Optimal. Leaf size=75 \[ \frac{a}{b n (b c-a d) \left (a+b x^n\right )}+\frac{c \log \left (a+b x^n\right )}{n (b c-a d)^2}-\frac{c \log \left (c+d x^n\right )}{n (b c-a d)^2} \]
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Rubi [A] time = 0.18436, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a}{b n (b c-a d) \left (a+b x^n\right )}+\frac{c \log \left (a+b x^n\right )}{n (b c-a d)^2}-\frac{c \log \left (c+d x^n\right )}{n (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 2*n)/((a + b*x^n)^2*(c + d*x^n)),x]
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Rubi in Sympy [A] time = 24.6123, size = 58, normalized size = 0.77 \[ - \frac{a}{b n \left (a + b x^{n}\right ) \left (a d - b c\right )} + \frac{c \log{\left (a + b x^{n} \right )}}{n \left (a d - b c\right )^{2}} - \frac{c \log{\left (c + d x^{n} \right )}}{n \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+2*n)/(a+b*x**n)**2/(c+d*x**n),x)
[Out]
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Mathematica [A] time = 0.121836, size = 75, normalized size = 1. \[ \frac{a}{b n (b c-a d) \left (a+b x^n\right )}+\frac{c \log \left (a+b x^n\right )}{n (b c-a d)^2}-\frac{c \log \left (c+d x^n\right )}{n (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 2*n)/((a + b*x^n)^2*(c + d*x^n)),x]
[Out]
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Maple [A] time = 0.048, size = 109, normalized size = 1.5 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{ \left ( ad-bc \right ) n \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{c\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }}-{\frac{c\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+2*n)/(a+b*x^n)^2/(c+d*x^n),x)
[Out]
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Maxima [A] time = 1.47418, size = 163, normalized size = 2.17 \[ \frac{c \log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} c^{2} n - 2 \, a b c d n + a^{2} d^{2} n} - \frac{c \log \left (\frac{d x^{n} + c}{d}\right )}{b^{2} c^{2} n - 2 \, a b c d n + a^{2} d^{2} n} + \frac{a}{a b^{2} c n - a^{2} b d n +{\left (b^{3} c n - a b^{2} d n\right )} x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234974, size = 162, normalized size = 2.16 \[ \frac{a b c - a^{2} d +{\left (b^{2} c x^{n} + a b c\right )} \log \left (b x^{n} + a\right ) -{\left (b^{2} c x^{n} + a b c\right )} \log \left (d x^{n} + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} n x^{n} +{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+2*n)/(a+b*x**n)**2/(c+d*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="giac")
[Out]