Optimal. Leaf size=103 \[ -\frac{b^3 x (3 b d-4 a e)}{e^4}+\frac{6 b^2 (b d-a e)^2 \log (d+e x)}{e^5}+\frac{4 b (b d-a e)^3}{e^5 (d+e x)}-\frac{(b d-a e)^4}{2 e^5 (d+e x)^2}+\frac{b^4 x^2}{2 e^3} \]
[Out]
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Rubi [A] time = 0.214793, antiderivative size = 103, 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 \[ -\frac{b^3 x (3 b d-4 a e)}{e^4}+\frac{6 b^2 (b d-a e)^2 \log (d+e x)}{e^5}+\frac{4 b (b d-a e)^3}{e^5 (d+e x)}-\frac{(b d-a e)^4}{2 e^5 (d+e x)^2}+\frac{b^4 x^2}{2 e^3} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{4} \int x\, dx}{e^{3}} + \frac{6 b^{2} \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{5}} - \frac{4 b \left (a e - b d\right )^{3}}{e^{5} \left (d + e x\right )} - \frac{\left (a e - b d\right )^{4}}{2 e^{5} \left (d + e x\right )^{2}} + \frac{\left (4 a e - 3 b d\right ) \int b^{7}\, dx}{b^{4} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.106452, size = 167, normalized size = 1.62 \[ \frac{-a^4 e^4-4 a^3 b e^3 (d+2 e x)+6 a^2 b^2 d e^2 (3 d+4 e x)+4 a b^3 e \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+12 b^2 (d+e x)^2 (b d-a e)^2 \log (d+e x)+b^4 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )}{2 e^5 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^3,x]
[Out]
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Maple [B] time = 0.012, size = 245, normalized size = 2.4 \[{\frac{{b}^{4}{x}^{2}}{2\,{e}^{3}}}+4\,{\frac{{b}^{3}xa}{{e}^{3}}}-3\,{\frac{{b}^{4}xd}{{e}^{4}}}+6\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{2}}{{e}^{3}}}-12\,{\frac{{b}^{3}\ln \left ( ex+d \right ) da}{{e}^{4}}}+6\,{\frac{{b}^{4}\ln \left ( ex+d \right ){d}^{2}}{{e}^{5}}}-{\frac{{a}^{4}}{2\,e \left ( ex+d \right ) ^{2}}}+2\,{\frac{{a}^{3}bd}{{e}^{2} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{d}^{2}{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{d}^{3}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{4}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-4\,{\frac{{a}^{3}b}{{e}^{2} \left ( ex+d \right ) }}+12\,{\frac{{a}^{2}d{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}-12\,{\frac{{d}^{2}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{d}^{3}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^3,x)
[Out]
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Maxima [A] time = 0.69675, size = 258, normalized size = 2.5 \[ \frac{7 \, b^{4} d^{4} - 20 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - a^{4} e^{4} + 8 \,{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac{b^{4} e x^{2} - 2 \,{\left (3 \, b^{4} d - 4 \, a b^{3} e\right )} x}{2 \, e^{4}} + \frac{6 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202794, size = 393, normalized size = 3.82 \[ \frac{b^{4} e^{4} x^{4} + 7 \, b^{4} d^{4} - 20 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - a^{4} e^{4} - 4 \,{\left (b^{4} d e^{3} - 2 \, a b^{3} e^{4}\right )} x^{3} -{\left (11 \, b^{4} d^{2} e^{2} - 16 \, a b^{3} d e^{3}\right )} x^{2} + 2 \,{\left (b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} + 12 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (b^{4} d^{3} e - 2 \, a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.85194, size = 184, normalized size = 1.79 \[ \frac{b^{4} x^{2}}{2 e^{3}} + \frac{6 b^{2} \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{4} e^{4} + 4 a^{3} b d e^{3} - 18 a^{2} b^{2} d^{2} e^{2} + 20 a b^{3} d^{3} e - 7 b^{4} d^{4} + x \left (8 a^{3} b e^{4} - 24 a^{2} b^{2} d e^{3} + 24 a b^{3} d^{2} e^{2} - 8 b^{4} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} + \frac{x \left (4 a b^{3} e - 3 b^{4} d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.210646, size = 236, normalized size = 2.29 \[ 6 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{4} x^{2} e^{3} - 6 \, b^{4} d x e^{2} + 8 \, a b^{3} x e^{3}\right )} e^{\left (-6\right )} + \frac{{\left (7 \, b^{4} d^{4} - 20 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - a^{4} e^{4} + 8 \,{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^3,x, algorithm="giac")
[Out]