Optimal. Leaf size=51 \[ -\frac {\left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac {3}{2}-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 60, 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 = {372, 371}
\begin {gather*} -\frac {\sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {\frac {b x^n}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^n}}{x^3} \, dx &=\frac {\sqrt {a+b x^n} \int \frac {\sqrt {1+\frac {b x^n}{a}}}{x^3} \, dx}{\sqrt {1+\frac {b x^n}{a}}}\\ &=-\frac {\sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {1+\frac {b x^n}{a}}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},-\frac {2}{n};1-\frac {2}{n};-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {1+\frac {b x^n}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +b \,x^{n}}}{x^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.64, size = 46, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a} \Gamma \left (- \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {2}{n} \\ 1 - \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n x^{2} \Gamma \left (1 - \frac {2}{n}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a+b\,x^n}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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