Optimal. Leaf size=120 \[ -\frac {a^2}{2 b^2 n (b c-a d) \left (a+b x^n\right )^2}+\frac {a (2 b c-a d)}{b^2 n (b c-a d)^2 \left (a+b x^n\right )}+\frac {c^2 \log \left (a+b x^n\right )}{n (b c-a d)^3}-\frac {c^2 \log \left (c+d x^n\right )}{n (b c-a d)^3} \]
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Rubi [A] time = 0.11, 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 = {446, 88} \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 88
Rule 446
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 {\operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^3 (c+d x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {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 = 0.27, size = 112, normalized size = 0.93 \begin {gather*} \frac {-\frac {a^2}{2 b^2 (b c-a d) \left (a+b x^n\right )^2}+\frac {a (2 b c-a d)}{b^2 (b c-a d)^2 \left (a+b x^n\right )}+\frac {c^2 \log \left (a+b x^n\right )}{(b c-a d)^3}-\frac {c^2 \log \left (c+d x^n\right )}{(b c-a d)^3}}{n} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 110, normalized size = 0.92 \begin {gather*} -\frac {a \left (a^2 d-3 a b c+2 a b d x^n-4 b^2 c x^n\right )}{2 b^2 n (b c-a d)^2 \left (a+b x^n\right )^2}+\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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fricas [B] time = 0.45, 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3 \, n - 1}}{{\left (b x^{n} + a\right )}^{3} {\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 = 0.12, size = 214, normalized size = 1.78 \begin {gather*} -\frac {c^{2} \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) n}+\frac {c^{2} \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) n}+\frac {\frac {\left (-a d +2 b c \right ) a \,{\mathrm e}^{n \ln \relax (x )}}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b n}+\frac {\left (-a d +3 b c \right ) a^{2}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} n}}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, 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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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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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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