Optimal. Leaf size=59 \[ \frac{2 b (b c-a d)}{d^3 (c+d x)}-\frac{(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d^3} \]
[Out]
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Rubi [A] time = 0.11379, antiderivative size = 59, 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 \[ \frac{2 b (b c-a d)}{d^3 (c+d x)}-\frac{(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 22.7649, size = 51, normalized size = 0.86 \[ \frac{b^{2} \log{\left (c + d x \right )}}{d^{3}} - \frac{2 b \left (a d - b c\right )}{d^{3} \left (c + d x\right )} - \frac{\left (a d - b c\right )^{2}}{2 d^{3} \left (c + d x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**5/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.04459, size = 48, normalized size = 0.81 \[ \frac{\frac{(b c-a d) (a d+3 b c+4 b d x)}{(c+d x)^2}+2 b^2 \log (c+d x)}{2 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^5/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]
[Out]
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Maple [A] time = 0.009, size = 92, normalized size = 1.6 \[ -2\,{\frac{ab}{{d}^{2} \left ( dx+c \right ) }}+2\,{\frac{{b}^{2}c}{{d}^{3} \left ( dx+c \right ) }}-{\frac{{a}^{2}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{abc}{{d}^{2} \left ( dx+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( dx+c \right ) }{{d}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^5/(a*c+(a*d+b*c)*x+x^2*b*d)^3,x)
[Out]
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Maxima [A] time = 0.765027, size = 108, normalized size = 1.83 \[ \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} + \frac{b^{2} \log \left (d x + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203066, size = 135, normalized size = 2.29 \[ \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right )}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.68478, size = 80, normalized size = 1.36 \[ \frac{b^{2} \log{\left (c + d x \right )}}{d^{3}} - \frac{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**5/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.217349, size = 93, normalized size = 1.58 \[ \frac{b^{2}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{3}} + \frac{4 \,{\left (b^{2} c - a b d\right )} x + \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}}{d}}{2 \,{\left (d x + c\right )}^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^5/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="giac")
[Out]