Optimal. Leaf size=45 \[ \frac{d^2 (b+2 c x)^5}{40 c^2}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^3}{24 c^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.103192, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{d^2 (b+2 c x)^5}{40 c^2}-\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^3}{24 c^2} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.551, size = 41, normalized size = 0.91 \[ \frac{d^{2} \left (b + 2 c x\right )^{5}}{40 c^{2}} - \frac{d^{2} \left (b + 2 c x\right )^{3} \left (- a c + \frac{b^{2}}{4}\right )}{6 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**2*(c*x**2+b*x+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0192051, size = 64, normalized size = 1.42 \[ d^2 \left (\frac{1}{3} c x^3 \left (4 a c+5 b^2\right )+\frac{1}{2} b x^2 \left (4 a c+b^2\right )+a b^2 x+2 b c^2 x^4+\frac{4 c^3 x^5}{5}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.001, size = 79, normalized size = 1.8 \[{\frac{4\,{c}^{3}{d}^{2}{x}^{5}}{5}}+2\,b{c}^{2}{d}^{2}{x}^{4}+{\frac{ \left ( 4\,{c}^{2}{d}^{2}a+5\,{b}^{2}{d}^{2}c \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,abc{d}^{2}+{b}^{3}{d}^{2} \right ){x}^{2}}{2}}+{b}^{2}{d}^{2}ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^2*(c*x^2+b*x+a),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.675368, size = 96, normalized size = 2.13 \[ \frac{4}{5} \, c^{3} d^{2} x^{5} + 2 \, b c^{2} d^{2} x^{4} + a b^{2} d^{2} x + \frac{1}{3} \,{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{3} + \frac{1}{2} \,{\left (b^{3} + 4 \, a b c\right )} d^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^2*(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.199993, size = 1, normalized size = 0.02 \[ \frac{4}{5} x^{5} d^{2} c^{3} + 2 x^{4} d^{2} c^{2} b + \frac{5}{3} x^{3} d^{2} c b^{2} + \frac{4}{3} x^{3} d^{2} c^{2} a + \frac{1}{2} x^{2} d^{2} b^{3} + 2 x^{2} d^{2} c b a + x d^{2} b^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^2*(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.129839, size = 85, normalized size = 1.89 \[ a b^{2} d^{2} x + 2 b c^{2} d^{2} x^{4} + \frac{4 c^{3} d^{2} x^{5}}{5} + x^{3} \left (\frac{4 a c^{2} d^{2}}{3} + \frac{5 b^{2} c d^{2}}{3}\right ) + x^{2} \left (2 a b c d^{2} + \frac{b^{3} d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**2*(c*x**2+b*x+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.213315, size = 107, normalized size = 2.38 \[ \frac{4}{5} \, c^{3} d^{2} x^{5} + 2 \, b c^{2} d^{2} x^{4} + \frac{5}{3} \, b^{2} c d^{2} x^{3} + \frac{4}{3} \, a c^{2} d^{2} x^{3} + \frac{1}{2} \, b^{3} d^{2} x^{2} + 2 \, a b c d^{2} x^{2} + a b^{2} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^2*(c*x^2 + b*x + a),x, algorithm="giac")
[Out]