^2 c^2 d+b^3 c^3\right )+\frac{1}{4} a x^4 \left (a^3 d^3+12 a^2 b c d^2+18 a b^2 c^2 d+4 b^3 c^3\right )+\frac{1}{7} b^3 d^2 x^7 (4 a d+3 b c)+\frac{1}{8} b^4 d^3 x^8 \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
[Out]
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Maple [B] time = 0.002, size = 391, normalized size = 4.3 \[{\frac{{d}^{3}{b}^{4}{x}^{8}}{8}}+{\frac{ \left ( a{b}^{3}{d}^{3}+3\,{b}^{3} \left ( ad+bc \right ){d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,a \left ( ad+bc \right ){b}^{2}{d}^{2}+b \left ( a{b}^{2}c{d}^{2}+2\, \left ( ad+bc \right ) ^{2}bd+bd \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( a \left ( a{b}^{2}c{d}^{2}+2\, \left ( ad+bc \right ) ^{2}bd+bd \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) +b \left ( 4\,ac \left ( ad+bc \right ) bd+ \left ( ad+bc \right ) \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( a \left ( 4\,ac \left ( ad+bc \right ) bd+ \left ( ad+bc \right ) \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ) +b \left ( ac \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\, \left ( ad+bc \right ) ^{2}ac+{a}^{2}b{c}^{2}d \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( a \left ( ac \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\, \left ( ad+bc \right ) ^{2}ac+{a}^{2}b{c}^{2}d \right ) +3\,b{a}^{2}{c}^{2} \left ( ad+bc \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{3}{c}^{2} \left ( ad+bc \right ) +{a}^{3}b{c}^{3} \right ){x}^{2}}{2}}+{a}^{4}{c}^{3}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(a*c+(a*d+b*c)*x+x^2*b*d)^3,x)
[Out]
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Maxima [A] time = 0.733903, size = 304, normalized size = 3.3 \[ \frac{1}{8} \, b^{4} d^{3} x^{8} + a^{4} c^{3} x + \frac{1}{7} \,{\left (3 \, b^{4} c d^{2} + 4 \, a b^{3} d^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b^{4} c^{2} d + 4 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} c^{3} + 12 \, a b^{3} c^{2} d + 18 \, a^{2} b^{2} c d^{2} + 4 \, a^{3} b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} c^{3} + 18 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} c^{3} + 4 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b c^{3} + 3 \, a^{4} c^{2} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3*(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.183732, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} d^{3} b^{4} + \frac{3}{7} x^{7} d^{2} c b^{4} + \frac{4}{7} x^{7} d^{3} b^{3} a + \frac{1}{2} x^{6} d c^{2} b^{4} + 2 x^{6} d^{2} c b^{3} a + x^{6} d^{3} b^{2} a^{2} + \frac{1}{5} x^{5} c^{3} b^{4} + \frac{12}{5} x^{5} d c^{2} b^{3} a + \frac{18}{5} x^{5} d^{2} c b^{2} a^{2} + \frac{4}{5} x^{5} d^{3} b a^{3} + x^{4} c^{3} b^{3} a + \frac{9}{2} x^{4} d c^{2} b^{2} a^{2} + 3 x^{4} d^{2} c b a^{3} + \frac{1}{4} x^{4} d^{3} a^{4} + 2 x^{3} c^{3} b^{2} a^{2} + 4 x^{3} d c^{2} b a^{3} + x^{3} d^{2} c a^{4} + 2 x^{2} c^{3} b a^{3} + \frac{3}{2} x^{2} d c^{2} a^{4} + x c^{3} a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3*(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.277997, size = 243, normalized size = 2.64 \[ a^{4} c^{3} x + \frac{b^{4} d^{3} x^{8}}{8} + x^{7} \left (\frac{4 a b^{3} d^{3}}{7} + \frac{3 b^{4} c d^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} d^{3} + 2 a b^{3} c d^{2} + \frac{b^{4} c^{2} d}{2}\right ) + x^{5} \left (\frac{4 a^{3} b d^{3}}{5} + \frac{18 a^{2} b^{2} c d^{2}}{5} + \frac{12 a b^{3} c^{2} d}{5} + \frac{b^{4} c^{3}}{5}\right ) + x^{4} \left (\frac{a^{4} d^{3}}{4} + 3 a^{3} b c d^{2} + \frac{9 a^{2} b^{2} c^{2} d}{2} + a b^{3} c^{3}\right ) + x^{3} \left (a^{4} c d^{2} + 4 a^{3} b c^{2} d + 2 a^{2} b^{2} c^{3}\right ) + x^{2} \left (\frac{3 a^{4} c^{2} d}{2} + 2 a^{3} b c^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.209841, size = 331, normalized size = 3.6 \[ \frac{1}{8} \, b^{4} d^{3} x^{8} + \frac{3}{7} \, b^{4} c d^{2} x^{7} + \frac{4}{7} \, a b^{3} d^{3} x^{7} + \frac{1}{2} \, b^{4} c^{2} d x^{6} + 2 \, a b^{3} c d^{2} x^{6} + a^{2} b^{2} d^{3} x^{6} + \frac{1}{5} \, b^{4} c^{3} x^{5} + \frac{12}{5} \, a b^{3} c^{2} d x^{5} + \frac{18}{5} \, a^{2} b^{2} c d^{2} x^{5} + \frac{4}{5} \, a^{3} b d^{3} x^{5} + a b^{3} c^{3} x^{4} + \frac{9}{2} \, a^{2} b^{2} c^{2} d x^{4} + 3 \, a^{3} b c d^{2} x^{4} + \frac{1}{4} \, a^{4} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{3} x^{3} + 4 \, a^{3} b c^{2} d x^{3} + a^{4} c d^{2} x^{3} + 2 \, a^{3} b c^{3} x^{2} + \frac{3}{2} \, a^{4} c^{2} d x^{2} + a^{4} c^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3*(b*x + a),x, algorithm="giac")
[Out]