Optimal. Leaf size=49 \[ -\frac {\left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac {3}{2}-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x} \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac {\sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{x \sqrt {\frac {b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^n}}{x^2} \, dx &=\frac {\sqrt {a+b x^n} \int \frac {\sqrt {1+\frac {b x^n}{a}}}{x^2} \, dx}{\sqrt {1+\frac {b x^n}{a}}}\\ &=-\frac {\sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{x \sqrt {1+\frac {b x^n}{a}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 55, normalized size = 1.12 \[ -\frac {\sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},-\frac {1}{n};1-\frac {1}{n};-\frac {b x^n}{a}\right )}{x \sqrt {\frac {b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\sqrt {b x^{n} + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \,x^{n}+a}}{x^{2}}\, dx \]
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 \[ \int \frac {\sqrt {b x^{n} + a}}{x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,x^n}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.25, size = 44, normalized size = 0.90 \[ \frac {\sqrt {a} \Gamma \left (- \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{n} \\ 1 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n x \Gamma \left (1 - \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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