Optimal. Leaf size=73 \[ \frac{(b d-a e)^3 \log (a+b x)}{b^4}+\frac{e x (b d-a e)^2}{b^3}+\frac{(d+e x)^2 (b d-a e)}{2 b^2}+\frac{(d+e x)^3}{3 b} \]
[Out]
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Rubi [A] time = 0.0708801, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{(b d-a e)^3 \log (a+b x)}{b^4}+\frac{e x (b d-a e)^2}{b^3}+\frac{(d+e x)^2 (b d-a e)}{2 b^2}+\frac{(d+e x)^3}{3 b} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^3)/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (d + e x\right )^{3}}{3 b} - \frac{\left (d + e x\right )^{2} \left (a e - b d\right )}{2 b^{2}} + \frac{\left (a e - b d\right )^{2} \int e\, dx}{b^{3}} - \frac{\left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0524379, size = 74, normalized size = 1.01 \[ \frac{b e x \left (6 a^2 e^2-3 a b e (6 d+e x)+b^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (b d-a e)^3 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^3)/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 133, normalized size = 1.8 \[{\frac{{e}^{3}{x}^{3}}{3\,b}}-{\frac{{e}^{3}{x}^{2}a}{2\,{b}^{2}}}+{\frac{3\,{e}^{2}{x}^{2}d}{2\,b}}+{\frac{{a}^{2}{e}^{3}x}{{b}^{3}}}-3\,{\frac{ad{e}^{2}x}{{b}^{2}}}+3\,{\frac{{d}^{2}ex}{b}}-{\frac{\ln \left ( bx+a \right ){a}^{3}{e}^{3}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx+a \right ){a}^{2}d{e}^{2}}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) a{d}^{2}e}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{3}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.708174, size = 154, normalized size = 2.11 \[ \frac{2 \, b^{2} e^{3} x^{3} + 3 \,{\left (3 \, b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x}{6 \, b^{3}} + \frac{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272595, size = 157, normalized size = 2.15 \[ \frac{2 \, b^{3} e^{3} x^{3} + 3 \,{\left (3 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.91058, size = 82, normalized size = 1.12 \[ \frac{e^{3} x^{3}}{3 b} - \frac{x^{2} \left (a e^{3} - 3 b d e^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} e^{3} - 3 a b d e^{2} + 3 b^{2} d^{2} e\right )}{b^{3}} - \frac{\left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.277577, size = 149, normalized size = 2.04 \[ \frac{2 \, b^{2} x^{3} e^{3} + 9 \, b^{2} d x^{2} e^{2} + 18 \, b^{2} d^{2} x e - 3 \, a b x^{2} e^{3} - 18 \, a b d x e^{2} + 6 \, a^{2} x e^{3}}{6 \, b^{3}} + \frac{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")
[Out]