Optimal. Leaf size=65 \[ -\frac{b (c+d x)^4 (b c-a d)}{2 d^3}+\frac{(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac{b^2 (c+d x)^5}{5 d^3} \]
[Out]
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Rubi [A] time = 0.167913, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{b (c+d x)^4 (b c-a d)}{2 d^3}+\frac{(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac{b^2 (c+d x)^5}{5 d^3} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 a c \left (a d + b c\right ) \int x\, dx + \frac{b^{2} d^{2} x^{5}}{5} + \frac{b d x^{4} \left (a d + b c\right )}{2} + c^{2} \int a^{2}\, dx + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.0210594, size = 79, normalized size = 1.22 \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{1}{2} b d x^4 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{5} b^2 d^2 x^5 \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 69, normalized size = 1.1 \[{\frac{{b}^{2}{d}^{2}{x}^{5}}{5}}+{\frac{ \left ( ad+bc \right ) bd{x}^{4}}{2}}+{\frac{ \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ){x}^{3}}{3}}+ac \left ( ad+bc \right ){x}^{2}+{a}^{2}{c}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)^2,x)
[Out]
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Maxima [A] time = 0.741145, size = 97, normalized size = 1.49 \[ \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{1}{2} \,{\left (b c + a d\right )} b d x^{4} + a^{2} c^{2} x + \frac{1}{3} \,{\left (b c + a d\right )}^{2} x^{3} + \frac{1}{3} \,{\left (2 \, b d x^{3} + 3 \,{\left (b c + a d\right )} x^{2}\right )} a c \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.179922, size = 1, normalized size = 0.02 \[ \frac{1}{5} x^{5} d^{2} b^{2} + \frac{1}{2} x^{4} d c b^{2} + \frac{1}{2} x^{4} d^{2} b a + \frac{1}{3} x^{3} c^{2} b^{2} + \frac{4}{3} x^{3} d c b a + \frac{1}{3} x^{3} d^{2} a^{2} + x^{2} c^{2} b a + x^{2} d c a^{2} + x c^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.154192, size = 87, normalized size = 1.34 \[ a^{2} c^{2} x + \frac{b^{2} d^{2} x^{5}}{5} + x^{4} \left (\frac{a b d^{2}}{2} + \frac{b^{2} c d}{2}\right ) + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{2} c d + a b c^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.207673, size = 120, normalized size = 1.85 \[ \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{1}{2} \, b^{2} c d x^{4} + \frac{1}{2} \, a b d^{2} x^{4} + \frac{1}{3} \, b^{2} c^{2} x^{3} + \frac{4}{3} \, a b c d x^{3} + \frac{1}{3} \, a^{2} d^{2} x^{3} + a b c^{2} x^{2} + a^{2} c d x^{2} + a^{2} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="giac")
[Out]