Optimal. Leaf size=115 \[ \frac{x (b c-a d) (a d (1-n)-b c (n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d^2 n}-\frac{b x (a d-b c (n+1))}{c d^2 n}-\frac{x (b c-a d) \left (a+b x^n\right )}{c d n \left (c+d x^n\right )} \]
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Rubi [A] time = 0.092344, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {413, 388, 245} \[ \frac{x (b c-a d) (a d (1-n)-b c (n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d^2 n}-\frac{b x (a d-b c (n+1))}{c d^2 n}-\frac{x (b c-a d) \left (a+b x^n\right )}{c d n \left (c+d x^n\right )} \]
Antiderivative was successfully verified.
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Rule 413
Rule 388
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (a+b x^n\right )^2}{\left (c+d x^n\right )^2} \, dx &=-\frac{(b c-a d) x \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}+\frac{\int \frac{a (b c-a d (1-n))-b (a d-b c (1+n)) x^n}{c+d x^n} \, dx}{c d n}\\ &=-\frac{b (a d-b c (1+n)) x}{c d^2 n}-\frac{(b c-a d) x \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}+\frac{((b c-a d) (a d (1-n)-b c (1+n))) \int \frac{1}{c+d x^n} \, dx}{c d^2 n}\\ &=-\frac{b (a d-b c (1+n)) x}{c d^2 n}-\frac{(b c-a d) x \left (a+b x^n\right )}{c d n \left (c+d x^n\right )}+\frac{(b c-a d) (a d (1-n)-b c (1+n)) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d^2 n}\\ \end{align*}
Mathematica [A] time = 0.113189, size = 95, normalized size = 0.83 \[ \frac{x \left (\frac{c \left (a^2 d^2-2 a b c d+b^2 c \left (c n+c+d n x^n\right )\right )}{c+d x^n}-(b c-a d) (a d (n-1)+b c (n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )\right )}{c^2 d^2 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.385, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{x}^{n} \right ) ^{2}}{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -{\left (b^{2} c^{2}{\left (n + 1\right )} - a^{2} d^{2}{\left (n - 1\right )} - 2 \, a b c d\right )} \int \frac{1}{c d^{3} n x^{n} + c^{2} d^{2} n}\,{d x} + \frac{b^{2} c d n x x^{n} +{\left (b^{2} c^{2}{\left (n + 1\right )} - 2 \, a b c d + a^{2} d^{2}\right )} x}{c d^{3} n x^{n} + c^{2} d^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{n}\right )^{2}}{\left (c + d x^{n}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{n} + a\right )}^{2}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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