Optimal. Leaf size=139 \[ \frac{1}{4} A b^3 d x^4+\frac{1}{7} c x^7 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+\frac{1}{6} b x^6 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+\frac{1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]
[Out]
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Rubi [A] time = 0.403917, antiderivative size = 139, 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 \[ \frac{1}{4} A b^3 d x^4+\frac{1}{7} c x^7 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+\frac{1}{6} b x^6 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+\frac{1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)*(b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 51.0924, size = 151, normalized size = 1.09 \[ \frac{A b^{3} d x^{4}}{4} + \frac{B c^{3} e x^{9}}{9} + \frac{b^{2} x^{5} \left (A b e + 3 A c d + B b d\right )}{5} + \frac{b x^{6} \left (3 A b c e + 3 A c^{2} d + B b^{2} e + 3 B b c d\right )}{6} + \frac{c^{2} x^{8} \left (A c e + 3 B b e + B c d\right )}{8} + \frac{c x^{7} \left (3 A b c e + A c^{2} d + 3 B b^{2} e + 3 B b c d\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**3,x)
[Out]
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Mathematica [A] time = 0.0770839, size = 141, normalized size = 1.01 \[ \frac{1}{4} A b^3 d x^4+\frac{1}{7} c x^7 \left (3 A b c e+A c^2 d+3 b^2 B e+3 b B c d\right )+\frac{1}{6} b x^6 \left (3 A b c e+3 A c^2 d+b^2 B e+3 b B c d\right )+\frac{1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)*(b*x + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.001, size = 138, normalized size = 1. \[{\frac{B{c}^{3}e{x}^{9}}{9}}+{\frac{ \left ( \left ( Ae+Bd \right ){c}^{3}+3\,Beb{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( Ad{c}^{3}+3\, \left ( Ae+Bd \right ) b{c}^{2}+3\,Be{b}^{2}c \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Adb{c}^{2}+3\,{b}^{2}c \left ( Ae+Bd \right ) +{b}^{3}Be \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,Ad{b}^{2}c+{b}^{3} \left ( Ae+Bd \right ) \right ){x}^{5}}{5}}+{\frac{A{b}^{3}d{x}^{4}}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)*(c*x^2+b*x)^3,x)
[Out]
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Maxima [A] time = 0.710359, size = 201, normalized size = 1.45 \[ \frac{1}{9} \, B c^{3} e x^{9} + \frac{1}{4} \, A b^{3} d x^{4} + \frac{1}{8} \,{\left (B c^{3} d +{\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac{1}{7} \,{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left (3 \,{\left (B b^{2} c + A b c^{2}\right )} d +{\left (B b^{3} + 3 \, A b^{2} c\right )} e\right )} x^{6} + \frac{1}{5} \,{\left (A b^{3} e +{\left (B b^{3} + 3 \, A b^{2} c\right )} d\right )} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3*(B*x + A)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26875, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e c^{3} B + \frac{1}{8} x^{8} d c^{3} B + \frac{3}{8} x^{8} e c^{2} b B + \frac{1}{8} x^{8} e c^{3} A + \frac{3}{7} x^{7} d c^{2} b B + \frac{3}{7} x^{7} e c b^{2} B + \frac{1}{7} x^{7} d c^{3} A + \frac{3}{7} x^{7} e c^{2} b A + \frac{1}{2} x^{6} d c b^{2} B + \frac{1}{6} x^{6} e b^{3} B + \frac{1}{2} x^{6} d c^{2} b A + \frac{1}{2} x^{6} e c b^{2} A + \frac{1}{5} x^{5} d b^{3} B + \frac{3}{5} x^{5} d c b^{2} A + \frac{1}{5} x^{5} e b^{3} A + \frac{1}{4} x^{4} d b^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3*(B*x + A)*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.176282, size = 177, normalized size = 1.27 \[ \frac{A b^{3} d x^{4}}{4} + \frac{B c^{3} e x^{9}}{9} + x^{8} \left (\frac{A c^{3} e}{8} + \frac{3 B b c^{2} e}{8} + \frac{B c^{3} d}{8}\right ) + x^{7} \left (\frac{3 A b c^{2} e}{7} + \frac{A c^{3} d}{7} + \frac{3 B b^{2} c e}{7} + \frac{3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac{A b^{2} c e}{2} + \frac{A b c^{2} d}{2} + \frac{B b^{3} e}{6} + \frac{B b^{2} c d}{2}\right ) + x^{5} \left (\frac{A b^{3} e}{5} + \frac{3 A b^{2} c d}{5} + \frac{B b^{3} d}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.277885, size = 239, normalized size = 1.72 \[ \frac{1}{9} \, B c^{3} x^{9} e + \frac{1}{8} \, B c^{3} d x^{8} + \frac{3}{8} \, B b c^{2} x^{8} e + \frac{1}{8} \, A c^{3} x^{8} e + \frac{3}{7} \, B b c^{2} d x^{7} + \frac{1}{7} \, A c^{3} d x^{7} + \frac{3}{7} \, B b^{2} c x^{7} e + \frac{3}{7} \, A b c^{2} x^{7} e + \frac{1}{2} \, B b^{2} c d x^{6} + \frac{1}{2} \, A b c^{2} d x^{6} + \frac{1}{6} \, B b^{3} x^{6} e + \frac{1}{2} \, A b^{2} c x^{6} e + \frac{1}{5} \, B b^{3} d x^{5} + \frac{3}{5} \, A b^{2} c d x^{5} + \frac{1}{5} \, A b^{3} x^{5} e + \frac{1}{4} \, A b^{3} d x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3*(B*x + A)*(e*x + d),x, algorithm="giac")
[Out]