Optimal. Leaf size=92 \[ -\frac{3 b^2 (d+e x)^8 (b d-a e)}{8 e^4}+\frac{3 b (d+e x)^7 (b d-a e)^2}{7 e^4}-\frac{(d+e x)^6 (b d-a e)^3}{6 e^4}+\frac{b^3 (d+e x)^9}{9 e^4} \]
[Out]
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Rubi [A] time = 0.313782, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{3 b^2 (d+e x)^8 (b d-a e)}{8 e^4}+\frac{3 b (d+e x)^7 (b d-a e)^2}{7 e^4}-\frac{(d+e x)^6 (b d-a e)^3}{6 e^4}+\frac{b^3 (d+e x)^9}{9 e^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 62.125, size = 82, normalized size = 0.89 \[ \frac{b^{3} \left (d + e x\right )^{9}}{9 e^{4}} + \frac{3 b^{2} \left (d + e x\right )^{8} \left (a e - b d\right )}{8 e^{4}} + \frac{3 b \left (d + e x\right )^{7} \left (a e - b d\right )^{2}}{7 e^{4}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{3}}{6 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [B] time = 0.071618, size = 267, normalized size = 2.9 \[ a^3 d^5 x+\frac{1}{7} b e^3 x^7 \left (3 a^2 e^2+15 a b d e+10 b^2 d^2\right )+\frac{1}{3} a d^3 x^3 \left (10 a^2 e^2+15 a b d e+3 b^2 d^2\right )+\frac{1}{2} a^2 d^4 x^2 (5 a e+3 b d)+\frac{1}{6} e^2 x^6 \left (a^3 e^3+15 a^2 b d e^2+30 a b^2 d^2 e+10 b^3 d^3\right )+d e x^5 \left (a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{4} d^2 x^4 \left (10 a^3 e^3+30 a^2 b d e^2+15 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{8} b^2 e^4 x^8 (3 a e+5 b d)+\frac{1}{9} b^3 e^5 x^9 \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [B] time = 0.004, size = 394, normalized size = 4.3 \[{\frac{{b}^{3}{e}^{5}{x}^{9}}{9}}+{\frac{ \left ( \left ( a{e}^{5}+5\,bd{e}^{4} \right ){b}^{2}+2\,{b}^{2}{e}^{5}a \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ){b}^{2}+2\, \left ( a{e}^{5}+5\,bd{e}^{4} \right ) ab+{a}^{2}b{e}^{5} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ){b}^{2}+2\, \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ) ab+ \left ( a{e}^{5}+5\,bd{e}^{4} \right ){a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ){b}^{2}+2\, \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ) ab+ \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 5\,a{d}^{4}e+b{d}^{5} \right ){b}^{2}+2\, \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ) ab+ \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( a{d}^{5}{b}^{2}+2\, \left ( 5\,a{d}^{4}e+b{d}^{5} \right ) ab+ \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{d}^{5}b+ \left ( 5\,a{d}^{4}e+b{d}^{5} \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{5}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.714869, size = 374, normalized size = 4.07 \[ \frac{1}{9} \, b^{3} e^{5} x^{9} + a^{3} d^{5} x + \frac{1}{8} \,{\left (5 \, b^{3} d e^{4} + 3 \, a b^{2} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, b^{3} d^{2} e^{3} + 15 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, b^{3} d^{3} e^{2} + 30 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} x^{6} +{\left (b^{3} d^{4} e + 6 \, a b^{2} d^{3} e^{2} + 6 \, a^{2} b d^{2} e^{3} + a^{3} d e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{5} + 15 \, a b^{2} d^{4} e + 30 \, a^{2} b d^{3} e^{2} + 10 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{2} d^{5} + 15 \, a^{2} b d^{4} e + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{5} + 5 \, a^{3} d^{4} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25527, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e^{5} b^{3} + \frac{5}{8} x^{8} e^{4} d b^{3} + \frac{3}{8} x^{8} e^{5} b^{2} a + \frac{10}{7} x^{7} e^{3} d^{2} b^{3} + \frac{15}{7} x^{7} e^{4} d b^{2} a + \frac{3}{7} x^{7} e^{5} b a^{2} + \frac{5}{3} x^{6} e^{2} d^{3} b^{3} + 5 x^{6} e^{3} d^{2} b^{2} a + \frac{5}{2} x^{6} e^{4} d b a^{2} + \frac{1}{6} x^{6} e^{5} a^{3} + x^{5} e d^{4} b^{3} + 6 x^{5} e^{2} d^{3} b^{2} a + 6 x^{5} e^{3} d^{2} b a^{2} + x^{5} e^{4} d a^{3} + \frac{1}{4} x^{4} d^{5} b^{3} + \frac{15}{4} x^{4} e d^{4} b^{2} a + \frac{15}{2} x^{4} e^{2} d^{3} b a^{2} + \frac{5}{2} x^{4} e^{3} d^{2} a^{3} + x^{3} d^{5} b^{2} a + 5 x^{3} e d^{4} b a^{2} + \frac{10}{3} x^{3} e^{2} d^{3} a^{3} + \frac{3}{2} x^{2} d^{5} b a^{2} + \frac{5}{2} x^{2} e d^{4} a^{3} + x d^{5} a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.28114, size = 308, normalized size = 3.35 \[ a^{3} d^{5} x + \frac{b^{3} e^{5} x^{9}}{9} + x^{8} \left (\frac{3 a b^{2} e^{5}}{8} + \frac{5 b^{3} d e^{4}}{8}\right ) + x^{7} \left (\frac{3 a^{2} b e^{5}}{7} + \frac{15 a b^{2} d e^{4}}{7} + \frac{10 b^{3} d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac{a^{3} e^{5}}{6} + \frac{5 a^{2} b d e^{4}}{2} + 5 a b^{2} d^{2} e^{3} + \frac{5 b^{3} d^{3} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{4} + 6 a^{2} b d^{2} e^{3} + 6 a b^{2} d^{3} e^{2} + b^{3} d^{4} e\right ) + x^{4} \left (\frac{5 a^{3} d^{2} e^{3}}{2} + \frac{15 a^{2} b d^{3} e^{2}}{2} + \frac{15 a b^{2} d^{4} e}{4} + \frac{b^{3} d^{5}}{4}\right ) + x^{3} \left (\frac{10 a^{3} d^{3} e^{2}}{3} + 5 a^{2} b d^{4} e + a b^{2} d^{5}\right ) + x^{2} \left (\frac{5 a^{3} d^{4} e}{2} + \frac{3 a^{2} b d^{5}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.276646, size = 393, normalized size = 4.27 \[ \frac{1}{9} \, b^{3} x^{9} e^{5} + \frac{5}{8} \, b^{3} d x^{8} e^{4} + \frac{10}{7} \, b^{3} d^{2} x^{7} e^{3} + \frac{5}{3} \, b^{3} d^{3} x^{6} e^{2} + b^{3} d^{4} x^{5} e + \frac{1}{4} \, b^{3} d^{5} x^{4} + \frac{3}{8} \, a b^{2} x^{8} e^{5} + \frac{15}{7} \, a b^{2} d x^{7} e^{4} + 5 \, a b^{2} d^{2} x^{6} e^{3} + 6 \, a b^{2} d^{3} x^{5} e^{2} + \frac{15}{4} \, a b^{2} d^{4} x^{4} e + a b^{2} d^{5} x^{3} + \frac{3}{7} \, a^{2} b x^{7} e^{5} + \frac{5}{2} \, a^{2} b d x^{6} e^{4} + 6 \, a^{2} b d^{2} x^{5} e^{3} + \frac{15}{2} \, a^{2} b d^{3} x^{4} e^{2} + 5 \, a^{2} b d^{4} x^{3} e + \frac{3}{2} \, a^{2} b d^{5} x^{2} + \frac{1}{6} \, a^{3} x^{6} e^{5} + a^{3} d x^{5} e^{4} + \frac{5}{2} \, a^{3} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{3} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{3} d^{4} x^{2} e + a^{3} d^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d)^5,x, algorithm="giac")
[Out]