Optimal. Leaf size=101 \[ -\frac {b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 n (b c-a d)^2}+\frac {\log (x)}{a^2 c}-\frac {d^2 \log \left (c+d x^n\right )}{c n (b c-a d)^2}+\frac {b}{a n (b c-a d) \left (a+b x^n\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 72} \begin {gather*} -\frac {b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 n (b c-a d)^2}+\frac {\log (x)}{a^2 c}-\frac {d^2 \log \left (c+d x^n\right )}{c n (b c-a d)^2}+\frac {b}{a n (b c-a d) \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2 (c+d x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2 c x}+\frac {b^2}{a (-b c+a d) (a+b x)^2}+\frac {b^2 (-b c+2 a d)}{a^2 (-b c+a d)^2 (a+b x)}-\frac {d^3}{c (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {b}{a (b c-a d) n \left (a+b x^n\right )}+\frac {\log (x)}{a^2 c}-\frac {b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 (b c-a d)^2 n}-\frac {d^2 \log \left (c+d x^n\right )}{c (b c-a d)^2 n}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 97, normalized size = 0.96 \begin {gather*} \frac {-\frac {b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 (b c-a d)^2}+\frac {n \log (x)}{a^2 c}-\frac {d^2 \log \left (c+d x^n\right )}{c (b c-a d)^2}+\frac {b}{a (b c-a d) \left (a+b x^n\right )}}{n} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 109, normalized size = 1.08 \begin {gather*} \frac {\left (2 a b d-b^2 c\right ) \log \left (a+b x^n\right )}{a^2 n (a d-b c)^2}+\frac {\log \left (x^n\right )}{a^2 c n}-\frac {d^2 \log \left (c+d x^n\right )}{c n (b c-a d)^2}-\frac {b}{a n (a d-b c) \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 224, normalized size = 2.22 \begin {gather*} \frac {a b^{2} c^{2} - a^{2} b c d + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} n x^{n} \log \relax (x) + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} n \log \relax (x) - {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + {\left (b^{3} c^{2} - 2 \, a b^{2} c d\right )} x^{n}\right )} \log \left (b x^{n} + a\right ) - {\left (a^{2} b d^{2} x^{n} + a^{3} d^{2}\right )} \log \left (d x^{n} + c\right )}{{\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} n x^{n} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c 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 {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 131, normalized size = 1.30 \begin {gather*} \frac {2 b d \ln \left (b \,x^{n}+a \right )}{\left (a d -b c \right )^{2} a n}-\frac {b^{2} c \ln \left (b \,x^{n}+a \right )}{\left (a d -b c \right )^{2} a^{2} n}-\frac {d^{2} \ln \left (d \,x^{n}+c \right )}{\left (a d -b c \right )^{2} c n}-\frac {b}{\left (a d -b c \right ) \left (b \,x^{n}+a \right ) a n}+\frac {\ln \left (x^{n}\right )}{a^{2} c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 151, normalized size = 1.50 \begin {gather*} -\frac {d^{2} \log \left (\frac {d x^{n} + c}{d}\right )}{b^{2} c^{3} n - 2 \, a b c^{2} d n + a^{2} c d^{2} n} - \frac {{\left (b^{2} c - 2 \, a b d\right )} \log \left (\frac {b x^{n} + a}{b}\right )}{a^{2} b^{2} c^{2} n - 2 \, a^{3} b c d n + a^{4} d^{2} n} + \frac {b}{a^{2} b c n - a^{3} d n + {\left (a b^{2} c n - a^{2} b d n\right )} x^{n}} + \frac {\log \relax (x)}{a^{2} c} \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 {1}{x\,{\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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