Optimal. Leaf size=137 \[ \frac{(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac{d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^9}{9 e^5} \]
[Out]
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Rubi [A] time = 0.376706, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac{c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac{d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac{c^2 (d+e x)^9}{9 e^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^4*(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 43.612, size = 124, normalized size = 0.91 \[ \frac{c^{2} \left (d + e x\right )^{9}}{9 e^{5}} + \frac{c \left (d + e x\right )^{8} \left (b e - 2 c d\right )}{4 e^{5}} + \frac{d^{2} \left (d + e x\right )^{5} \left (b e - c d\right )^{2}}{5 e^{5}} - \frac{d \left (d + e x\right )^{6} \left (b e - 2 c d\right ) \left (b e - c d\right )}{3 e^{5}} + \frac{\left (d + e x\right )^{7} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**4*(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.0441727, size = 159, normalized size = 1.16 \[ \frac{1}{7} e^2 x^7 \left (b^2 e^2+8 b c d e+6 c^2 d^2\right )+\frac{2}{3} d e x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac{1}{5} d^2 x^5 \left (6 b^2 e^2+8 b c d e+c^2 d^2\right )+\frac{1}{3} b^2 d^4 x^3+\frac{1}{2} b d^3 x^4 (2 b e+c d)+\frac{1}{4} c e^3 x^8 (b e+2 c d)+\frac{1}{9} c^2 e^4 x^9 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^4*(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 166, normalized size = 1.2 \[{\frac{{e}^{4}{c}^{2}{x}^{9}}{9}}+{\frac{ \left ( 2\,{e}^{4}bc+4\,d{e}^{3}{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{4}{b}^{2}+8\,d{e}^{3}bc+6\,{d}^{2}{e}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,d{e}^{3}{b}^{2}+12\,{d}^{2}{e}^{2}bc+4\,{d}^{3}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{b}^{2}+8\,{d}^{3}ebc+{c}^{2}{d}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{d}^{3}e{b}^{2}+2\,{d}^{4}bc \right ){x}^{4}}{4}}+{\frac{{d}^{4}{b}^{2}{x}^{3}}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^4*(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.704783, size = 217, normalized size = 1.58 \[ \frac{1}{9} \, c^{2} e^{4} x^{9} + \frac{1}{3} \, b^{2} d^{4} x^{3} + \frac{1}{4} \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + b^{2} e^{4}\right )} x^{7} + \frac{2}{3} \,{\left (c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{4} + 8 \, b c d^{3} e + 6 \, b^{2} d^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (b c d^{4} + 2 \, b^{2} d^{3} e\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.191231, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e^{4} c^{2} + \frac{1}{2} x^{8} e^{3} d c^{2} + \frac{1}{4} x^{8} e^{4} c b + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} + \frac{8}{7} x^{7} e^{3} d c b + \frac{1}{7} x^{7} e^{4} b^{2} + \frac{2}{3} x^{6} e d^{3} c^{2} + 2 x^{6} e^{2} d^{2} c b + \frac{2}{3} x^{6} e^{3} d b^{2} + \frac{1}{5} x^{5} d^{4} c^{2} + \frac{8}{5} x^{5} e d^{3} c b + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} + \frac{1}{2} x^{4} d^{4} c b + x^{4} e d^{3} b^{2} + \frac{1}{3} x^{3} d^{4} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.196425, size = 178, normalized size = 1.3 \[ \frac{b^{2} d^{4} x^{3}}{3} + \frac{c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac{b c e^{4}}{4} + \frac{c^{2} d e^{3}}{2}\right ) + x^{7} \left (\frac{b^{2} e^{4}}{7} + \frac{8 b c d e^{3}}{7} + \frac{6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac{2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac{6 b^{2} d^{2} e^{2}}{5} + \frac{8 b c d^{3} e}{5} + \frac{c^{2} d^{4}}{5}\right ) + x^{4} \left (b^{2} d^{3} e + \frac{b c d^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**4*(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.205257, size = 228, normalized size = 1.66 \[ \frac{1}{9} \, c^{2} x^{9} e^{4} + \frac{1}{2} \, c^{2} d x^{8} e^{3} + \frac{6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac{2}{3} \, c^{2} d^{3} x^{6} e + \frac{1}{5} \, c^{2} d^{4} x^{5} + \frac{1}{4} \, b c x^{8} e^{4} + \frac{8}{7} \, b c d x^{7} e^{3} + 2 \, b c d^{2} x^{6} e^{2} + \frac{8}{5} \, b c d^{3} x^{5} e + \frac{1}{2} \, b c d^{4} x^{4} + \frac{1}{7} \, b^{2} x^{7} e^{4} + \frac{2}{3} \, b^{2} d x^{6} e^{3} + \frac{6}{5} \, b^{2} d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + \frac{1}{3} \, b^{2} d^{4} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(e*x + d)^4,x, algorithm="giac")
[Out]