Optimal. Leaf size=304 \[ -\frac{(d+e x)^6 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{6 e^6}-\frac{(d+e x)^5 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{5 e^6}-\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{4 e^6}-\frac{(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac{c (d+e x)^7 (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac{B c^2 (d+e x)^8}{8 e^6} \]
[Out]
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Rubi [A] time = 1.15539, antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{(d+e x)^6 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{6 e^6}-\frac{(d+e x)^5 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{5 e^6}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{4 e^6}-\frac{(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac{c (d+e x)^7 (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac{B c^2 (d+e x)^8}{8 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^2*(a + b*x + c*x^2)^2,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((B*x+A)*(e*x+d)**2*(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.295362, size = 301, normalized size = 0.99 \[ a^2 A d^2 x+\frac{1}{6} x^6 \left (B \left (2 c e (a e+2 b d)+b^2 e^2+c^2 d^2\right )+2 A c e (b e+c d)\right )+\frac{1}{5} x^5 \left (2 b \left (a B e^2+2 A c d e+B c d^2\right )+c \left (2 a A e^2+4 a B d e+A c d^2\right )+b^2 e (A e+2 B d)\right )+\frac{1}{4} x^4 \left (2 b \left (a A e^2+2 a B d e+A c d^2\right )+a \left (a B e^2+4 A c d e+2 B c d^2\right )+b^2 d (2 A e+B d)\right )+\frac{1}{3} x^3 \left (A \left (4 a b d e+a \left (a e^2+2 c d^2\right )+b^2 d^2\right )+2 a B d (a e+b d)\right )+\frac{1}{2} a d x^2 (2 A (a e+b d)+a B d)+\frac{1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac{1}{8} B c^2 e^2 x^8 \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^2*(a + b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0., size = 293, normalized size = 1. \[{\frac{B{c}^{2}{e}^{2}{x}^{8}}{8}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){c}^{2}+2\,B{e}^{2}bc \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){c}^{2}+2\, \left ( A{e}^{2}+2\,Bde \right ) bc+B{e}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}{d}^{2}+2\, \left ( 2\,Ade+B{d}^{2} \right ) bc+ \left ( A{e}^{2}+2\,Bde \right ) \left ( 2\,ac+{b}^{2} \right ) +2\,B{e}^{2}ab \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,A{d}^{2}bc+ \left ( 2\,Ade+B{d}^{2} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( A{e}^{2}+2\,Bde \right ) ab+{a}^{2}B{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{2} \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 2\,Ade+B{d}^{2} \right ) ab+ \left ( A{e}^{2}+2\,Bde \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,A{d}^{2}ab+ \left ( 2\,Ade+B{d}^{2} \right ){a}^{2} \right ){x}^{2}}{2}}+A{d}^{2}{a}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^2*(c*x^2+b*x+a)^2,x)
[Out]
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Maxima [A] time = 0.696989, size = 405, normalized size = 1.33 \[ \frac{1}{8} \, B c^{2} e^{2} x^{8} + \frac{1}{7} \,{\left (2 \, B c^{2} d e +{\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{2} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac{1}{5} \,{\left ({\left (2 \, B b c + A c^{2}\right )} d^{2} + 2 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left ({\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} + 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a^{2} e^{2} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a^{2} d e +{\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240038, size = 1, normalized size = 0. \[ \frac{1}{8} x^{8} e^{2} c^{2} B + \frac{2}{7} x^{7} e d c^{2} B + \frac{2}{7} x^{7} e^{2} c b B + \frac{1}{7} x^{7} e^{2} c^{2} A + \frac{1}{6} x^{6} d^{2} c^{2} B + \frac{2}{3} x^{6} e d c b B + \frac{1}{6} x^{6} e^{2} b^{2} B + \frac{1}{3} x^{6} e^{2} c a B + \frac{1}{3} x^{6} e d c^{2} A + \frac{1}{3} x^{6} e^{2} c b A + \frac{2}{5} x^{5} d^{2} c b B + \frac{2}{5} x^{5} e d b^{2} B + \frac{4}{5} x^{5} e d c a B + \frac{2}{5} x^{5} e^{2} b a B + \frac{1}{5} x^{5} d^{2} c^{2} A + \frac{4}{5} x^{5} e d c b A + \frac{1}{5} x^{5} e^{2} b^{2} A + \frac{2}{5} x^{5} e^{2} c a A + \frac{1}{4} x^{4} d^{2} b^{2} B + \frac{1}{2} x^{4} d^{2} c a B + x^{4} e d b a B + \frac{1}{4} x^{4} e^{2} a^{2} B + \frac{1}{2} x^{4} d^{2} c b A + \frac{1}{2} x^{4} e d b^{2} A + x^{4} e d c a A + \frac{1}{2} x^{4} e^{2} b a A + \frac{2}{3} x^{3} d^{2} b a B + \frac{2}{3} x^{3} e d a^{2} B + \frac{1}{3} x^{3} d^{2} b^{2} A + \frac{2}{3} x^{3} d^{2} c a A + \frac{4}{3} x^{3} e d b a A + \frac{1}{3} x^{3} e^{2} a^{2} A + \frac{1}{2} x^{2} d^{2} a^{2} B + x^{2} d^{2} b a A + x^{2} e d a^{2} A + x d^{2} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.280919, size = 405, normalized size = 1.33 \[ A a^{2} d^{2} x + \frac{B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac{A c^{2} e^{2}}{7} + \frac{2 B b c e^{2}}{7} + \frac{2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac{A b c e^{2}}{3} + \frac{A c^{2} d e}{3} + \frac{B a c e^{2}}{3} + \frac{B b^{2} e^{2}}{6} + \frac{2 B b c d e}{3} + \frac{B c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac{2 A a c e^{2}}{5} + \frac{A b^{2} e^{2}}{5} + \frac{4 A b c d e}{5} + \frac{A c^{2} d^{2}}{5} + \frac{2 B a b e^{2}}{5} + \frac{4 B a c d e}{5} + \frac{2 B b^{2} d e}{5} + \frac{2 B b c d^{2}}{5}\right ) + x^{4} \left (\frac{A a b e^{2}}{2} + A a c d e + \frac{A b^{2} d e}{2} + \frac{A b c d^{2}}{2} + \frac{B a^{2} e^{2}}{4} + B a b d e + \frac{B a c d^{2}}{2} + \frac{B b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac{A a^{2} e^{2}}{3} + \frac{4 A a b d e}{3} + \frac{2 A a c d^{2}}{3} + \frac{A b^{2} d^{2}}{3} + \frac{2 B a^{2} d e}{3} + \frac{2 B a b d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + A a b d^{2} + \frac{B a^{2} d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**2*(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.254171, size = 535, normalized size = 1.76 \[ \frac{1}{8} \, B c^{2} x^{8} e^{2} + \frac{2}{7} \, B c^{2} d x^{7} e + \frac{1}{6} \, B c^{2} d^{2} x^{6} + \frac{2}{7} \, B b c x^{7} e^{2} + \frac{1}{7} \, A c^{2} x^{7} e^{2} + \frac{2}{3} \, B b c d x^{6} e + \frac{1}{3} \, A c^{2} d x^{6} e + \frac{2}{5} \, B b c d^{2} x^{5} + \frac{1}{5} \, A c^{2} d^{2} x^{5} + \frac{1}{6} \, B b^{2} x^{6} e^{2} + \frac{1}{3} \, B a c x^{6} e^{2} + \frac{1}{3} \, A b c x^{6} e^{2} + \frac{2}{5} \, B b^{2} d x^{5} e + \frac{4}{5} \, B a c d x^{5} e + \frac{4}{5} \, A b c d x^{5} e + \frac{1}{4} \, B b^{2} d^{2} x^{4} + \frac{1}{2} \, B a c d^{2} x^{4} + \frac{1}{2} \, A b c d^{2} x^{4} + \frac{2}{5} \, B a b x^{5} e^{2} + \frac{1}{5} \, A b^{2} x^{5} e^{2} + \frac{2}{5} \, A a c x^{5} e^{2} + B a b d x^{4} e + \frac{1}{2} \, A b^{2} d x^{4} e + A a c d x^{4} e + \frac{2}{3} \, B a b d^{2} x^{3} + \frac{1}{3} \, A b^{2} d^{2} x^{3} + \frac{2}{3} \, A a c d^{2} x^{3} + \frac{1}{4} \, B a^{2} x^{4} e^{2} + \frac{1}{2} \, A a b x^{4} e^{2} + \frac{2}{3} \, B a^{2} d x^{3} e + \frac{4}{3} \, A a b d x^{3} e + \frac{1}{2} \, B a^{2} d^{2} x^{2} + A a b d^{2} x^{2} + \frac{1}{3} \, A a^{2} x^{3} e^{2} + A a^{2} d x^{2} e + A a^{2} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*(e*x + d)^2,x, algorithm="giac")
[Out]