Optimal. Leaf size=193 \[ \frac {d (b c+a d) x}{a c (b c-a d)^2 n \left (c+d x^n\right )}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac {b^2 (a d (1-3 n)-b (c-c n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^3 n}-\frac {d^2 (b c (1-3 n)-a d (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 (b c-a d)^3 n} \]
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Rubi [A]
time = 0.20, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {425, 541, 536,
251} \begin {gather*} \frac {b^2 x (a d (1-3 n)-b (c-c n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 n (b c-a d)^3}-\frac {d^2 x (b c (1-3 n)-a d (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 n (b c-a d)^3}+\frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac {d x (a d+b c)}{a c n (b c-a d)^2 \left (c+d x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 425
Rule 536
Rule 541
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx &=\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )}-\frac {\int \frac {a d n+b (c-c n)+b d (1-2 n) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )^2} \, dx}{a (b c-a d) n}\\ &=\frac {d (b c+a d) x}{a c (b c-a d)^2 n \left (c+d x^n\right )}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )}-\frac {\int \frac {n \left (b^2 c^2 (1-n)+a^2 d^2 (1-n)+2 a b c d n\right )+b d (b c+a d) (1-n) n x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{a c (b c-a d)^2 n^2}\\ &=\frac {d (b c+a d) x}{a c (b c-a d)^2 n \left (c+d x^n\right )}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac {\left (d^2 (a d (1-n)-b (c-3 c n))\right ) \int \frac {1}{c+d x^n} \, dx}{c (b c-a d)^3 n}+\frac {\left (b^2 (a d (1-3 n)-b (c-c n))\right ) \int \frac {1}{a+b x^n} \, dx}{a (b c-a d)^3 n}\\ &=\frac {d (b c+a d) x}{a c (b c-a d)^2 n \left (c+d x^n\right )}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac {b^2 (a d (1-3 n)-b (c-c n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^3 n}-\frac {d^2 (b c (1-3 n)-a d (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 (b c-a d)^3 n}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 147, normalized size = 0.76 \begin {gather*} \frac {x \left (\frac {b^2 (b c-a d)}{a \left (a+b x^n\right )}+\frac {d^2 (b c-a d)}{c \left (c+d x^n\right )}+\frac {b^2 (a d (1-3 n)+b c (-1+n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2}+\frac {d^2 (-a d (-1+n)+b c (-1+3 n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2}\right )}{(b c-a d)^3 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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}{{\left (a+b\,x^n\right )}^2\,{\left (c+d\,x^n\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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