Optimal. Leaf size=49 \[ -\frac {\left (a+b x^n\right )^{5/2} \, _2F_1\left (1,\frac {5}{2}-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x} \]
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Rubi [A]
time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.20, 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 {a \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{2},-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{x \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 {\left (a+b x^n\right )^{3/2}}{x^2} \, dx &=\frac {\left (a \sqrt {a+b x^n}\right ) \int \frac {\left (1+\frac {b x^n}{a}\right )^{3/2}}{x^2} \, dx}{\sqrt {1+\frac {b x^n}{a}}}\\ &=-\frac {a \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{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 = 0.03, size = 56, normalized size = 1.14 \begin {gather*} -\frac {a \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{2},-\frac {1}{n};1-\frac {1}{n};-\frac {b x^n}{a}\right )}{x \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 \[\int \frac {\left (a +b \,x^{n}\right )^{\frac {3}{2}}}{x^{2}}\, dx\]
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 \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 = 0.89, size = 44, normalized size = 0.90 \begin {gather*} \frac {a^{\frac {3}{2}} \Gamma \left (- \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{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 )} \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 {{\left (a+b\,x^n\right )}^{3/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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