Optimal. Leaf size=77 \[ \frac{a (b c-a d)^2}{b^4 (a+b x)}+\frac{(b c-3 a d) (b c-a d) \log (a+b x)}{b^4}+\frac{2 d x (b c-a d)}{b^3}+\frac{d^2 x^2}{2 b^2} \]
[Out]
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Rubi [A] time = 0.15119, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a (b c-a d)^2}{b^4 (a+b x)}+\frac{(b c-3 a d) (b c-a d) \log (a+b x)}{b^4}+\frac{2 d x (b c-a d)}{b^3}+\frac{d^2 x^2}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x)^2)/(a + b*x)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a \left (a d - b c\right )^{2}}{b^{4} \left (a + b x\right )} + \frac{d^{2} \int x\, dx}{b^{2}} - \frac{2 d x \left (a d - b c\right )}{b^{3}} + \frac{\left (a d - b c\right ) \left (3 a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x+c)**2/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0847245, size = 81, normalized size = 1.05 \[ \frac{2 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)+\frac{2 a (b c-a d)^2}{a+b x}+4 b d x (b c-a d)+b^2 d^2 x^2}{2 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x)^2)/(a + b*x)^2,x]
[Out]
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Maple [A] time = 0.011, size = 124, normalized size = 1.6 \[{\frac{{d}^{2}{x}^{2}}{2\,{b}^{2}}}-2\,{\frac{{d}^{2}ax}{{b}^{3}}}+2\,{\frac{dxc}{{b}^{2}}}+3\,{\frac{\ln \left ( bx+a \right ){a}^{2}{d}^{2}}{{b}^{4}}}-4\,{\frac{\ln \left ( bx+a \right ) acd}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ){c}^{2}}{{b}^{2}}}+{\frac{{a}^{3}{d}^{2}}{ \left ( bx+a \right ){b}^{4}}}-2\,{\frac{{a}^{2}cd}{ \left ( bx+a \right ){b}^{3}}}+{\frac{a{c}^{2}}{ \left ( bx+a \right ){b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x+c)^2/(b*x+a)^2,x)
[Out]
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Maxima [A] time = 1.34898, size = 134, normalized size = 1.74 \[ \frac{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}}{b^{5} x + a b^{4}} + \frac{b d^{2} x^{2} + 4 \,{\left (b c d - a d^{2}\right )} x}{2 \, b^{3}} + \frac{{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*x/(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206083, size = 205, normalized size = 2.66 \[ \frac{b^{3} d^{2} x^{3} + 2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} +{\left (4 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{2} + 4 \,{\left (a b^{2} c d - a^{2} b d^{2}\right )} x + 2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} b c d + 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*x/(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.05784, size = 90, normalized size = 1.17 \[ \frac{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}}{a b^{4} + b^{5} x} + \frac{d^{2} x^{2}}{2 b^{2}} - \frac{x \left (2 a d^{2} - 2 b c d\right )}{b^{3}} + \frac{\left (a d - b c\right ) \left (3 a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x+c)**2/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.266182, size = 201, normalized size = 2.61 \[ \frac{\frac{{\left (d^{2} + \frac{2 \,{\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )}}{{\left (b x + a\right )} b}\right )}{\left (b x + a\right )}^{2}}{b^{3}} - \frac{2 \,{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{2 \,{\left (\frac{a b^{4} c^{2}}{b x + a} - \frac{2 \, a^{2} b^{3} c d}{b x + a} + \frac{a^{3} b^{2} d^{2}}{b x + a}\right )}}{b^{5}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*x/(b*x + a)^2,x, algorithm="giac")
[Out]