Optimal. Leaf size=65 \[ -\frac {\left (a+b x^n\right ) \, _2F_1\left (3,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a^3 x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1355, 364} \[ -\frac {\left (a+b x^n\right ) \, _2F_1\left (3,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a^3 x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^n\right )^3} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=-\frac {\left (a+b x^n\right ) \, _2F_1\left (3,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a^3 x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.82 \[ -\frac {\left (a+b x^n\right )^3 \, _2F_1\left (3,-\frac {1}{n};1-\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3 x \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{b^{4} x^{2} x^{4 \, n} + 4 \, a^{2} b^{2} x^{2} x^{2 \, n} + 4 \, a^{3} b x^{2} x^{n} + a^{4} x^{2} + 2 \, {\left (2 \, a b^{3} x^{2} x^{n} + a^{2} b^{2} x^{2}\right )} x^{2 \, n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (2 a b \,x^{n}+b^{2} x^{2 n}+a^{2}\right )^{\frac {3}{2}} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (2 \, n^{2} + 3 \, n + 1\right )} \int \frac {1}{2 \, {\left (a^{2} b n^{2} x^{2} x^{n} + a^{3} n^{2} x^{2}\right )}}\,{d x} + \frac {b {\left (2 \, n + 1\right )} x^{n} + a {\left (3 \, n + 1\right )}}{2 \, {\left (a^{2} b^{2} n^{2} x x^{2 \, n} + 2 \, a^{3} b n^{2} x x^{n} + a^{4} n^{2} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^2\,{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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