Optimal. Leaf size=105 \[ \frac {a}{2 b n (b c-a d) \left (a+b x^n\right )^2}-\frac {c}{n (b c-a d)^2 \left (a+b x^n\right )}-\frac {c d \log \left (a+b x^n\right )}{n (b c-a d)^3}+\frac {c d \log \left (c+d x^n\right )}{n (b c-a d)^3} \]
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Rubi [A] time = 0.09, antiderivative size = 105, 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, 77} \[ \frac {a}{2 b n (b c-a d) \left (a+b x^n\right )^2}-\frac {c}{n (b c-a d)^2 \left (a+b x^n\right )}-\frac {c d \log \left (a+b x^n\right )}{n (b c-a d)^3}+\frac {c d \log \left (c+d x^n\right )}{n (b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{(a+b x)^3 (c+d x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{(b c-a d) (a+b x)^3}+\frac {b c}{(b c-a d)^2 (a+b x)^2}-\frac {b c d}{(b c-a d)^3 (a+b x)}+\frac {c d^2}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a}{2 b (b c-a d) n \left (a+b x^n\right )^2}-\frac {c}{(b c-a d)^2 n \left (a+b x^n\right )}-\frac {c d \log \left (a+b x^n\right )}{(b c-a d)^3 n}+\frac {c d \log \left (c+d x^n\right )}{(b c-a d)^3 n}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 97, normalized size = 0.92 \[ \frac {\frac {a}{2 b (b c-a d) \left (a+b x^n\right )^2}-\frac {c}{(b c-a d)^2 \left (a+b x^n\right )}-\frac {c d \log \left (a+b x^n\right )}{(b c-a d)^3}+\frac {c d \log \left (c+d x^n\right )}{(b c-a d)^3}}{n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 267, normalized size = 2.54 \[ -\frac {a b^{2} c^{2} - a^{3} d^{2} + 2 \, {\left (b^{3} c^{2} - a b^{2} c d\right )} x^{n} + 2 \, {\left (b^{3} c d x^{2 \, n} + 2 \, a b^{2} c d x^{n} + a^{2} b c d\right )} \log \left (b x^{n} + a\right ) - 2 \, {\left (b^{3} c d x^{2 \, n} + 2 \, a b^{2} c d x^{n} + a^{2} b c d\right )} \log \left (d x^{n} + c\right )}{2 \, {\left ({\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} n x^{2 \, n} + 2 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} n x^{n} + {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} n\right )}} \]
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 \[ \int \frac {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3} {\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 203, normalized size = 1.93 \[ \frac {c d \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) n}-\frac {c d \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) n}+\frac {-\frac {b c \,{\mathrm e}^{n \ln \relax (x )}}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) n}+\frac {\left (-a b d -b^{2} c \right ) a}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} n}}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 243, normalized size = 2.31 \[ -\frac {c d \log \left (\frac {b x^{n} + a}{b}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} + \frac {c d \log \left (\frac {d x^{n} + c}{d}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} - \frac {2 \, b^{2} c x^{n} + a b c + a^{2} d}{2 \, {\left (a^{2} b^{3} c^{2} n - 2 \, a^{3} b^{2} c d n + a^{4} b d^{2} n + {\left (b^{5} c^{2} n - 2 \, a b^{4} c d n + a^{2} b^{3} d^{2} n\right )} x^{2 \, n} + 2 \, {\left (a b^{4} c^{2} n - 2 \, a^{2} b^{3} c d n + a^{3} b^{2} d^{2} n\right )} x^{n}\right )}} \]
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 \[ \int \frac {x^{2\,n-1}}{{\left (a+b\,x^n\right )}^3\,\left (c+d\,x^n\right )} \,d x \]
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 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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