Optimal. Leaf size=138 \[ -\frac{5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac{5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac{10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac{5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac{(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac{e^5 \log (a+b x)}{b^6} \]
[Out]
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Rubi [A] time = 0.286943, antiderivative size = 138, 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 \[ -\frac{5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac{5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac{10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac{5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac{(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac{e^5 \log (a+b x)}{b^6} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 72.2238, size = 126, normalized size = 0.91 \[ \frac{e^{5} \log{\left (a + b x \right )}}{b^{6}} + \frac{5 e^{4} \left (a e - b d\right )}{b^{6} \left (a + b x\right )} - \frac{5 e^{3} \left (a e - b d\right )^{2}}{b^{6} \left (a + b x\right )^{2}} + \frac{10 e^{2} \left (a e - b d\right )^{3}}{3 b^{6} \left (a + b x\right )^{3}} - \frac{5 e \left (a e - b d\right )^{4}}{4 b^{6} \left (a + b x\right )^{4}} + \frac{\left (a e - b d\right )^{5}}{5 b^{6} \left (a + b x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.175491, size = 171, normalized size = 1.24 \[ \frac{e^5 \log (a+b x)}{b^6}-\frac{(b d-a e) \left (137 a^4 e^4+a^3 b e^3 (77 d+625 e x)+a^2 b^2 e^2 \left (47 d^2+325 d e x+1100 e^2 x^2\right )+a b^3 e \left (27 d^3+175 d^2 e x+500 d e^2 x^2+900 e^3 x^3\right )+b^4 \left (12 d^4+75 d^3 e x+200 d^2 e^2 x^2+300 d e^3 x^3+300 e^4 x^4\right )\right )}{60 b^6 (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [B] time = 0.013, size = 377, normalized size = 2.7 \[{\frac{{a}^{5}{e}^{5}}{5\,{b}^{6} \left ( bx+a \right ) ^{5}}}-{\frac{d{e}^{4}{a}^{4}}{{b}^{5} \left ( bx+a \right ) ^{5}}}+2\,{\frac{{d}^{2}{e}^{3}{a}^{3}}{{b}^{4} \left ( bx+a \right ) ^{5}}}-2\,{\frac{{d}^{3}{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) ^{5}}}+{\frac{{d}^{4}ea}{{b}^{2} \left ( bx+a \right ) ^{5}}}-{\frac{{d}^{5}}{5\,b \left ( bx+a \right ) ^{5}}}-{\frac{5\,{e}^{5}{a}^{4}}{4\,{b}^{6} \left ( bx+a \right ) ^{4}}}+5\,{\frac{d{e}^{4}{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{4}}}-{\frac{15\,{d}^{2}{e}^{3}{a}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{4}}}+5\,{\frac{{d}^{3}{e}^{2}a}{{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{5\,{d}^{4}e}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}}+{\frac{{e}^{5}\ln \left ( bx+a \right ) }{{b}^{6}}}+5\,{\frac{a{e}^{5}}{{b}^{6} \left ( bx+a \right ) }}-5\,{\frac{d{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}-5\,{\frac{{a}^{2}{e}^{5}}{{b}^{6} \left ( bx+a \right ) ^{2}}}+10\,{\frac{d{e}^{4}a}{{b}^{5} \left ( bx+a \right ) ^{2}}}-5\,{\frac{{d}^{2}{e}^{3}}{{b}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{10\,{a}^{3}{e}^{5}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-10\,{\frac{{a}^{2}d{e}^{4}}{{b}^{5} \left ( bx+a \right ) ^{3}}}+10\,{\frac{{d}^{2}{e}^{3}a}{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{10\,{d}^{3}{e}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
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Maxima [A] time = 0.70019, size = 419, normalized size = 3.04 \[ -\frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \,{\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac{e^{5} \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202915, size = 502, normalized size = 3.64 \[ -\frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \,{\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \,{\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (b x + a\right )}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 35.496, size = 326, normalized size = 2.36 \[ \frac{137 a^{5} e^{5} - 60 a^{4} b d e^{4} - 30 a^{3} b^{2} d^{2} e^{3} - 20 a^{2} b^{3} d^{3} e^{2} - 15 a b^{4} d^{4} e - 12 b^{5} d^{5} + x^{4} \left (300 a b^{4} e^{5} - 300 b^{5} d e^{4}\right ) + x^{3} \left (900 a^{2} b^{3} e^{5} - 600 a b^{4} d e^{4} - 300 b^{5} d^{2} e^{3}\right ) + x^{2} \left (1100 a^{3} b^{2} e^{5} - 600 a^{2} b^{3} d e^{4} - 300 a b^{4} d^{2} e^{3} - 200 b^{5} d^{3} e^{2}\right ) + x \left (625 a^{4} b e^{5} - 300 a^{3} b^{2} d e^{4} - 150 a^{2} b^{3} d^{2} e^{3} - 100 a b^{4} d^{3} e^{2} - 75 b^{5} d^{4} e\right )}{60 a^{5} b^{6} + 300 a^{4} b^{7} x + 600 a^{3} b^{8} x^{2} + 600 a^{2} b^{9} x^{3} + 300 a b^{10} x^{4} + 60 b^{11} x^{5}} + \frac{e^{5} \log{\left (a + b x \right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.212995, size = 335, normalized size = 2.43 \[ \frac{e^{5}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{300 \,{\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} x^{4} + 300 \,{\left (b^{4} d^{2} e^{3} + 2 \, a b^{3} d e^{4} - 3 \, a^{2} b^{2} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{4} d^{3} e^{2} + 3 \, a b^{3} d^{2} e^{3} + 6 \, a^{2} b^{2} d e^{4} - 11 \, a^{3} b e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{4} d^{4} e + 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} + 12 \, a^{3} b d e^{4} - 25 \, a^{4} e^{5}\right )} x + \frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5}}{b}}{60 \,{\left (b x + a\right )}^{5} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]