Optimal. Leaf size=101 \[ \frac {b}{a (b c-a d) n \left (a+b x^n\right )}+\frac {\log (x)}{a^2 c}-\frac {b (b c-2 a d) \log \left (a+b x^n\right )}{a^2 (b c-a d)^2 n}-\frac {d^2 \log \left (c+d x^n\right )}{c (b c-a d)^2 n} \]
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Rubi [A]
time = 0.08, 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 = {457, 84}
\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 84
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^2 (c+d x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {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 = 0.15, size = 107, normalized size = 1.06 \begin {gather*} -\frac {b}{a (-b c+a d) n \left (a+b x^n\right )}+\frac {\log \left (x^n\right )}{a^2 c n}+\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 100, normalized size = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (x^{n}\right )}{a^{2} c}-\frac {b}{a \left (a d -b c \right ) \left (a +b \,x^{n}\right )}+\frac {b \left (2 a d -b c \right ) \ln \left (a +b \,x^{n}\right )}{\left (a d -b c \right )^{2} a^{2}}-\frac {d^{2} \ln \left (c +d \,x^{n}\right )}{\left (a d -b c \right )^{2} c}}{n}\) | \(100\) |
default | \(\frac {\frac {\ln \left (x^{n}\right )}{a^{2} c}-\frac {b}{a \left (a d -b c \right ) \left (a +b \,x^{n}\right )}+\frac {b \left (2 a d -b c \right ) \ln \left (a +b \,x^{n}\right )}{\left (a d -b c \right )^{2} a^{2}}-\frac {d^{2} \ln \left (c +d \,x^{n}\right )}{\left (a d -b c \right )^{2} c}}{n}\) | \(100\) |
norman | \(\frac {\frac {b^{2} {\mathrm e}^{n \ln \left (x \right )}}{n \,a^{2} \left (a d -b c \right )}+\frac {\ln \left (x \right )}{a c}+\frac {b \ln \left (x \right ) {\mathrm e}^{n \ln \left (x \right )}}{a^{2} c}}{a +b \,{\mathrm e}^{n \ln \left (x \right )}}+\frac {b \left (2 a d -b c \right ) \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a^{2} n}-\frac {d^{2} \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{c n \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(158\) |
risch | \(\frac {\ln \left (x \right ) d^{2}}{c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {2 \ln \left (x \right ) b d}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a}+\frac {\ln \left (x \right ) b^{2} c}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a^{2}}-\frac {b}{\left (a d -b c \right ) a n \left (a +b \,x^{n}\right )}-\frac {d^{2} \ln \left (x^{n}+\frac {c}{d}\right )}{c n \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 b \ln \left (x^{n}+\frac {a}{b}\right ) d}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a n}-\frac {b^{2} \ln \left (x^{n}+\frac {a}{b}\right ) c}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a^{2} n}\) | \(259\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, 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 \left (x\right )}{a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs.
\(2 (101) = 202\).
time = 0.50, 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 \left (x\right ) + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} n \log \left (x\right ) - {\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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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