Optimal. Leaf size=142 \[ \frac{b x^3 (a d (3-2 n)-b c (3-n)) \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )}{3 a^2 n (b c-a d)^2}+\frac{d^2 x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right )}{3 c (b c-a d)^2}+\frac{b x^3}{a n (b c-a d) \left (a+b x^n\right )} \]
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Rubi [A] time = 0.224274, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {504, 597, 364} \[ \frac{b x^3 (a d (3-2 n)-b c (3-n)) \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )}{3 a^2 n (b c-a d)^2}+\frac{d^2 x^3 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right )}{3 c (b c-a d)^2}+\frac{b x^3}{a n (b c-a d) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 504
Rule 597
Rule 364
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac{b x^3}{a (b c-a d) n \left (a+b x^n\right )}-\frac{\int \frac{x^2 \left (b c (3-n)+a d n+b d (3-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 x^3}{a (b c-a d) n \left (a+b x^n\right )}-\frac{\int \left (\frac{b (-a d (3-2 n)+b c (3-n)) x^2}{(b c-a d) \left (a+b x^n\right )}+\frac{a d^2 n x^2}{(-b c+a d) \left (c+d x^n\right )}\right ) \, dx}{a (b c-a d) n}\\ &=\frac{b x^3}{a (b c-a d) n \left (a+b x^n\right )}+\frac{d^2 \int \frac{x^2}{c+d x^n} \, dx}{(b c-a d)^2}+\frac{(b (a d (3-2 n)-b c (3-n))) \int \frac{x^2}{a+b x^n} \, dx}{a (b c-a d)^2 n}\\ &=\frac{b x^3}{a (b c-a d) n \left (a+b x^n\right )}+\frac{b (a d (3-2 n)-b c (3-n)) x^3 \, _2F_1\left (1,\frac{3}{n};\frac{3+n}{n};-\frac{b x^n}{a}\right )}{3 a^2 (b c-a d)^2 n}+\frac{d^2 x^3 \, _2F_1\left (1,\frac{3}{n};\frac{3+n}{n};-\frac{d x^n}{c}\right )}{3 c (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.159323, size = 135, normalized size = 0.95 \[ \frac{x^3 \left (a \left (a d^2 n \left (a+b x^n\right ) \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{d x^n}{c}\right )+3 b c (b c-a d)\right )+b c \left (a+b x^n\right ) (a d (3-2 n)+b c (n-3)) \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )\right )}{3 a^2 c n (b c-a d)^2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ \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*} \frac{b x^{3}}{a^{2} b c n - a^{3} d n +{\left (a b^{2} c n - a^{2} b d n\right )} x^{n}} + d^{2} \int \frac{x^{2}}{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} -{\left (a b d{\left (2 \, n - 3\right )} - b^{2} c{\left (n - 3\right )}\right )} \int \frac{x^{2}}{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{x^{2}}{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{x^{2}}{\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{x^{2}}{{\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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