Optimal. Leaf size=155 \[ -\frac{e^5 \log (a+b x)}{(b d-a e)^6}+\frac{e^5 \log (d+e x)}{(b d-a e)^6}-\frac{e^4}{(a+b x) (b d-a e)^5}+\frac{e^3}{2 (a+b x)^2 (b d-a e)^4}-\frac{e^2}{3 (a+b x)^3 (b d-a e)^3}+\frac{e}{4 (a+b x)^4 (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (b d-a e)} \]
[Out]
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Rubi [A] time = 0.278365, antiderivative size = 155, 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 \[ -\frac{e^5 \log (a+b x)}{(b d-a e)^6}+\frac{e^5 \log (d+e x)}{(b d-a e)^6}-\frac{e^4}{(a+b x) (b d-a e)^5}+\frac{e^3}{2 (a+b x)^2 (b d-a e)^4}-\frac{e^2}{3 (a+b x)^3 (b d-a e)^3}+\frac{e}{4 (a+b x)^4 (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 108.882, size = 129, normalized size = 0.83 \[ - \frac{e^{5} \log{\left (a + b x \right )}}{\left (a e - b d\right )^{6}} + \frac{e^{5} \log{\left (d + e x \right )}}{\left (a e - b d\right )^{6}} + \frac{e^{4}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{e^{3}}{2 \left (a + b x\right )^{2} \left (a e - b d\right )^{4}} + \frac{e^{2}}{3 \left (a + b x\right )^{3} \left (a e - b d\right )^{3}} + \frac{e}{4 \left (a + b x\right )^{4} \left (a e - b d\right )^{2}} + \frac{1}{5 \left (a + b x\right )^{5} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.140264, size = 152, normalized size = 0.98 \[ \frac{60 e^5 (a+b x)^5 \log (d+e x)+60 e^4 (a+b x)^4 (a e-b d)+30 e^3 (a+b x)^3 (b d-a e)^2+20 e^2 (a+b x)^2 (a e-b d)^3+15 e (a+b x) (b d-a e)^4-12 (b d-a e)^5-60 e^5 (a+b x)^5 \log (a+b x)}{60 (a+b x)^5 (b d-a e)^6} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
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Maple [A] time = 0.019, size = 147, normalized size = 1. \[{\frac{1}{ \left ( 5\,ae-5\,bd \right ) \left ( bx+a \right ) ^{5}}}+{\frac{e}{4\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{4}}}+{\frac{{e}^{2}}{3\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{3}}}+{\frac{{e}^{3}}{2\, \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{2}}}+{\frac{{e}^{4}}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }}-{\frac{{e}^{5}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{6}}}+{\frac{{e}^{5}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
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Maxima [A] time = 0.730664, size = 1087, normalized size = 7.01 \[ -\frac{e^{5} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac{e^{5} \log \left (e x + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} - \frac{60 \, b^{4} e^{4} x^{4} + 12 \, b^{4} d^{4} - 63 \, a b^{3} d^{3} e + 137 \, a^{2} b^{2} d^{2} e^{2} - 163 \, a^{3} b d e^{3} + 137 \, a^{4} e^{4} - 30 \,{\left (b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 10 \,{\left (2 \, b^{4} d^{2} e^{2} - 13 \, a b^{3} d e^{3} + 47 \, a^{2} b^{2} e^{4}\right )} x^{2} - 5 \,{\left (3 \, b^{4} d^{3} e - 17 \, a b^{3} d^{2} e^{2} + 43 \, a^{2} b^{2} d e^{3} - 77 \, a^{3} b e^{4}\right )} x}{60 \,{\left (a^{5} b^{5} d^{5} - 5 \, a^{6} b^{4} d^{4} e + 10 \, a^{7} b^{3} d^{3} e^{2} - 10 \, a^{8} b^{2} d^{2} e^{3} + 5 \, a^{9} b d e^{4} - a^{10} e^{5} +{\left (b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}\right )} x^{5} + 5 \,{\left (a b^{9} d^{5} - 5 \, a^{2} b^{8} d^{4} e + 10 \, a^{3} b^{7} d^{3} e^{2} - 10 \, a^{4} b^{6} d^{2} e^{3} + 5 \, a^{5} b^{5} d e^{4} - a^{6} b^{4} e^{5}\right )} x^{4} + 10 \,{\left (a^{2} b^{8} d^{5} - 5 \, a^{3} b^{7} d^{4} e + 10 \, a^{4} b^{6} d^{3} e^{2} - 10 \, a^{5} b^{5} d^{2} e^{3} + 5 \, a^{6} b^{4} d e^{4} - a^{7} b^{3} e^{5}\right )} x^{3} + 10 \,{\left (a^{3} b^{7} d^{5} - 5 \, a^{4} b^{6} d^{4} e + 10 \, a^{5} b^{5} d^{3} e^{2} - 10 \, a^{6} b^{4} d^{2} e^{3} + 5 \, a^{7} b^{3} d e^{4} - a^{8} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (a^{4} b^{6} d^{5} - 5 \, a^{5} b^{5} d^{4} e + 10 \, a^{6} b^{4} d^{3} e^{2} - 10 \, a^{7} b^{3} d^{2} e^{3} + 5 \, a^{8} b^{2} d e^{4} - a^{9} b e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21964, size = 1246, normalized size = 8.04 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 15.7374, size = 1081, normalized size = 6.97 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.213661, size = 568, normalized size = 3.66 \[ -\frac{b e^{5}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac{e^{6}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac{12 \, b^{5} d^{5} - 75 \, a b^{4} d^{4} e + 200 \, a^{2} b^{3} d^{3} e^{2} - 300 \, a^{3} b^{2} d^{2} e^{3} + 300 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 60 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \,{\left (b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 9 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \,{\left (2 \, b^{5} d^{3} e^{2} - 15 \, a b^{4} d^{2} e^{3} + 60 \, a^{2} b^{3} d e^{4} - 47 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \,{\left (3 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 60 \, a^{2} b^{3} d^{2} e^{3} - 120 \, a^{3} b^{2} d e^{4} + 77 \, a^{4} b e^{5}\right )} x}{60 \,{\left (b d - a e\right )}^{6}{\left (b x + a\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)),x, algorithm="giac")
[Out]