Optimal. Leaf size=59 \[ \frac {4 \left (x+\sqrt {a+x^2}\right )^{2+n} \, _2F_1\left (2,\frac {2+n}{2};\frac {4+n}{2};-\frac {\left (x+\sqrt {a+x^2}\right )^2}{a}\right )}{a^2 (2+n)} \]
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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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2147, 371}
\begin {gather*} \frac {4 \left (\sqrt {a+x^2}+x\right )^{n+2} \, _2F_1\left (2,\frac {n+2}{2};\frac {n+4}{2};-\frac {\left (x+\sqrt {x^2+a}\right )^2}{a}\right )}{a^2 (n+2)} \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}{\left (a+x^2\right )^{3/2}} \, dx &=4 \text {Subst}\left (\int \frac {x^{1+n}}{\left (a+x^2\right )^2} \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {4 \left (x+\sqrt {a+x^2}\right )^{2+n} \, _2F_1\left (2,\frac {2+n}{2};\frac {4+n}{2};-\frac {\left (x+\sqrt {a+x^2}\right )^2}{a}\right )}{a^2 (2+n)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 61, normalized size = 1.03 \begin {gather*} \frac {4 \left (x+\sqrt {a+x^2}\right )^{2+n} \, _2F_1\left (2,\frac {2+n}{2};1+\frac {2+n}{2};-\frac {\left (x+\sqrt {a+x^2}\right )^2}{a}\right )}{a^2 (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x +\sqrt {x^{2}+a}\right )^{n}}{\left (x^{2}+a \right )^{\frac {3}{2}}}\, dx\]
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]
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}}{\left (a + x^{2}\right )^{\frac {3}{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}{{\left (x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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