Optimal. Leaf size=49 \[ \frac{(b c-a d)^2 \log (a+b x)}{b^3}+\frac{d x (b c-a d)}{b^2}+\frac{(c+d x)^2}{2 b} \]
[Out]
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Rubi [A] time = 0.0730761, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(b c-a d)^2 \log (a+b x)}{b^3}+\frac{d x (b c-a d)}{b^2}+\frac{(c+d x)^2}{2 b} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (c + d x\right )^{2}}{2 b} - \frac{\left (a d - b c\right ) \int d\, dx}{b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.0303833, size = 43, normalized size = 0.88 \[ \frac{b d x (-2 a d+4 b c+b d x)+2 (b c-a d)^2 \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^3,x]
[Out]
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Maple [A] time = 0.004, size = 74, normalized size = 1.5 \[{\frac{{d}^{2}{x}^{2}}{2\,b}}-{\frac{a{d}^{2}x}{{b}^{2}}}+2\,{\frac{cdx}{b}}+{\frac{\ln \left ( bx+a \right ){a}^{2}{d}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx+a \right ) cad}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)^2/(b*x+a)^3,x)
[Out]
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Maxima [A] time = 0.722328, size = 82, normalized size = 1.67 \[ \frac{b d^{2} x^{2} + 2 \,{\left (2 \, b c d - a d^{2}\right )} x}{2 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202876, size = 85, normalized size = 1.73 \[ \frac{b^{2} d^{2} x^{2} + 2 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.60214, size = 44, normalized size = 0.9 \[ \frac{d^{2} x^{2}}{2 b} - \frac{x \left (a d^{2} - 2 b c d\right )}{b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.212789, size = 81, normalized size = 1.65 \[ \frac{b d^{2} x^{2} + 4 \, b c d x - 2 \, a d^{2} x}{2 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^3,x, algorithm="giac")
[Out]