Optimal. Leaf size=139 \[ \frac{\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}}}{x}-\frac{d \left (b x^{n/2}-a\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}} \left (1-\frac{b^2 x^n}{a^2}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};-\frac{1-n}{n};\frac{b^2 x^n}{a^2}\right )}{b^2 x} \]
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Rubi [A] time = 0.117721, antiderivative size = 167, normalized size of antiderivative = 1.2, number of steps used = 4, number of rules used = 4, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073, Rules used = {519, 452, 365, 364} \[ \frac{a^2 d \left (b x^{n/2}-a\right )^{\frac{1}{n}-1} \left (a+b x^{n/2}\right )^{\frac{1}{n}-1} \left (1-\frac{b^2 x^n}{a^2}\right )^{-\frac{1-n}{n}} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};-\frac{1-n}{n};\frac{b^2 x^n}{a^2}\right )}{b^2 x}-\frac{\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac{1}{n}-1} \left (a+b x^{n/2}\right )^{\frac{1}{n}-1} \left (a^2-b^2 x^n\right )}{x} \]
Antiderivative was successfully verified.
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Rule 519
Rule 452
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (-a+b x^{n/2}\right )^{\frac{1-n}{n}} \left (a+b x^{n/2}\right )^{\frac{1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx &=\left (\left (-a+b x^{n/2}\right )^{\frac{1-n}{n}} \left (a+b x^{n/2}\right )^{\frac{1-n}{n}} \left (-a^2+b^2 x^n\right )^{-\frac{1-n}{n}}\right ) \int \frac{\left (-a^2+b^2 x^n\right )^{\frac{1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx\\ &=-\frac{\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{-1+\frac{1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac{1}{n}} \left (a^2-b^2 x^n\right )}{x}+\frac{\left (d \left (-a+b x^{n/2}\right )^{\frac{1-n}{n}} \left (a+b x^{n/2}\right )^{\frac{1-n}{n}} \left (-a^2+b^2 x^n\right )^{-\frac{1-n}{n}}\right ) \int \frac{\left (-a^2+b^2 x^n\right )^{1+\frac{1-n}{n}}}{x^2} \, dx}{b^2}\\ &=-\frac{\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{-1+\frac{1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac{1}{n}} \left (a^2-b^2 x^n\right )}{x}-\frac{\left (a^2 d \left (-a+b x^{n/2}\right )^{\frac{1-n}{n}} \left (a+b x^{n/2}\right )^{\frac{1-n}{n}} \left (1-\frac{b^2 x^n}{a^2}\right )^{-\frac{1-n}{n}}\right ) \int \frac{\left (1-\frac{b^2 x^n}{a^2}\right )^{1+\frac{1-n}{n}}}{x^2} \, dx}{b^2}\\ &=-\frac{\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{-1+\frac{1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac{1}{n}} \left (a^2-b^2 x^n\right )}{x}+\frac{a^2 d \left (-a+b x^{n/2}\right )^{-1+\frac{1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac{1}{n}} \left (1-\frac{b^2 x^n}{a^2}\right )^{-\frac{1-n}{n}} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};-\frac{1-n}{n};\frac{b^2 x^n}{a^2}\right )}{b^2 x}\\ \end{align*}
Mathematica [A] time = 0.0521359, size = 124, normalized size = 0.89 \[ \frac{\left (b x^{n/2}-a\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}} \left (1-\frac{b^2 x^n}{a^2}\right )^{-1/n} \left (c (n-1) \left (1-\frac{b^2 x^n}{a^2}\right )^{\frac{1}{n}}-d x^n \, _2F_1\left (\frac{n-1}{n},\frac{n-1}{n};2-\frac{1}{n};\frac{b^2 x^n}{a^2}\right )\right )}{a^2 (n-1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.8, size = 0, normalized size = 0. \begin{align*} \int{\frac{c+d{x}^{n}}{{x}^{2}} \left ( -a+b{x}^{{\frac{n}{2}}} \right ) ^{{\frac{1-n}{n}}} \left ( a+b{x}^{{\frac{n}{2}}} \right ) ^{{\frac{1-n}{n}}}}\, 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{d x^{n} + c}{{\left (b x^{\frac{1}{2} \, n} + a\right )}^{\frac{n - 1}{n}}{\left (b x^{\frac{1}{2} \, n} - a\right )}^{\frac{n - 1}{n}} x^{2}}\,{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{d x^{n} + c}{{\left (b x^{\frac{1}{2} \, n} + a\right )}^{\frac{n - 1}{n}}{\left (b x^{\frac{1}{2} \, n} - a\right )}^{\frac{n - 1}{n}} x^{2}}, 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{d x^{n} + c}{{\left (b x^{\frac{1}{2} \, n} + a\right )}^{\frac{n - 1}{n}}{\left (b x^{\frac{1}{2} \, n} - a\right )}^{\frac{n - 1}{n}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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