Optimal. Leaf size=411 \[ \frac{(d+e x)^6 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{6 e^8}+\frac{3 c^2 (d+e x)^8 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{8 e^8}-\frac{5 c (d+e x)^7 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}-\frac{3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^8}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^8}-\frac{(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac{7 c^3 (d+e x)^9 (2 c d-b e)}{9 e^8}+\frac{c^4 (d+e x)^{10}}{5 e^8} \]
[Out]
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Rubi [A] time = 1.30114, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{(d+e x)^6 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{6 e^8}+\frac{3 c^2 (d+e x)^8 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{8 e^8}-\frac{5 c (d+e x)^7 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}-\frac{3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^8}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^8}-\frac{(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac{7 c^3 (d+e x)^9 (2 c d-b e)}{9 e^8}+\frac{c^4 (d+e x)^{10}}{5 e^8} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)*(d + e*x)^2*(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(e*x+d)**2*(c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.268771, size = 413, normalized size = 1. \[ a^3 b d^2 x+\frac{1}{2} a^2 d x^2 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac{1}{3} a x^3 \left (4 a^2 c d e+6 a b^2 d e+a b \left (a e^2+9 c d^2\right )+3 b^3 d^2\right )+\frac{1}{5} x^5 \left (12 a^2 c^2 d e+b^3 \left (3 a e^2+5 c d^2\right )+24 a b^2 c d e+3 a b c \left (3 a e^2+5 c d^2\right )+2 b^4 d e\right )+\frac{1}{4} x^4 \left (18 a^2 b c d e+2 a^2 c \left (a e^2+3 c d^2\right )+6 a b^3 d e+3 a b^2 \left (a e^2+4 c d^2\right )+b^4 d^2\right )+\frac{1}{8} c^2 x^8 \left (2 c e (3 a e+7 b d)+9 b^2 e^2+2 c^2 d^2\right )+\frac{1}{7} c x^7 \left (b c \left (15 a e^2+7 c d^2\right )+12 a c^2 d e+5 b^3 e^2+18 b^2 c d e\right )+\frac{1}{6} x^6 \left (3 b^2 c \left (4 a e^2+3 c d^2\right )+30 a b c^2 d e+6 a c^2 \left (a e^2+c d^2\right )+b^4 e^2+10 b^3 c d e\right )+\frac{1}{9} c^3 e x^9 (7 b e+4 c d)+\frac{1}{5} c^4 e^2 x^{10} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)*(d + e*x)^2*(a + b*x + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.002, size = 608, normalized size = 1.5 \[{\frac{{c}^{4}{e}^{2}{x}^{10}}{5}}+{\frac{ \left ( \left ( b{e}^{2}+4\,dec \right ){c}^{3}+6\,{c}^{3}{e}^{2}b \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,bde+2\,c{d}^{2} \right ){c}^{3}+3\, \left ( b{e}^{2}+4\,dec \right ) b{c}^{2}+2\,c{e}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( b{d}^{2}{c}^{3}+3\, \left ( 2\,bde+2\,c{d}^{2} \right ) b{c}^{2}+ \left ( b{e}^{2}+4\,dec \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,c{e}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{2}{d}^{2}{c}^{2}+ \left ( 2\,bde+2\,c{d}^{2} \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( b{e}^{2}+4\,dec \right ) \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,c{e}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( b{d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( 2\,bde+2\,c{d}^{2} \right ) \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( b{e}^{2}+4\,dec \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +6\,c{e}^{2}{a}^{2}b \right ){x}^{5}}{5}}+{\frac{ \left ( b{d}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( 2\,bde+2\,c{d}^{2} \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +3\, \left ( b{e}^{2}+4\,dec \right ){a}^{2}b+2\,c{e}^{2}{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +3\, \left ( 2\,bde+2\,c{d}^{2} \right ){a}^{2}b+ \left ( b{e}^{2}+4\,dec \right ){a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{b}^{2}{d}^{2}{a}^{2}+ \left ( 2\,bde+2\,c{d}^{2} \right ){a}^{3} \right ){x}^{2}}{2}}+b{d}^{2}{a}^{3}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(e*x+d)^2*(c*x^2+b*x+a)^3,x)
[Out]
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Maxima [A] time = 0.735175, size = 563, normalized size = 1.37 \[ \frac{1}{5} \, c^{4} e^{2} x^{10} + \frac{1}{9} \,{\left (4 \, c^{4} d e + 7 \, b c^{3} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (2 \, c^{4} d^{2} + 14 \, b c^{3} d e + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (7 \, b c^{3} d^{2} + 6 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e + 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{2}\right )} x^{7} + a^{3} b d^{2} x + \frac{1}{6} \,{\left (3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} + 10 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} + 2 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} + 6 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (a^{3} b e^{2} + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} + 2 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a^{3} b d e +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(2*c*x + b)*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240841, size = 1, normalized size = 0. \[ \frac{1}{5} x^{10} e^{2} c^{4} + \frac{4}{9} x^{9} e d c^{4} + \frac{7}{9} x^{9} e^{2} c^{3} b + \frac{1}{4} x^{8} d^{2} c^{4} + \frac{7}{4} x^{8} e d c^{3} b + \frac{9}{8} x^{8} e^{2} c^{2} b^{2} + \frac{3}{4} x^{8} e^{2} c^{3} a + x^{7} d^{2} c^{3} b + \frac{18}{7} x^{7} e d c^{2} b^{2} + \frac{5}{7} x^{7} e^{2} c b^{3} + \frac{12}{7} x^{7} e d c^{3} a + \frac{15}{7} x^{7} e^{2} c^{2} b a + \frac{3}{2} x^{6} d^{2} c^{2} b^{2} + \frac{5}{3} x^{6} e d c b^{3} + \frac{1}{6} x^{6} e^{2} b^{4} + x^{6} d^{2} c^{3} a + 5 x^{6} e d c^{2} b a + 2 x^{6} e^{2} c b^{2} a + x^{6} e^{2} c^{2} a^{2} + x^{5} d^{2} c b^{3} + \frac{2}{5} x^{5} e d b^{4} + 3 x^{5} d^{2} c^{2} b a + \frac{24}{5} x^{5} e d c b^{2} a + \frac{3}{5} x^{5} e^{2} b^{3} a + \frac{12}{5} x^{5} e d c^{2} a^{2} + \frac{9}{5} x^{5} e^{2} c b a^{2} + \frac{1}{4} x^{4} d^{2} b^{4} + 3 x^{4} d^{2} c b^{2} a + \frac{3}{2} x^{4} e d b^{3} a + \frac{3}{2} x^{4} d^{2} c^{2} a^{2} + \frac{9}{2} x^{4} e d c b a^{2} + \frac{3}{4} x^{4} e^{2} b^{2} a^{2} + \frac{1}{2} x^{4} e^{2} c a^{3} + x^{3} d^{2} b^{3} a + 3 x^{3} d^{2} c b a^{2} + 2 x^{3} e d b^{2} a^{2} + \frac{4}{3} x^{3} e d c a^{3} + \frac{1}{3} x^{3} e^{2} b a^{3} + \frac{3}{2} x^{2} d^{2} b^{2} a^{2} + x^{2} d^{2} c a^{3} + x^{2} e d b a^{3} + x d^{2} b a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(2*c*x + b)*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.338229, size = 503, normalized size = 1.22 \[ a^{3} b d^{2} x + \frac{c^{4} e^{2} x^{10}}{5} + x^{9} \left (\frac{7 b c^{3} e^{2}}{9} + \frac{4 c^{4} d e}{9}\right ) + x^{8} \left (\frac{3 a c^{3} e^{2}}{4} + \frac{9 b^{2} c^{2} e^{2}}{8} + \frac{7 b c^{3} d e}{4} + \frac{c^{4} d^{2}}{4}\right ) + x^{7} \left (\frac{15 a b c^{2} e^{2}}{7} + \frac{12 a c^{3} d e}{7} + \frac{5 b^{3} c e^{2}}{7} + \frac{18 b^{2} c^{2} d e}{7} + b c^{3} d^{2}\right ) + x^{6} \left (a^{2} c^{2} e^{2} + 2 a b^{2} c e^{2} + 5 a b c^{2} d e + a c^{3} d^{2} + \frac{b^{4} e^{2}}{6} + \frac{5 b^{3} c d e}{3} + \frac{3 b^{2} c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac{9 a^{2} b c e^{2}}{5} + \frac{12 a^{2} c^{2} d e}{5} + \frac{3 a b^{3} e^{2}}{5} + \frac{24 a b^{2} c d e}{5} + 3 a b c^{2} d^{2} + \frac{2 b^{4} d e}{5} + b^{3} c d^{2}\right ) + x^{4} \left (\frac{a^{3} c e^{2}}{2} + \frac{3 a^{2} b^{2} e^{2}}{4} + \frac{9 a^{2} b c d e}{2} + \frac{3 a^{2} c^{2} d^{2}}{2} + \frac{3 a b^{3} d e}{2} + 3 a b^{2} c d^{2} + \frac{b^{4} d^{2}}{4}\right ) + x^{3} \left (\frac{a^{3} b e^{2}}{3} + \frac{4 a^{3} c d e}{3} + 2 a^{2} b^{2} d e + 3 a^{2} b c d^{2} + a b^{3} d^{2}\right ) + x^{2} \left (a^{3} b d e + a^{3} c d^{2} + \frac{3 a^{2} b^{2} d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(e*x+d)**2*(c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.269352, size = 678, normalized size = 1.65 \[ \frac{1}{5} \, c^{4} x^{10} e^{2} + \frac{4}{9} \, c^{4} d x^{9} e + \frac{1}{4} \, c^{4} d^{2} x^{8} + \frac{7}{9} \, b c^{3} x^{9} e^{2} + \frac{7}{4} \, b c^{3} d x^{8} e + b c^{3} d^{2} x^{7} + \frac{9}{8} \, b^{2} c^{2} x^{8} e^{2} + \frac{3}{4} \, a c^{3} x^{8} e^{2} + \frac{18}{7} \, b^{2} c^{2} d x^{7} e + \frac{12}{7} \, a c^{3} d x^{7} e + \frac{3}{2} \, b^{2} c^{2} d^{2} x^{6} + a c^{3} d^{2} x^{6} + \frac{5}{7} \, b^{3} c x^{7} e^{2} + \frac{15}{7} \, a b c^{2} x^{7} e^{2} + \frac{5}{3} \, b^{3} c d x^{6} e + 5 \, a b c^{2} d x^{6} e + b^{3} c d^{2} x^{5} + 3 \, a b c^{2} d^{2} x^{5} + \frac{1}{6} \, b^{4} x^{6} e^{2} + 2 \, a b^{2} c x^{6} e^{2} + a^{2} c^{2} x^{6} e^{2} + \frac{2}{5} \, b^{4} d x^{5} e + \frac{24}{5} \, a b^{2} c d x^{5} e + \frac{12}{5} \, a^{2} c^{2} d x^{5} e + \frac{1}{4} \, b^{4} d^{2} x^{4} + 3 \, a b^{2} c d^{2} x^{4} + \frac{3}{2} \, a^{2} c^{2} d^{2} x^{4} + \frac{3}{5} \, a b^{3} x^{5} e^{2} + \frac{9}{5} \, a^{2} b c x^{5} e^{2} + \frac{3}{2} \, a b^{3} d x^{4} e + \frac{9}{2} \, a^{2} b c d x^{4} e + a b^{3} d^{2} x^{3} + 3 \, a^{2} b c d^{2} x^{3} + \frac{3}{4} \, a^{2} b^{2} x^{4} e^{2} + \frac{1}{2} \, a^{3} c x^{4} e^{2} + 2 \, a^{2} b^{2} d x^{3} e + \frac{4}{3} \, a^{3} c d x^{3} e + \frac{3}{2} \, a^{2} b^{2} d^{2} x^{2} + a^{3} c d^{2} x^{2} + \frac{1}{3} \, a^{3} b x^{3} e^{2} + a^{3} b d x^{2} e + a^{3} b d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(2*c*x + b)*(e*x + d)^2,x, algorithm="giac")
[Out]