Optimal. Leaf size=68 \[ \frac {1}{2} f x \sqrt {a+\frac {e^2 x^2}{f^2}}+\frac {a f^2 \tanh ^{-1}\left (\frac {e x}{f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}+d x+\frac {e x^2}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {195, 217, 206} \[ \frac {1}{2} f x \sqrt {a+\frac {e^2 x^2}{f^2}}+\frac {a f^2 \tanh ^{-1}\left (\frac {e x}{f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}+d x+\frac {e x^2}{2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right ) \, dx &=d x+\frac {e x^2}{2}+f \int \sqrt {a+\frac {e^2 x^2}{f^2}} \, dx\\ &=d x+\frac {e x^2}{2}+\frac {1}{2} f x \sqrt {a+\frac {e^2 x^2}{f^2}}+\frac {1}{2} (a f) \int \frac {1}{\sqrt {a+\frac {e^2 x^2}{f^2}}} \, dx\\ &=d x+\frac {e x^2}{2}+\frac {1}{2} f x \sqrt {a+\frac {e^2 x^2}{f^2}}+\frac {1}{2} (a f) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e^2 x^2}{f^2}} \, dx,x,\frac {x}{\sqrt {a+\frac {e^2 x^2}{f^2}}}\right )\\ &=d x+\frac {e x^2}{2}+\frac {1}{2} f x \sqrt {a+\frac {e^2 x^2}{f^2}}+\frac {a f^2 \tanh ^{-1}\left (\frac {e x}{f \sqrt {a+\frac {e^2 x^2}{f^2}}}\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 81, normalized size = 1.19 \[ \frac {1}{2} f x \sqrt {\frac {a f^2+e^2 x^2}{f^2}}+\frac {a f^2 \log \left (e f \sqrt {\frac {a f^2+e^2 x^2}{f^2}}+e^2 x\right )}{2 e}+d x+\frac {e x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 74, normalized size = 1.09 \[ \frac {e^{2} x^{2} - a f^{2} \log \left (-e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + e f x \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + 2 \, d e x}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 65, normalized size = 0.96 \[ \frac {1}{2} \, x^{2} e + d x - \frac {{\left (a f^{2} e^{\left (-1\right )} \log \left ({\left | -x e + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right ) - \sqrt {a f^{2} + x^{2} e^{2}} x\right )} {\left | f \right |}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 75, normalized size = 1.10 \[ \frac {a f \ln \left (\frac {e^{2} x}{\sqrt {\frac {e^{2}}{f^{2}}}\, f^{2}}+\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}+\frac {e \,x^{2}}{2}+d x +\frac {\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\, f x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 46, normalized size = 0.68 \[ \frac {1}{2} \, e x^{2} + \frac {1}{2} \, {\left (\frac {a f \operatorname {arsinh}\left (\frac {e x}{\sqrt {a} f}\right )}{e} + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} x\right )} f + d x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.89, size = 136, normalized size = 2.00 \[ \left \{\begin {array}{cl} x\,\left (d+\sqrt {a}\,f\right ) & \text {\ if\ \ }e=0\\ d\,x+\frac {e\,x^2}{2}+\frac {f\,x\,\sqrt {a+\frac {e^2\,x^2}{f^2}}}{2}+\frac {a\,e^2\,\ln \left (x\,\sqrt {\frac {e^2}{f^2}}+\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}{f\,{\left (\frac {e^2}{f^2}\right )}^{3/2}}-\frac {a\,e^2\,\ln \left (2\,x\,\sqrt {\frac {e^2}{f^2}}+2\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}{2\,f\,{\left (\frac {e^2}{f^2}\right )}^{3/2}} & \text {\ if\ \ }e\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.21, size = 54, normalized size = 0.79 \[ d x + \frac {e x^{2}}{2} + f \left (\frac {\sqrt {a} x \sqrt {1 + \frac {e^{2} x^{2}}{a f^{2}}}}{2} + \frac {a f \operatorname {asinh}{\left (\frac {e x}{\sqrt {a} f} \right )}}{2 e}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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