Optimal. Leaf size=59 \[ \frac {2 \left (x+\sqrt {a+x^2}\right )^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\frac {\left (x+\sqrt {a+x^2}\right )^2}{a}\right )}{a (1+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2147, 371}
\begin {gather*} \frac {2 \left (\sqrt {a+x^2}+x\right )^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\frac {\left (x+\sqrt {x^2+a}\right )^2}{a}\right )}{a (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2147
Rubi steps
\begin {align*} \int \frac {\left (x+\sqrt {a+x^2}\right )^n}{a+x^2} \, dx &=2 \text {Subst}\left (\int \frac {x^n}{a+x^2} \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {2 \left (x+\sqrt {a+x^2}\right )^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\frac {\left (x+\sqrt {a+x^2}\right )^2}{a}\right )}{a (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 61, normalized size = 1.03 \begin {gather*} \frac {2 \left (x+\sqrt {a+x^2}\right )^{1+n} \, _2F_1\left (1,\frac {1+n}{2};1+\frac {1+n}{2};-\frac {\left (x+\sqrt {a+x^2}\right )^2}{a}\right )}{a (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x +\sqrt {x^{2}+a}\right )^{n}}{x^{2}+a}\, dx\]
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x + \sqrt {a + x^{2}}\right )^{n}}{a + x^{2}}\, dx \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 (x+\sqrt {x^2+a}\right )}^n}{x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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