Optimal. Leaf size=39 \[ \frac {x \, _2F_1\left (1,-\frac {1}{2}+\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a \sqrt {a+b x^n}} \]
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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.31, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {252, 251}
\begin {gather*} \frac {x \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {3}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^n\right )^{3/2}} \, dx &=\frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {1}{\left (1+\frac {b x^n}{a}\right )^{3/2}} \, dx}{a \sqrt {a+b x^n}}\\ &=\frac {x \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {3}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a \sqrt {a+b x^n}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 51, normalized size = 1.31 \begin {gather*} \frac {x \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {3}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,x^{n}\right )^{\frac {3}{2}}}\, dx\]
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 = 0.60, size = 39, normalized size = 1.00 \begin {gather*} \frac {x \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 )}}{a^{\frac {3}{2}} n \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 [B]
time = 1.28, size = 43, normalized size = 1.10 \begin {gather*} \frac {x\,{\left (\frac {b\,x^n}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{{\left (a+b\,x^n\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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