Optimal. Leaf size=38 \[ \frac{(a+b x)^5 (b c-a d)}{5 b^2}+\frac{d (a+b x)^6}{6 b^2} \]
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Rubi [A] time = 0.0734121, antiderivative size = 38, 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 \[ \frac{(a+b x)^5 (b c-a d)}{5 b^2}+\frac{d (a+b x)^6}{6 b^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 19.108, size = 31, normalized size = 0.82 \[ \frac{d \left (a + b x\right )^{6}}{6 b^{2}} - \frac{\left (a + b x\right )^{5} \left (a d - b c\right )}{5 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3*(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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Mathematica [B] time = 0.0332613, size = 84, normalized size = 2.21 \[ \frac{1}{30} x \left (15 a^4 (2 c+d x)+20 a^3 b x (3 c+2 d x)+15 a^2 b^2 x^2 (4 c+3 d x)+6 a b^3 x^3 (5 c+4 d x)+b^4 x^4 (6 c+5 d x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2),x]
[Out]
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Maple [B] time = 0.002, size = 133, normalized size = 3.5 \[{\frac{d{b}^{4}{x}^{6}}{6}}+{\frac{ \left ( 3\,ad{b}^{3}+{b}^{3} \left ( ad+bc \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{a}^{2}d{b}^{2}+3\,a{b}^{2} \left ( ad+bc \right ) +ac{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3}bd+3\,{a}^{2}b \left ( ad+bc \right ) +3\,{a}^{2}{b}^{2}c \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{3} \left ( ad+bc \right ) +3\,{a}^{3}bc \right ){x}^{2}}{2}}+{a}^{4}cx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3*(a*c+(a*d+b*c)*x+x^2*b*d),x)
[Out]
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Maxima [A] time = 0.729409, size = 130, normalized size = 3.42 \[ \frac{1}{6} \, b^{4} d x^{6} + a^{4} c x + \frac{1}{5} \,{\left (b^{4} c + 4 \, a b^{3} d\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b c + a^{4} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.178062, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} d b^{4} + \frac{1}{5} x^{5} c b^{4} + \frac{4}{5} x^{5} d b^{3} a + x^{4} c b^{3} a + \frac{3}{2} x^{4} d b^{2} a^{2} + 2 x^{3} c b^{2} a^{2} + \frac{4}{3} x^{3} d b a^{3} + 2 x^{2} c b a^{3} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.154522, size = 100, normalized size = 2.63 \[ a^{4} c x + \frac{b^{4} d x^{6}}{6} + x^{5} \left (\frac{4 a b^{3} d}{5} + \frac{b^{4} c}{5}\right ) + x^{4} \left (\frac{3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \left (\frac{4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac{a^{4} d}{2} + 2 a^{3} b c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**3*(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.208606, size = 131, normalized size = 3.45 \[ \frac{1}{6} \, b^{4} d x^{6} + \frac{1}{5} \, b^{4} c x^{5} + \frac{4}{5} \, a b^{3} d x^{5} + a b^{3} c x^{4} + \frac{3}{2} \, a^{2} b^{2} d x^{4} + 2 \, a^{2} b^{2} c x^{3} + \frac{4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^3,x, algorithm="giac")
[Out]