Optimal. Leaf size=120 \[ -\frac {a^2}{2 b^2 (b c-a d) n \left (a+b x^n\right )^2}+\frac {a (2 b c-a d)}{b^2 (b c-a d)^2 n \left (a+b x^n\right )}+\frac {c^2 \log \left (a+b x^n\right )}{(b c-a d)^3 n}-\frac {c^2 \log \left (c+d x^n\right )}{(b c-a d)^3 n} \]
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Rubi [A]
time = 0.07, antiderivative size = 120, 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, 90}
\begin {gather*} -\frac {a^2}{2 b^2 n (b c-a d) \left (a+b x^n\right )^2}+\frac {a (2 b c-a d)}{b^2 n (b c-a d)^2 \left (a+b x^n\right )}+\frac {c^2 \log \left (a+b x^n\right )}{n (b c-a d)^3}-\frac {c^2 \log \left (c+d x^n\right )}{n (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{(a+b x)^3 (c+d x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2}{b (b c-a d) (a+b x)^3}+\frac {a (-2 b c+a d)}{b (b c-a d)^2 (a+b x)^2}+\frac {b c^2}{(b c-a d)^3 (a+b x)}-\frac {c^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^2}{2 b^2 (b c-a d) n \left (a+b x^n\right )^2}+\frac {a (2 b c-a d)}{b^2 (b c-a d)^2 n \left (a+b x^n\right )}+\frac {c^2 \log \left (a+b x^n\right )}{(b c-a d)^3 n}-\frac {c^2 \log \left (c+d x^n\right )}{(b c-a d)^3 n}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 94, normalized size = 0.78 \begin {gather*} \frac {\frac {a (-b c+a d) \left (-3 a b c+a^2 d-4 b^2 c x^n+2 a b d x^n\right )}{b^2 \left (a+b x^n\right )^2}+2 c^2 \log \left (a+b x^n\right )-2 c^2 \log \left (c+d x^n\right )}{2 (b c-a d)^3 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 169, normalized size = 1.41
method | result | size |
risch | \(-\frac {a \left (2 a b d \,x^{n}-4 b^{2} c \,x^{n}+a^{2} d -3 a b c \right )}{2 \left (a d -b c \right )^{2} b^{2} n \left (a +b \,x^{n}\right )^{2}}+\frac {c^{2} \ln \left (x^{n}+\frac {c}{d}\right )}{n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c^{2} \ln \left (x^{n}+\frac {a}{b}\right )}{n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(169\) |
norman | \(\frac {\frac {\left (-a d +2 b c \right ) a \,{\mathrm e}^{n \ln \left (x \right )}}{n b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a^{2} \left (-a d +3 b c \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} n}}{\left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}}+\frac {c^{2} \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c^{2} \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(214\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 262 vs.
\(2 (118) = 236\).
time = 0.30, size = 262, normalized size = 2.18 \begin {gather*} \frac {c^{2} \log \left (\frac {b x^{n} + a}{b}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} - \frac {c^{2} \log \left (\frac {d x^{n} + c}{d}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} + \frac {3 \, a^{2} b c - a^{3} d + 2 \, {\left (2 \, a b^{2} c - a^{2} b d\right )} x^{n}}{2 \, {\left (a^{2} b^{4} c^{2} n - 2 \, a^{3} b^{3} c d n + a^{4} b^{2} d^{2} n + {\left (b^{6} c^{2} n - 2 \, a b^{5} c d n + a^{2} b^{4} d^{2} n\right )} x^{2 \, n} + 2 \, {\left (a b^{5} c^{2} n - 2 \, a^{2} b^{4} c d n + a^{3} b^{3} d^{2} n\right )} x^{n}\right )}} \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. 301 vs.
\(2 (118) = 236\).
time = 2.41, size = 301, normalized size = 2.51 \begin {gather*} \frac {3 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + a^{4} d^{2} + 2 \, {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{n} + 2 \, {\left (b^{4} c^{2} x^{2 \, n} + 2 \, a b^{3} c^{2} x^{n} + a^{2} b^{2} c^{2}\right )} \log \left (b x^{n} + a\right ) - 2 \, {\left (b^{4} c^{2} x^{2 \, n} + 2 \, a b^{3} c^{2} x^{n} + a^{2} b^{2} c^{2}\right )} \log \left (d x^{n} + c\right )}{2 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} n x^{2 \, n} + 2 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} n x^{n} + {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} n\right )}} \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 )}^3\,\left (c+d\,x^n\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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