Optimal. Leaf size=65 \[ -\frac {\left (a+b x^n\right ) \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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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} \[ -\frac {\left (a+b x^n\right ) \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx &=\frac {\left (a b+b^2 x^n\right ) \int \frac {1}{x^2 \left (a b+b^2 x^n\right )} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=-\frac {\left (a+b x^n\right ) \, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 0.78 \[ -\frac {\left (a+b x^n\right ) \, _2F_1\left (1,-\frac {1}{n};1-\frac {1}{n};-\frac {b x^n}{a}\right )}{a x \sqrt {\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{b^{2} x^{2} x^{2 \, n} + 2 \, a b x^{2} x^{n} + a^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 a b \,x^{n}+b^{2} x^{2 n}+a^{2}}\, x^{2}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}} \,d x \]
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 \[ \int \frac {1}{x^{2} \sqrt {\left (a + b x^{n}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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