Optimal. Leaf size=134 \[ -\frac{(d+e x)^4 \left (A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )\right )}{4 e^4}-\frac{(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac{(d+e x)^5 (-A c e-b B e+3 B c d)}{5 e^4}+\frac{B c (d+e x)^6}{6 e^4} \]
[Out]
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Rubi [A] time = 0.345438, antiderivative size = 133, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{(d+e x)^4 \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{4 e^4}-\frac{(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac{(d+e x)^5 (-A c e-b B e+3 B c d)}{5 e^4}+\frac{B c (d+e x)^6}{6 e^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^2*(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 49.2664, size = 128, normalized size = 0.96 \[ \frac{B c \left (d + e x\right )^{6}}{6 e^{4}} + \frac{\left (d + e x\right )^{5} \left (A c e + B b e - 3 B c d\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{4} \left (A b e^{2} - 2 A c d e + B a e^{2} - 2 B b d e + 3 B c d^{2}\right )}{4 e^{4}} + \frac{\left (d + e x\right )^{3} \left (A e - B d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{4}} \]
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),x)
[Out]
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Mathematica [A] time = 0.120838, size = 137, normalized size = 1.02 \[ \frac{1}{3} x^3 \left (a A e^2+2 a B d e+b d (2 A e+B d)+A c d^2\right )+\frac{1}{4} x^4 \left (B e (a e+2 b d)+A e (b e+2 c d)+B c d^2\right )+\frac{1}{2} d x^2 (2 a A e+a B d+A b d)+a A d^2 x+\frac{1}{5} e x^5 (A c e+b B e+2 B c d)+\frac{1}{6} B c e^2 x^6 \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^2*(a + b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.003, size = 145, normalized size = 1.1 \[{\frac{Bc{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ) c+B{e}^{2}b \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ) c+ \left ( A{e}^{2}+2\,Bde \right ) b+aB{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( Ac{d}^{2}+ \left ( 2\,Ade+B{d}^{2} \right ) b+ \left ( A{e}^{2}+2\,Bde \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ( A{d}^{2}b+ \left ( 2\,Ade+B{d}^{2} \right ) a \right ){x}^{2}}{2}}+A{d}^{2}ax \]
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),x)
[Out]
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Maxima [A] time = 0.700465, size = 178, normalized size = 1.33 \[ \frac{1}{6} \, B c e^{2} x^{6} + \frac{1}{5} \,{\left (2 \, B c d e +{\left (B b + A c\right )} e^{2}\right )} x^{5} + A a d^{2} x + \frac{1}{4} \,{\left (B c d^{2} + 2 \,{\left (B b + A c\right )} d e +{\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a e^{2} +{\left (B b + A c\right )} d^{2} + 2 \,{\left (B a + A b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a d e +{\left (B a + A b\right )} d^{2}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238342, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e^{2} c B + \frac{2}{5} x^{5} e d c B + \frac{1}{5} x^{5} e^{2} b B + \frac{1}{5} x^{5} e^{2} c A + \frac{1}{4} x^{4} d^{2} c B + \frac{1}{2} x^{4} e d b B + \frac{1}{4} x^{4} e^{2} a B + \frac{1}{2} x^{4} e d c A + \frac{1}{4} x^{4} e^{2} b A + \frac{1}{3} x^{3} d^{2} b B + \frac{2}{3} x^{3} e d a B + \frac{1}{3} x^{3} d^{2} c A + \frac{2}{3} x^{3} e d b A + \frac{1}{3} x^{3} e^{2} a A + \frac{1}{2} x^{2} d^{2} a B + \frac{1}{2} x^{2} d^{2} b A + x^{2} e d a A + x d^{2} a A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.168539, size = 172, normalized size = 1.28 \[ A a d^{2} x + \frac{B c e^{2} x^{6}}{6} + x^{5} \left (\frac{A c e^{2}}{5} + \frac{B b e^{2}}{5} + \frac{2 B c d e}{5}\right ) + x^{4} \left (\frac{A b e^{2}}{4} + \frac{A c d e}{2} + \frac{B a e^{2}}{4} + \frac{B b d e}{2} + \frac{B c d^{2}}{4}\right ) + x^{3} \left (\frac{A a e^{2}}{3} + \frac{2 A b d e}{3} + \frac{A c d^{2}}{3} + \frac{2 B a d e}{3} + \frac{B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac{A b d^{2}}{2} + \frac{B a d^{2}}{2}\right ) \]
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),x)
[Out]
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GIAC/XCAS [A] time = 0.278948, size = 231, normalized size = 1.72 \[ \frac{1}{6} \, B c x^{6} e^{2} + \frac{2}{5} \, B c d x^{5} e + \frac{1}{4} \, B c d^{2} x^{4} + \frac{1}{5} \, B b x^{5} e^{2} + \frac{1}{5} \, A c x^{5} e^{2} + \frac{1}{2} \, B b d x^{4} e + \frac{1}{2} \, A c d x^{4} e + \frac{1}{3} \, B b d^{2} x^{3} + \frac{1}{3} \, A c d^{2} x^{3} + \frac{1}{4} \, B a x^{4} e^{2} + \frac{1}{4} \, A b x^{4} e^{2} + \frac{2}{3} \, B a d x^{3} e + \frac{2}{3} \, A b d x^{3} e + \frac{1}{2} \, B a d^{2} x^{2} + \frac{1}{2} \, A b d^{2} x^{2} + \frac{1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)*(e*x + d)^2,x, algorithm="giac")
[Out]