Optimal. Leaf size=133 \[ \frac{b x \left (c+d x^n\right )^{3-\frac{1}{n}}}{3 a n (b c-a d) \left (a+b x^n\right )^3}-\frac{c^2 x \left (c+d x^n\right )^{-1/n} (3 a d n+b c (1-3 n)) \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{3 a^4 n (b c-a d)} \]
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Rubi [A] time = 0.0546743, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {382, 379} \[ \frac{b x \left (c+d x^n\right )^{3-\frac{1}{n}}}{3 a n (b c-a d) \left (a+b x^n\right )^3}-\frac{c^2 x \left (c+d x^n\right )^{-1/n} (3 a d n+b c (1-3 n)) \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{3 a^4 n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 382
Rule 379
Rubi steps
\begin{align*} \int \frac{\left (c+d x^n\right )^{2-\frac{1}{n}}}{\left (a+b x^n\right )^4} \, dx &=\frac{b x \left (c+d x^n\right )^{3-\frac{1}{n}}}{3 a (b c-a d) n \left (a+b x^n\right )^3}-\frac{(b c-3 (b c-a d) n) \int \frac{\left (c+d x^n\right )^{2-\frac{1}{n}}}{\left (a+b x^n\right )^3} \, dx}{3 a (b c-a d) n}\\ &=\frac{b x \left (c+d x^n\right )^{3-\frac{1}{n}}}{3 a (b c-a d) n \left (a+b x^n\right )^3}-\frac{c^2 (b c (1-3 n)+3 a d n) x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{3 a^4 (b c-a d) n}\\ \end{align*}
Mathematica [F] time = 180.006, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.717, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{4}} \left ( c+d{x}^{n} \right ) ^{2-{n}^{-1}}}\, 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*} \int \frac{{\left (d x^{n} + c\right )}^{-\frac{1}{n} + 2}}{{\left (b x^{n} + a\right )}^{4}}\,{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 (d x^{n} + c\right )}^{\frac{2 \, n - 1}{n}}}{b^{4} x^{4 \, n} + 4 \, a b^{3} x^{3 \, n} + 6 \, a^{2} b^{2} x^{2 \, n} + 4 \, a^{3} b x^{n} + a^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (d x^{n} + c\right )}^{-\frac{1}{n} + 2}}{{\left (b x^{n} + a\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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