Optimal. Leaf size=105 \[ \frac{a}{2 b n (b c-a d) \left (a+b x^n\right )^2}-\frac{c}{n (b c-a d)^2 \left (a+b x^n\right )}-\frac{c d \log \left (a+b x^n\right )}{n (b c-a d)^3}+\frac{c d \log \left (c+d x^n\right )}{n (b c-a d)^3} \]
[Out]
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Rubi [A] time = 0.240954, antiderivative size = 105, 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 \[ \frac{a}{2 b n (b c-a d) \left (a+b x^n\right )^2}-\frac{c}{n (b c-a d)^2 \left (a+b x^n\right )}-\frac{c d \log \left (a+b x^n\right )}{n (b c-a d)^3}+\frac{c d \log \left (c+d x^n\right )}{n (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 2*n)/((a + b*x^n)^3*(c + d*x^n)),x]
[Out]
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Rubi in Sympy [A] time = 32.5667, size = 83, normalized size = 0.79 \[ - \frac{a}{2 b n \left (a + b x^{n}\right )^{2} \left (a d - b c\right )} + \frac{c d \log{\left (a + b x^{n} \right )}}{n \left (a d - b c\right )^{3}} - \frac{c d \log{\left (c + d x^{n} \right )}}{n \left (a d - b c\right )^{3}} - \frac{c}{n \left (a + b x^{n}\right ) \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+2*n)/(a+b*x**n)**3/(c+d*x**n),x)
[Out]
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Mathematica [A] time = 0.195277, size = 102, normalized size = 0.97 \[ \frac{(a d-b c) \left (a^2 d+a b c+2 b^2 c x^n\right )-2 b c d \left (a+b x^n\right )^2 \log \left (a+b x^n\right )+2 b c d \left (a+b x^n\right )^2 \log \left (c+d x^n\right )}{2 b n (b c-a d)^3 \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 2*n)/((a + b*x^n)^3*(c + d*x^n)),x]
[Out]
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Maple [A] time = 0.085, size = 203, normalized size = 1.9 \[{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( -{\frac{bc{{\rm e}^{n\ln \left ( x \right ) }}}{ \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) n}}+{\frac{a \left ( -adb-{b}^{2}c \right ) }{ \left ( 2\,{a}^{2}{d}^{2}-4\,cabd+2\,{b}^{2}{c}^{2} \right ){b}^{2}n}} \right ) }+{\frac{cd\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{3}{d}^{3}-3\,{a}^{2}c{d}^{2}b+3\,a{c}^{2}d{b}^{2}-{c}^{3}{b}^{3} \right ) }}-{\frac{cd\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n \left ({a}^{3}{d}^{3}-3\,{a}^{2}c{d}^{2}b+3\,a{c}^{2}d{b}^{2}-{c}^{3}{b}^{3} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x)
[Out]
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Maxima [A] time = 1.41719, size = 328, normalized size = 3.12 \[ -\frac{c d \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} + \frac{c d \log \left (\frac{d x^{n} + c}{d}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} - \frac{2 \, b^{2} c x^{n} + a b c + a^{2} d}{2 \,{\left (a^{2} b^{3} c^{2} n - 2 \, a^{3} b^{2} c d n + a^{4} b d^{2} n +{\left (b^{5} c^{2} n - 2 \, a b^{4} c d n + a^{2} b^{3} d^{2} n\right )} x^{2 \, n} + 2 \,{\left (a b^{4} c^{2} n - 2 \, a^{2} b^{3} c d n + a^{3} b^{2} d^{2} n\right )} x^{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/((b*x^n + a)^3*(d*x^n + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241559, size = 360, normalized size = 3.43 \[ -\frac{a b^{2} c^{2} - a^{3} d^{2} + 2 \,{\left (b^{3} c^{2} - a b^{2} c d\right )} x^{n} + 2 \,{\left (b^{3} c d x^{2 \, n} + 2 \, a b^{2} c d x^{n} + a^{2} b c d\right )} \log \left (b x^{n} + a\right ) - 2 \,{\left (b^{3} c d x^{2 \, n} + 2 \, a b^{2} c d x^{n} + a^{2} b c d\right )} \log \left (d x^{n} + c\right )}{2 \,{\left ({\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} n x^{2 \, n} + 2 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} n x^{n} +{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/((b*x^n + a)^3*(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)**3/(c+d*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/((b*x^n + a)^3*(d*x^n + c)),x, algorithm="giac")
[Out]