Optimal. Leaf size=145 \[ \frac{b (2 a d (n+1)-b c (n+2)) \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^2 n x^2 (b c-a d)^2}-\frac{d^2 \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c x^2 (b c-a d)^2}+\frac{b}{a n x^2 (b c-a d) \left (a+b x^n\right )} \]
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Rubi [A] time = 0.21373, antiderivative size = 145, 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} \[ \frac{b (2 a d (n+1)-b c (n+2)) \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^2 n x^2 (b c-a d)^2}-\frac{d^2 \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c x^2 (b c-a d)^2}+\frac{b}{a n x^2 (b c-a d) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 504
Rule 597
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac{b}{a (b c-a d) n x^2 \left (a+b x^n\right )}-\frac{\int \frac{a d n-b c (2+n)-b d (2+n) x^n}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{a (b c-a d) n}\\ &=\frac{b}{a (b c-a d) n x^2 \left (a+b x^n\right )}-\frac{\int \left (\frac{b (2 a d (1+n)-b c (2+n))}{(b c-a d) x^3 \left (a+b x^n\right )}+\frac{a d^2 n}{(-b c+a d) x^3 \left (c+d x^n\right )}\right ) \, dx}{a (b c-a d) n}\\ &=\frac{b}{a (b c-a d) n x^2 \left (a+b x^n\right )}+\frac{d^2 \int \frac{1}{x^3 \left (c+d x^n\right )} \, dx}{(b c-a d)^2}-\frac{(b (2 a d (1+n)-b c (2+n))) \int \frac{1}{x^3 \left (a+b x^n\right )} \, dx}{a (b c-a d)^2 n}\\ &=\frac{b}{a (b c-a d) n x^2 \left (a+b x^n\right )}+\frac{b (2 a d (1+n)-b c (2+n)) \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^2 (b c-a d)^2 n x^2}-\frac{d^2 \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)^2 x^2}\\ \end{align*}
Mathematica [A] time = 0.152146, size = 136, normalized size = 0.94 \[ \frac{b c \left (a+b x^n\right ) (2 a d (n+1)-b c (n+2)) \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )-a \left (a d^2 n \left (a+b x^n\right ) \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{d x^n}{c}\right )+2 b c (a d-b c)\right )}{2 a^2 c n x^2 (b c-a d)^2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \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} \int \frac{1}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} x^{n} +{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{3}}\,{d x} +{\left (b^{2} c{\left (n + 2\right )} - 2 \, a b d{\left (n + 1\right )}\right )} \int \frac{1}{{\left (a b^{3} c^{2} n - 2 \, a^{2} b^{2} c d n + a^{3} b d^{2} n\right )} x^{3} x^{n} +{\left (a^{2} b^{2} c^{2} n - 2 \, a^{3} b c d n + a^{4} d^{2} n\right )} x^{3}}\,{d x} + \frac{b}{{\left (a b^{2} c n - a^{2} b d n\right )} x^{2} x^{n} +{\left (a^{2} b c n - a^{3} d n\right )} x^{2}} \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{1}{b^{2} d x^{3} x^{3 \, n} + a^{2} c x^{3} +{\left (b^{2} c + 2 \, a b d\right )} x^{3} x^{2 \, n} +{\left (2 \, a b c + a^{2} d\right )} x^{3} 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{1}{x^{3} \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{1}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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