Optimal. Leaf size=71 \[ \frac {x^n}{b d n}+\frac {a^2 \log \left (a+b x^n\right )}{b^2 (b c-a d) n}-\frac {c^2 \log \left (c+d x^n\right )}{d^2 (b c-a d) n} \]
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Rubi [A]
time = 0.05, antiderivative size = 71, 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 = {457, 84}
\begin {gather*} \frac {a^2 \log \left (a+b x^n\right )}{b^2 n (b c-a d)}-\frac {c^2 \log \left (c+d x^n\right )}{d^2 n (b c-a d)}+\frac {x^n}{b d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n}}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{(a+b x) (c+d x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{b d}+\frac {a^2}{b (b c-a d) (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {x^n}{b d n}+\frac {a^2 \log \left (a+b x^n\right )}{b^2 (b c-a d) n}-\frac {c^2 \log \left (c+d x^n\right )}{d^2 (b c-a d) n}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 66, normalized size = 0.93 \begin {gather*} \frac {a^2 d^2 \log \left (a+b x^n\right )+b \left (d (b c-a d) x^n-b c^2 \log \left (c+d x^n\right )\right )}{b^2 d^2 (b c-a d) n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 78, normalized size = 1.10
method | result | size |
norman | \(\frac {{\mathrm e}^{n \ln \left (x \right )}}{b d n}+\frac {c^{2} \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{d^{2} n \left (a d -b c \right )}-\frac {a^{2} \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{\left (a d -b c \right ) b^{2} n}\) | \(78\) |
risch | \(-\frac {\ln \left (x \right ) a}{b^{2} d}-\frac {\ln \left (x \right ) c}{b \,d^{2}}+\frac {x^{n}}{b d n}-\frac {\ln \left (x \right ) c^{2}}{d^{2} \left (a d -b c \right )}+\frac {\ln \left (x \right ) a^{2}}{\left (a d -b c \right ) b^{2}}+\frac {c^{2} \ln \left (x^{n}+\frac {c}{d}\right )}{d^{2} n \left (a d -b c \right )}-\frac {a^{2} \ln \left (x^{n}+\frac {a}{b}\right )}{\left (a d -b c \right ) b^{2} n}\) | \(137\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 81, normalized size = 1.14 \begin {gather*} \frac {a^{2} \log \left (\frac {b x^{n} + a}{b}\right )}{b^{3} c n - a b^{2} d n} - \frac {c^{2} \log \left (\frac {d x^{n} + c}{d}\right )}{b c d^{2} n - a d^{3} n} + \frac {x^{n}}{b d n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.29, size = 74, normalized size = 1.04 \begin {gather*} \frac {a^{2} d^{2} \log \left (b x^{n} + a\right ) - b^{2} c^{2} \log \left (d x^{n} + c\right ) + {\left (b^{2} c d - a b d^{2}\right )} x^{n}}{{\left (b^{3} c d^{2} - a b^{2} d^{3}\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 {x^{3\,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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