Optimal. Leaf size=121 \[ \frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n}}{2 e (1+n)}+\frac {a f^2 \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n} \, _2F_1\left (2,1+n;2+n;\frac {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}{d}\right )}{2 d^2 e (1+n)} \]
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Rubi [A]
time = 0.08, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2142, 961, 66}
\begin {gather*} \frac {a f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+a}}{d}\right )}{2 d^2 e (n+1)}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 961
Rule 2142
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^n \, dx &=\frac {\text {Subst}\left (\int \frac {x^n \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\text {Subst}\left (\int \left (x^n+\frac {a f^2 x^n}{(d-x)^2}\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n}}{2 e (1+n)}+\frac {\left (a f^2\right ) \text {Subst}\left (\int \frac {x^n}{(d-x)^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n}}{2 e (1+n)}+\frac {a f^2 \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n} \, _2F_1\left (2,1+n;2+n;\frac {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}{d}\right )}{2 d^2 e (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 86, normalized size = 0.71 \begin {gather*} \frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n} \left (d^2+a f^2 \, _2F_1\left (2,1+n;2+n;\frac {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}{d}\right )\right )}{2 d^2 e (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (d +e x +f \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx\]
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 \left (d + e x + f \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}\right )^{n}\, 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 = -0.01 \begin {gather*} \int {\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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