Optimal. Leaf size=175 \[ \frac{b (e x)^{m+1} (a d (m-2 n+1)-b c (m-n+1)) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 e (m+1) n (b c-a d)^2}+\frac{d^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)^2}+\frac{b (e x)^{m+1}}{a e n (b c-a d) \left (a+b x^n\right )} \]
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Rubi [A] time = 0.288443, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {504, 597, 364} \[ \frac{b (e x)^{m+1} (a d (m-2 n+1)-b c (m-n+1)) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 e (m+1) n (b c-a d)^2}+\frac{d^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)^2}+\frac{b (e x)^{m+1}}{a e n (b c-a d) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 504
Rule 597
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac{b (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}-\frac{\int \frac{(e x)^m \left (b c (1+m-n)+a d n+b d (1+m-n) x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{a (b c-a d) n}\\ &=\frac{b (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}-\frac{\int \left (\frac{b (-a d (1+m-2 n)+b c (1+m-n)) (e x)^m}{(b c-a d) \left (a+b x^n\right )}+\frac{a d^2 n (e x)^m}{(-b c+a d) \left (c+d x^n\right )}\right ) \, dx}{a (b c-a d) n}\\ &=\frac{b (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}+\frac{d^2 \int \frac{(e x)^m}{c+d x^n} \, dx}{(b c-a d)^2}+\frac{(b (a d (1+m-2 n)-b c (1+m-n))) \int \frac{(e x)^m}{a+b x^n} \, dx}{a (b c-a d)^2 n}\\ &=\frac{b (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right )}+\frac{b (a d (1+m-2 n)-b c (1+m-n)) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{a^2 (b c-a d)^2 e (1+m) n}+\frac{d^2 (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{d x^n}{c}\right )}{c (b c-a d)^2 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.185273, size = 141, normalized size = 0.81 \[ \frac{x (e x)^m \left (\frac{b^2 c-a b d}{a^2 n+a b n x^n}+\frac{b (a d (m-2 n+1)-b c (m-n+1)) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a^2 (m+1) n}+\frac{d^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c m+c}\right )}{(b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.081, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) }}\, 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*} d^{2} e^{m} \int \frac{x^{m}}{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{n}}\,{d x} + \frac{b e^{m} x x^{m}}{a^{2} b c n - a^{3} d n +{\left (a b^{2} c n - a^{2} b d n\right )} x^{n}} -{\left (b^{2} c e^{m}{\left (m - n + 1\right )} - a b d e^{m}{\left (m - 2 \, n + 1\right )}\right )} \int \frac{x^{m}}{a^{2} b^{2} c^{2} n - 2 \, a^{3} b c d n + a^{4} d^{2} n +{\left (a b^{3} c^{2} n - 2 \, a^{2} b^{2} c d n + a^{3} b d^{2} n\right )} x^{n}}\,{d x} \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{\left (e x\right )^{m}}{b^{2} d x^{3 \, n} + a^{2} c +{\left (b^{2} c + 2 \, a b d\right )} x^{2 \, n} +{\left (2 \, a b c + a^{2} d\right )} x^{n}}, 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 (e x\right )^{m}}{\left (a + b x^{n}\right )^{2} \left (c + d x^{n}\right )}\, 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 (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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