Optimal. Leaf size=54 \[ \frac {c \log \left (c+d x^n\right )}{d n (b c-a d)}-\frac {a \log \left (a+b x^n\right )}{b n (b c-a d)} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {446, 72} \begin {gather*} \frac {c \log \left (c+d x^n\right )}{d n (b c-a d)}-\frac {a \log \left (a+b x^n\right )}{b n (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{(a+b x) (c+d x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{(b c-a d) (a+b x)}+\frac {c}{(b c-a d) (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a \log \left (a+b x^n\right )}{b (b c-a d) n}+\frac {c \log \left (c+d x^n\right )}{d (b c-a d) n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 0.81 \begin {gather*} -\frac {a d \log \left (a+b x^n\right )-b c \log \left (c+d x^n\right )}{b^2 c d n-a b d^2 n} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 55, normalized size = 1.02 \begin {gather*} -\frac {a \log \left (a+b x^n\right )}{b n (b c-a d)}-\frac {c \log \left (c+d x^n\right )}{d n (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 45, normalized size = 0.83 \begin {gather*} -\frac {a d \log \left (b x^{n} + a\right ) - b c \log \left (d x^{n} + c\right )}{{\left (b^{2} c d - a b d^{2}\right )} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 59, normalized size = 1.09 \begin {gather*} \frac {a \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{\left (a d -b c \right ) b n}-\frac {c \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}{\left (a d -b c \right ) d n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 60, normalized size = 1.11 \begin {gather*} -\frac {a \log \left (\frac {b x^{n} + a}{b}\right )}{b^{2} c n - a b d n} + \frac {c \log \left (\frac {d x^{n} + c}{d}\right )}{b c d n - a d^{2} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{2\,n-1}}{\left (a+b\,x^n\right )\,\left (c+d\,x^n\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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