Optimal. Leaf size=86 \[ -\frac{3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac{3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac{(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac{e^3 \log (a+b x)}{b^4} \]
[Out]
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Rubi [A] time = 0.152776, antiderivative size = 86, 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 \[ -\frac{3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac{3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac{(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac{e^3 \log (a+b x)}{b^4} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 40.6885, size = 76, normalized size = 0.88 \[ \frac{e^{3} \log{\left (a + b x \right )}}{b^{4}} + \frac{3 e^{2} \left (a e - b d\right )}{b^{4} \left (a + b x\right )} - \frac{3 e \left (a e - b d\right )^{2}}{2 b^{4} \left (a + b x\right )^{2}} + \frac{\left (a e - b d\right )^{3}}{3 b^{4} \left (a + b x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.0790848, size = 80, normalized size = 0.93 \[ \frac{6 e^3 \log (a+b x)-\frac{(b d-a e) \left (11 a^2 e^2+a b e (5 d+27 e x)+b^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )}{(a+b x)^3}}{6 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Maple [B] time = 0.01, size = 166, normalized size = 1.9 \[{\frac{{e}^{3}\ln \left ( bx+a \right ) }{{b}^{4}}}+3\,{\frac{a{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{d{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{{a}^{3}{e}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{d{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{{d}^{2}ea}{{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{3}}{3\,b \left ( bx+a \right ) ^{3}}}-{\frac{3\,{a}^{2}{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+3\,{\frac{ad{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{d}^{2}e}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
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Maxima [A] time = 0.694279, size = 192, normalized size = 2.23 \[ -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac{e^{3} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203049, size = 238, normalized size = 2.77 \[ -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.05712, size = 148, normalized size = 1.72 \[ \frac{11 a^{3} e^{3} - 6 a^{2} b d e^{2} - 3 a b^{2} d^{2} e - 2 b^{3} d^{3} + x^{2} \left (18 a b^{2} e^{3} - 18 b^{3} d e^{2}\right ) + x \left (27 a^{2} b e^{3} - 18 a b^{2} d e^{2} - 9 b^{3} d^{2} e\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac{e^{3} \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.211822, size = 153, normalized size = 1.78 \[ \frac{e^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{18 \,{\left (b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 9 \,{\left (b^{2} d^{2} e + 2 \, a b d e^{2} - 3 \, a^{2} e^{3}\right )} x + \frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3}}{b}}{6 \,{\left (b x + a\right )}^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="giac")
[Out]