Optimal. Leaf size=107 \[ \frac{d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )}-\frac{2 b^2 c d \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^2}-\frac{b \log (a-b x)}{2 a (a d+b c)^2}+\frac{b \log (a+b x)}{2 a (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.191868, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )}-\frac{2 b^2 c d \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^2}-\frac{b \log (a-b x)}{2 a (a d+b c)^2}+\frac{b \log (a+b x)}{2 a (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a - b*x)*(a + b*x)*(c + d*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 48.9866, size = 90, normalized size = 0.84 \[ - \frac{2 b^{2} c d \log{\left (c + d x \right )}}{\left (a^{2} d^{2} - b^{2} c^{2}\right )^{2}} - \frac{d}{\left (c + d x\right ) \left (a^{2} d^{2} - b^{2} c^{2}\right )} - \frac{b \log{\left (a - b x \right )}}{2 a \left (a d + b c\right )^{2}} + \frac{b \log{\left (a + b x \right )}}{2 a \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-b*x+a)/(b*x+a)/(d*x+c)**2,x)
[Out]
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Mathematica [A] time = 0.338671, size = 102, normalized size = 0.95 \[ \frac{1}{2} \left (\frac{\frac{b \log (a+b x)}{a}-\frac{2 d \left (a^2 d^2+b^2 \left (-c^2\right )+2 b^2 c (c+d x) \log (c+d x)\right )}{(c+d x) (a d+b c)^2}}{(b c-a d)^2}-\frac{b \log (a-b x)}{a (a d+b c)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((a - b*x)*(a + b*x)*(c + d*x)^2),x]
[Out]
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Maple [A] time = 0.029, size = 108, normalized size = 1. \[ -{\frac{d}{ \left ( ad+bc \right ) \left ( ad-bc \right ) \left ( dx+c \right ) }}-2\,{\frac{{b}^{2}dc\ln \left ( dx+c \right ) }{ \left ( ad+bc \right ) ^{2} \left ( ad-bc \right ) ^{2}}}+{\frac{b\ln \left ( bx+a \right ) }{2\,a \left ( ad-bc \right ) ^{2}}}-{\frac{b\ln \left ( bx-a \right ) }{2\,a \left ( ad+bc \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-b*x+a)/(b*x+a)/(d*x+c)^2,x)
[Out]
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Maxima [A] time = 1.35573, size = 212, normalized size = 1.98 \[ -\frac{2 \, b^{2} c d \log \left (d x + c\right )}{b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}} + \frac{b \log \left (b x + a\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} - \frac{b \log \left (b x - a\right )}{2 \,{\left (a b^{2} c^{2} + 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac{d}{b^{2} c^{3} - a^{2} c d^{2} +{\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x + a)*(b*x - a)*(d*x + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.712819, size = 329, normalized size = 3.07 \[ \frac{2 \, a b^{2} c^{2} d - 2 \, a^{3} d^{3} +{\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) -{\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x - a\right ) - 4 \,{\left (a b^{2} c d^{2} x + a b^{2} c^{2} d\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{4} c^{5} - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{5} c d^{4} +{\left (a b^{4} c^{4} d - 2 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x + a)*(b*x - a)*(d*x + c)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 59.6231, size = 1232, normalized size = 11.51 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-b*x+a)/(b*x+a)/(d*x+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.232711, size = 385, normalized size = 3.6 \[ \frac{b^{2} c d{\rm ln}\left ({\left | b^{2} - \frac{2 \, b^{2} c}{d x + c} + \frac{b^{2} c^{2}}{{\left (d x + c\right )}^{2}} - \frac{a^{2} d^{2}}{{\left (d x + c\right )}^{2}} \right |}\right )}{b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}} + \frac{d^{3}}{{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}{\left (d x + c\right )}} - \frac{{\left (b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )}{\rm ln}\left (\frac{{\left | 2 \, b^{2} c d - \frac{2 \, b^{2} c^{2} d}{d x + c} + \frac{2 \, a^{2} d^{3}}{d x + c} - 2 \, d^{2}{\left | a \right |}{\left | b \right |} \right |}}{{\left | 2 \, b^{2} c d - \frac{2 \, b^{2} c^{2} d}{d x + c} + \frac{2 \, a^{2} d^{3}}{d x + c} + 2 \, d^{2}{\left | a \right |}{\left | b \right |} \right |}}\right )}{2 \,{\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} d^{2}{\left | a \right |}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x + a)*(b*x - a)*(d*x + c)^2),x, algorithm="giac")
[Out]