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 = 0.06, antiderivative size = 75, 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, 77} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{(a+b x)^2 (c+d x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{(b c-a d) (a+b x)^2}+\frac {b c}{(b c-a d)^2 (a+b x)}-\frac {c d}{(b c-a d)^2 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a}{b (b c-a d) n \left (a+b x^n\right )}+\frac {c \log \left (a+b x^n\right )}{(b c-a d)^2 n}-\frac {c \log \left (c+d x^n\right )}{(b c-a d)^2 n}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 58, normalized size = 0.77 \begin {gather*} \frac {\frac {a (b c-a d)}{b \left (a+b x^n\right )}+c \log \left (a+b x^n\right )-c \log \left (c+d x^n\right )}{n (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 75, normalized size = 1.00 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 120, normalized size = 1.60 \begin {gather*} \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} \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 )}^{2} {\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 = 0.09, size = 109, normalized size = 1.45 \begin {gather*} \frac {c \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) n}-\frac {c \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) n}+\frac {{\mathrm e}^{n \ln \relax (x )}}{\left (a d -b c \right ) \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right ) n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 121, normalized size = 1.61 \begin {gather*} \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}} \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 = -0.01 \begin {gather*} \int \frac {x^{2\,n-1}}{{\left (a+b\,x^n\right )}^2\,\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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