Optimal. Leaf size=118 \[ -\frac{(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac{(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac{d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]
[Out]
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Rubi [A] time = 0.475131, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac{(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac{d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^4*(b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 47.1134, size = 114, normalized size = 0.97 \[ \frac{B c \left (d + e x\right )^{8}}{8 e^{4}} - \frac{d \left (d + e x\right )^{5} \left (A e - B d\right ) \left (b e - c d\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{7} \left (A c e + B b e - 3 B c d\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{6} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{6 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**4*(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.10362, size = 177, normalized size = 1.5 \[ \frac{1}{3} d^3 x^3 (4 A b e+A c d+b B d)+\frac{1}{4} d^2 x^4 (2 A e (3 b e+2 c d)+B d (4 b e+c d))+\frac{1}{7} e^3 x^7 (A c e+b B e+4 B c d)+\frac{1}{6} e^2 x^6 (A e (b e+4 c d)+2 B d (2 b e+3 c d))+\frac{2}{5} d e x^5 (A e (2 b e+3 c d)+B d (3 b e+2 c d))+\frac{1}{2} A b d^4 x^2+\frac{1}{8} B c e^4 x^8 \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^4*(b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.001, size = 200, normalized size = 1.7 \[{\frac{B{e}^{4}c{x}^{8}}{8}}+{\frac{ \left ( \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) c+B{e}^{4}b \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) c+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) b \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) c+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) b \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) c+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) b \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{4}c+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) b \right ){x}^{3}}{3}}+{\frac{A{d}^{4}b{x}^{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^4*(c*x^2+b*x),x)
[Out]
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Maxima [A] time = 0.688193, size = 240, normalized size = 2.03 \[ \frac{1}{8} \, B c e^{4} x^{8} + \frac{1}{2} \, A b d^{4} x^{2} + \frac{1}{7} \,{\left (4 \, B c d e^{3} +{\left (B b + A c\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, B c d^{2} e^{2} + A b e^{4} + 4 \,{\left (B b + A c\right )} d e^{3}\right )} x^{6} + \frac{2}{5} \,{\left (2 \, B c d^{3} e + 2 \, A b d e^{3} + 3 \,{\left (B b + A c\right )} d^{2} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{4} + 6 \, A b d^{2} e^{2} + 4 \,{\left (B b + A c\right )} d^{3} e\right )} x^{4} + \frac{1}{3} \,{\left (4 \, A b d^{3} e +{\left (B b + A c\right )} d^{4}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(B*x + A)*(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282567, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e^{4} c B + \frac{4}{7} x^{7} e^{3} d c B + \frac{1}{7} x^{7} e^{4} b B + \frac{1}{7} x^{7} e^{4} c A + x^{6} e^{2} d^{2} c B + \frac{2}{3} x^{6} e^{3} d b B + \frac{2}{3} x^{6} e^{3} d c A + \frac{1}{6} x^{6} e^{4} b A + \frac{4}{5} x^{5} e d^{3} c B + \frac{6}{5} x^{5} e^{2} d^{2} b B + \frac{6}{5} x^{5} e^{2} d^{2} c A + \frac{4}{5} x^{5} e^{3} d b A + \frac{1}{4} x^{4} d^{4} c B + x^{4} e d^{3} b B + x^{4} e d^{3} c A + \frac{3}{2} x^{4} e^{2} d^{2} b A + \frac{1}{3} x^{3} d^{4} b B + \frac{1}{3} x^{3} d^{4} c A + \frac{4}{3} x^{3} e d^{3} b A + \frac{1}{2} x^{2} d^{4} b A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(B*x + A)*(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.192616, size = 230, normalized size = 1.95 \[ \frac{A b d^{4} x^{2}}{2} + \frac{B c e^{4} x^{8}}{8} + x^{7} \left (\frac{A c e^{4}}{7} + \frac{B b e^{4}}{7} + \frac{4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac{A b e^{4}}{6} + \frac{2 A c d e^{3}}{3} + \frac{2 B b d e^{3}}{3} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac{4 A b d e^{3}}{5} + \frac{6 A c d^{2} e^{2}}{5} + \frac{6 B b d^{2} e^{2}}{5} + \frac{4 B c d^{3} e}{5}\right ) + x^{4} \left (\frac{3 A b d^{2} e^{2}}{2} + A c d^{3} e + B b d^{3} e + \frac{B c d^{4}}{4}\right ) + x^{3} \left (\frac{4 A b d^{3} e}{3} + \frac{A c d^{4}}{3} + \frac{B b d^{4}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**4*(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.278567, size = 284, normalized size = 2.41 \[ \frac{1}{8} \, B c x^{8} e^{4} + \frac{4}{7} \, B c d x^{7} e^{3} + B c d^{2} x^{6} e^{2} + \frac{4}{5} \, B c d^{3} x^{5} e + \frac{1}{4} \, B c d^{4} x^{4} + \frac{1}{7} \, B b x^{7} e^{4} + \frac{1}{7} \, A c x^{7} e^{4} + \frac{2}{3} \, B b d x^{6} e^{3} + \frac{2}{3} \, A c d x^{6} e^{3} + \frac{6}{5} \, B b d^{2} x^{5} e^{2} + \frac{6}{5} \, A c d^{2} x^{5} e^{2} + B b d^{3} x^{4} e + A c d^{3} x^{4} e + \frac{1}{3} \, B b d^{4} x^{3} + \frac{1}{3} \, A c d^{4} x^{3} + \frac{1}{6} \, A b x^{6} e^{4} + \frac{4}{5} \, A b d x^{5} e^{3} + \frac{3}{2} \, A b d^{2} x^{4} e^{2} + \frac{4}{3} \, A b d^{3} x^{3} e + \frac{1}{2} \, A b d^{4} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(B*x + A)*(e*x + d)^4,x, algorithm="giac")
[Out]