Optimal. Leaf size=124 \[ \frac{(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac{c (d+e x)^6 (2 c d-b e)}{2 e^4}+\frac{2 c^2 (d+e x)^7}{7 e^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.333575, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac{c (d+e x)^6 (2 c d-b e)}{2 e^4}+\frac{2 c^2 (d+e x)^7}{7 e^4} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 45.5667, size = 117, normalized size = 0.94 \[ \frac{2 c^{2} \left (d + e x\right )^{7}}{7 e^{4}} + \frac{c \left (d + e x\right )^{6} \left (b e - 2 c d\right )}{2 e^{4}} + \frac{\left (d + e x\right )^{5} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{4} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{4 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0919801, size = 175, normalized size = 1.41 \[ \frac{1}{5} e x^5 \left (c e (2 a e+9 b d)+b^2 e^2+6 c^2 d^2\right )+d x^3 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac{1}{2} d^2 x^2 \left (3 a b e+2 a c d+b^2 d\right )+\frac{1}{4} x^4 \left (3 c d e (2 a e+3 b d)+b e^2 (a e+3 b d)+2 c^2 d^3\right )+a b d^3 x+\frac{1}{2} c e^2 x^6 (b e+2 c d)+\frac{2}{7} c^2 e^3 x^7 \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.002, size = 221, normalized size = 1.8 \[{\frac{2\,{e}^{3}{c}^{2}{x}^{7}}{7}}+{\frac{ \left ( \left ( b{e}^{3}+6\,d{e}^{2}c \right ) c+2\,bc{e}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,bd{e}^{2}+6\,{d}^{2}ec \right ) c+ \left ( b{e}^{3}+6\,d{e}^{2}c \right ) b+2\,ac{e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) c+ \left ( 3\,bd{e}^{2}+6\,{d}^{2}ec \right ) b+ \left ( b{e}^{3}+6\,d{e}^{2}c \right ) a \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{3}bc+ \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) b+ \left ( 3\,bd{e}^{2}+6\,{d}^{2}ec \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ({b}^{2}{d}^{3}+ \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) a \right ){x}^{2}}{2}}+b{d}^{3}ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.702267, size = 235, normalized size = 1.9 \[ \frac{2}{7} \, c^{2} e^{3} x^{7} + \frac{1}{2} \,{\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{6} + a b d^{3} x + \frac{1}{5} \,{\left (6 \, c^{2} d^{2} e + 9 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, c^{2} d^{3} + 9 \, b c d^{2} e + a b e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{4} +{\left (b c d^{3} + a b d e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a b d^{2} e +{\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(2*c*x + b)*(e*x + d)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.24661, size = 1, normalized size = 0.01 \[ \frac{2}{7} x^{7} e^{3} c^{2} + x^{6} e^{2} d c^{2} + \frac{1}{2} x^{6} e^{3} c b + \frac{6}{5} x^{5} e d^{2} c^{2} + \frac{9}{5} x^{5} e^{2} d c b + \frac{1}{5} x^{5} e^{3} b^{2} + \frac{2}{5} x^{5} e^{3} c a + \frac{1}{2} x^{4} d^{3} c^{2} + \frac{9}{4} x^{4} e d^{2} c b + \frac{3}{4} x^{4} e^{2} d b^{2} + \frac{3}{2} x^{4} e^{2} d c a + \frac{1}{4} x^{4} e^{3} b a + x^{3} d^{3} c b + x^{3} e d^{2} b^{2} + 2 x^{3} e d^{2} c a + x^{3} e^{2} d b a + \frac{1}{2} x^{2} d^{3} b^{2} + x^{2} d^{3} c a + \frac{3}{2} x^{2} e d^{2} b a + x d^{3} b a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(2*c*x + b)*(e*x + d)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.202566, size = 211, normalized size = 1.7 \[ a b d^{3} x + \frac{2 c^{2} e^{3} x^{7}}{7} + x^{6} \left (\frac{b c e^{3}}{2} + c^{2} d e^{2}\right ) + x^{5} \left (\frac{2 a c e^{3}}{5} + \frac{b^{2} e^{3}}{5} + \frac{9 b c d e^{2}}{5} + \frac{6 c^{2} d^{2} e}{5}\right ) + x^{4} \left (\frac{a b e^{3}}{4} + \frac{3 a c d e^{2}}{2} + \frac{3 b^{2} d e^{2}}{4} + \frac{9 b c d^{2} e}{4} + \frac{c^{2} d^{3}}{2}\right ) + x^{3} \left (a b d e^{2} + 2 a c d^{2} e + b^{2} d^{2} e + b c d^{3}\right ) + x^{2} \left (\frac{3 a b d^{2} e}{2} + a c d^{3} + \frac{b^{2} d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.270912, size = 278, normalized size = 2.24 \[ \frac{2}{7} \, c^{2} x^{7} e^{3} + c^{2} d x^{6} e^{2} + \frac{6}{5} \, c^{2} d^{2} x^{5} e + \frac{1}{2} \, c^{2} d^{3} x^{4} + \frac{1}{2} \, b c x^{6} e^{3} + \frac{9}{5} \, b c d x^{5} e^{2} + \frac{9}{4} \, b c d^{2} x^{4} e + b c d^{3} x^{3} + \frac{1}{5} \, b^{2} x^{5} e^{3} + \frac{2}{5} \, a c x^{5} e^{3} + \frac{3}{4} \, b^{2} d x^{4} e^{2} + \frac{3}{2} \, a c d x^{4} e^{2} + b^{2} d^{2} x^{3} e + 2 \, a c d^{2} x^{3} e + \frac{1}{2} \, b^{2} d^{3} x^{2} + a c d^{3} x^{2} + \frac{1}{4} \, a b x^{4} e^{3} + a b d x^{3} e^{2} + \frac{3}{2} \, a b d^{2} x^{2} e + a b d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(2*c*x + b)*(e*x + d)^3,x, algorithm="giac")
[Out]