Optimal. Leaf size=119 \[ \frac{2 e^3 (a+b x)^{10} (b d-a e)}{5 b^5}+\frac{2 e^2 (a+b x)^9 (b d-a e)^2}{3 b^5}+\frac{e (a+b x)^8 (b d-a e)^3}{2 b^5}+\frac{(a+b x)^7 (b d-a e)^4}{7 b^5}+\frac{e^4 (a+b x)^{11}}{11 b^5} \]
[Out]
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Rubi [A] time = 0.54603, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 e^3 (a+b x)^{10} (b d-a e)}{5 b^5}+\frac{2 e^2 (a+b x)^9 (b d-a e)^2}{3 b^5}+\frac{e (a+b x)^8 (b d-a e)^3}{2 b^5}+\frac{(a+b x)^7 (b d-a e)^4}{7 b^5}+\frac{e^4 (a+b x)^{11}}{11 b^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 71.698, size = 105, normalized size = 0.88 \[ \frac{e^{4} \left (a + b x\right )^{11}}{11 b^{5}} - \frac{2 e^{3} \left (a + b x\right )^{10} \left (a e - b d\right )}{5 b^{5}} + \frac{2 e^{2} \left (a + b x\right )^{9} \left (a e - b d\right )^{2}}{3 b^{5}} - \frac{e \left (a + b x\right )^{8} \left (a e - b d\right )^{3}}{2 b^{5}} + \frac{\left (a + b x\right )^{7} \left (a e - b d\right )^{4}}{7 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [B] time = 0.113734, size = 398, normalized size = 3.34 \[ a^6 d^4 x+a^5 d^3 x^2 (2 a e+3 b d)+\frac{1}{3} b^4 e^2 x^9 \left (5 a^2 e^2+8 a b d e+2 b^2 d^2\right )+a^4 d^2 x^3 \left (2 a^2 e^2+8 a b d e+5 b^2 d^2\right )+\frac{1}{2} b^3 e x^8 \left (5 a^3 e^3+15 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )+a^3 d x^4 \left (a^3 e^3+9 a^2 b d e^2+15 a b^2 d^2 e+5 b^3 d^3\right )+\frac{1}{7} b^2 x^7 \left (15 a^4 e^4+80 a^3 b d e^3+90 a^2 b^2 d^2 e^2+24 a b^3 d^3 e+b^4 d^4\right )+a b x^6 \left (a^4 e^4+10 a^3 b d e^3+20 a^2 b^2 d^2 e^2+10 a b^3 d^3 e+b^4 d^4\right )+\frac{1}{5} a^2 x^5 \left (a^4 e^4+24 a^3 b d e^3+90 a^2 b^2 d^2 e^2+80 a b^3 d^3 e+15 b^4 d^4\right )+\frac{1}{5} b^5 e^3 x^{10} (3 a e+2 b d)+\frac{1}{11} b^6 e^4 x^{11} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [B] time = 0.002, size = 427, normalized size = 3.6 \[{\frac{{e}^{4}{b}^{6}{x}^{11}}{11}}+{\frac{ \left ( 6\,{e}^{4}a{b}^{5}+4\,d{e}^{3}{b}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 15\,{e}^{4}{a}^{2}{b}^{4}+24\,d{e}^{3}a{b}^{5}+6\,{d}^{2}{e}^{2}{b}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ( 20\,{e}^{4}{a}^{3}{b}^{3}+60\,d{e}^{3}{a}^{2}{b}^{4}+36\,{d}^{2}{e}^{2}a{b}^{5}+4\,{d}^{3}e{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 15\,{e}^{4}{b}^{2}{a}^{4}+80\,d{e}^{3}{a}^{3}{b}^{3}+90\,{d}^{2}{e}^{2}{a}^{2}{b}^{4}+24\,{d}^{3}ea{b}^{5}+{d}^{4}{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{e}^{4}{a}^{5}b+60\,d{e}^{3}{b}^{2}{a}^{4}+120\,{d}^{2}{e}^{2}{a}^{3}{b}^{3}+60\,{d}^{3}e{a}^{2}{b}^{4}+6\,{d}^{4}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ({e}^{4}{a}^{6}+24\,d{e}^{3}{a}^{5}b+90\,{d}^{2}{e}^{2}{b}^{2}{a}^{4}+80\,{d}^{3}e{a}^{3}{b}^{3}+15\,{d}^{4}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{6}+36\,{d}^{2}{e}^{2}{a}^{5}b+60\,{d}^{3}e{b}^{2}{a}^{4}+20\,{d}^{4}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{6}+24\,{d}^{3}e{a}^{5}b+15\,{d}^{4}{b}^{2}{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{d}^{3}e{a}^{6}+6\,{d}^{4}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{4}{a}^{6}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
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Maxima [A] time = 0.694287, size = 564, normalized size = 4.74 \[ \frac{1}{11} \, b^{6} e^{4} x^{11} + a^{6} d^{4} x + \frac{1}{5} \,{\left (2 \, b^{6} d e^{3} + 3 \, a b^{5} e^{4}\right )} x^{10} + \frac{1}{3} \,{\left (2 \, b^{6} d^{2} e^{2} + 8 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{9} + \frac{1}{2} \,{\left (b^{6} d^{3} e + 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} + 5 \, a^{3} b^{3} e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{4} + 24 \, a b^{5} d^{3} e + 90 \, a^{2} b^{4} d^{2} e^{2} + 80 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x^{7} +{\left (a b^{5} d^{4} + 10 \, a^{2} b^{4} d^{3} e + 20 \, a^{3} b^{3} d^{2} e^{2} + 10 \, a^{4} b^{2} d e^{3} + a^{5} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (15 \, a^{2} b^{4} d^{4} + 80 \, a^{3} b^{3} d^{3} e + 90 \, a^{4} b^{2} d^{2} e^{2} + 24 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} x^{5} +{\left (5 \, a^{3} b^{3} d^{4} + 15 \, a^{4} b^{2} d^{3} e + 9 \, a^{5} b d^{2} e^{2} + a^{6} d e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{4} + 8 \, a^{5} b d^{3} e + 2 \, a^{6} d^{2} e^{2}\right )} x^{3} +{\left (3 \, a^{5} b d^{4} + 2 \, a^{6} d^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.18011, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} e^{4} b^{6} + \frac{2}{5} x^{10} e^{3} d b^{6} + \frac{3}{5} x^{10} e^{4} b^{5} a + \frac{2}{3} x^{9} e^{2} d^{2} b^{6} + \frac{8}{3} x^{9} e^{3} d b^{5} a + \frac{5}{3} x^{9} e^{4} b^{4} a^{2} + \frac{1}{2} x^{8} e d^{3} b^{6} + \frac{9}{2} x^{8} e^{2} d^{2} b^{5} a + \frac{15}{2} x^{8} e^{3} d b^{4} a^{2} + \frac{5}{2} x^{8} e^{4} b^{3} a^{3} + \frac{1}{7} x^{7} d^{4} b^{6} + \frac{24}{7} x^{7} e d^{3} b^{5} a + \frac{90}{7} x^{7} e^{2} d^{2} b^{4} a^{2} + \frac{80}{7} x^{7} e^{3} d b^{3} a^{3} + \frac{15}{7} x^{7} e^{4} b^{2} a^{4} + x^{6} d^{4} b^{5} a + 10 x^{6} e d^{3} b^{4} a^{2} + 20 x^{6} e^{2} d^{2} b^{3} a^{3} + 10 x^{6} e^{3} d b^{2} a^{4} + x^{6} e^{4} b a^{5} + 3 x^{5} d^{4} b^{4} a^{2} + 16 x^{5} e d^{3} b^{3} a^{3} + 18 x^{5} e^{2} d^{2} b^{2} a^{4} + \frac{24}{5} x^{5} e^{3} d b a^{5} + \frac{1}{5} x^{5} e^{4} a^{6} + 5 x^{4} d^{4} b^{3} a^{3} + 15 x^{4} e d^{3} b^{2} a^{4} + 9 x^{4} e^{2} d^{2} b a^{5} + x^{4} e^{3} d a^{6} + 5 x^{3} d^{4} b^{2} a^{4} + 8 x^{3} e d^{3} b a^{5} + 2 x^{3} e^{2} d^{2} a^{6} + 3 x^{2} d^{4} b a^{5} + 2 x^{2} e d^{3} a^{6} + x d^{4} a^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.316312, size = 462, normalized size = 3.88 \[ a^{6} d^{4} x + \frac{b^{6} e^{4} x^{11}}{11} + x^{10} \left (\frac{3 a b^{5} e^{4}}{5} + \frac{2 b^{6} d e^{3}}{5}\right ) + x^{9} \left (\frac{5 a^{2} b^{4} e^{4}}{3} + \frac{8 a b^{5} d e^{3}}{3} + \frac{2 b^{6} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac{5 a^{3} b^{3} e^{4}}{2} + \frac{15 a^{2} b^{4} d e^{3}}{2} + \frac{9 a b^{5} d^{2} e^{2}}{2} + \frac{b^{6} d^{3} e}{2}\right ) + x^{7} \left (\frac{15 a^{4} b^{2} e^{4}}{7} + \frac{80 a^{3} b^{3} d e^{3}}{7} + \frac{90 a^{2} b^{4} d^{2} e^{2}}{7} + \frac{24 a b^{5} d^{3} e}{7} + \frac{b^{6} d^{4}}{7}\right ) + x^{6} \left (a^{5} b e^{4} + 10 a^{4} b^{2} d e^{3} + 20 a^{3} b^{3} d^{2} e^{2} + 10 a^{2} b^{4} d^{3} e + a b^{5} d^{4}\right ) + x^{5} \left (\frac{a^{6} e^{4}}{5} + \frac{24 a^{5} b d e^{3}}{5} + 18 a^{4} b^{2} d^{2} e^{2} + 16 a^{3} b^{3} d^{3} e + 3 a^{2} b^{4} d^{4}\right ) + x^{4} \left (a^{6} d e^{3} + 9 a^{5} b d^{2} e^{2} + 15 a^{4} b^{2} d^{3} e + 5 a^{3} b^{3} d^{4}\right ) + x^{3} \left (2 a^{6} d^{2} e^{2} + 8 a^{5} b d^{3} e + 5 a^{4} b^{2} d^{4}\right ) + x^{2} \left (2 a^{6} d^{3} e + 3 a^{5} b d^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.209041, size = 616, normalized size = 5.18 \[ \frac{1}{11} \, b^{6} x^{11} e^{4} + \frac{2}{5} \, b^{6} d x^{10} e^{3} + \frac{2}{3} \, b^{6} d^{2} x^{9} e^{2} + \frac{1}{2} \, b^{6} d^{3} x^{8} e + \frac{1}{7} \, b^{6} d^{4} x^{7} + \frac{3}{5} \, a b^{5} x^{10} e^{4} + \frac{8}{3} \, a b^{5} d x^{9} e^{3} + \frac{9}{2} \, a b^{5} d^{2} x^{8} e^{2} + \frac{24}{7} \, a b^{5} d^{3} x^{7} e + a b^{5} d^{4} x^{6} + \frac{5}{3} \, a^{2} b^{4} x^{9} e^{4} + \frac{15}{2} \, a^{2} b^{4} d x^{8} e^{3} + \frac{90}{7} \, a^{2} b^{4} d^{2} x^{7} e^{2} + 10 \, a^{2} b^{4} d^{3} x^{6} e + 3 \, a^{2} b^{4} d^{4} x^{5} + \frac{5}{2} \, a^{3} b^{3} x^{8} e^{4} + \frac{80}{7} \, a^{3} b^{3} d x^{7} e^{3} + 20 \, a^{3} b^{3} d^{2} x^{6} e^{2} + 16 \, a^{3} b^{3} d^{3} x^{5} e + 5 \, a^{3} b^{3} d^{4} x^{4} + \frac{15}{7} \, a^{4} b^{2} x^{7} e^{4} + 10 \, a^{4} b^{2} d x^{6} e^{3} + 18 \, a^{4} b^{2} d^{2} x^{5} e^{2} + 15 \, a^{4} b^{2} d^{3} x^{4} e + 5 \, a^{4} b^{2} d^{4} x^{3} + a^{5} b x^{6} e^{4} + \frac{24}{5} \, a^{5} b d x^{5} e^{3} + 9 \, a^{5} b d^{2} x^{4} e^{2} + 8 \, a^{5} b d^{3} x^{3} e + 3 \, a^{5} b d^{4} x^{2} + \frac{1}{5} \, a^{6} x^{5} e^{4} + a^{6} d x^{4} e^{3} + 2 \, a^{6} d^{2} x^{3} e^{2} + 2 \, a^{6} d^{3} x^{2} e + a^{6} d^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^4,x, algorithm="giac")
[Out]