Optimal. Leaf size=51 \[ -\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2} \]
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Rubi [A] time = 0.109883, antiderivative size = 51, 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 \[ -\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 b \left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{3}} + \frac{\int b^{2}\, dx}{d^{2}} - \frac{\left (a d - b c\right )^{2}}{d^{3} \left (c + d x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.069052, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{c+d x}+2 b (a d-b c) \log (c+d x)+b^2 d x}{d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Maple [A] time = 0.01, size = 86, normalized size = 1.7 \[{\frac{{b}^{2}x}{{d}^{2}}}-{\frac{{a}^{2}}{d \left ( dx+c \right ) }}+2\,{\frac{abc}{{d}^{2} \left ( dx+c \right ) }}-{\frac{{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+2\,{\frac{b\ln \left ( dx+c \right ) a}{{d}^{2}}}-2\,{\frac{{b}^{2}\ln \left ( dx+c \right ) c}{{d}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^4/(a*c+(a*d+b*c)*x+x^2*b*d)^2,x)
[Out]
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Maxima [A] time = 0.737027, size = 90, normalized size = 1.76 \[ \frac{b^{2} x}{d^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{4} x + c d^{3}} - \frac{2 \,{\left (b^{2} c - a b d\right )} \log \left (d x + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24142, size = 124, normalized size = 2.43 \[ \frac{b^{2} d^{2} x^{2} + b^{2} c d x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \,{\left (b^{2} c^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{d^{4} x + c d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.14145, size = 60, normalized size = 1.18 \[ \frac{b^{2} x}{d^{2}} + \frac{2 b \left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{3}} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{c d^{3} + d^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.215643, size = 88, normalized size = 1.73 \[ \frac{b^{2} x}{d^{2}} - \frac{2 \,{\left (b^{2} c - a b d\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{3}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{{\left (d x + c\right )} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="giac")
[Out]