Optimal. Leaf size=117 \[ -\frac {a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 d^2 e}+\frac {\left (\frac {a f^2}{d^2}+1\right ) \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}{2 e}-\frac {a f^2}{2 d e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2117, 893} \[ -\frac {a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 d^2 e}+\frac {\left (\frac {a f^2}{d^2}+1\right ) \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}{2 e}-\frac {a f^2}{2 d e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )} \]
Antiderivative was successfully verified.
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Rule 893
Rule 2117
Rubi steps
\begin {align*} \int \frac {1}{d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {d^2+a f^2-2 d x+x^2}{(d-x)^2 x} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a f^2}{d (d-x)^2}+\frac {a f^2}{d^2 (d-x)}+\frac {d^2+a f^2}{d^2 x}\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=-\frac {a f^2}{2 d e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {a f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 d^2 e}+\frac {\left (1+\frac {a f^2}{d^2}\right ) \log \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 109, normalized size = 0.93 \[ \frac {-\frac {a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{d^2}+\left (\frac {a f^2}{d^2}+1\right ) \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )+\frac {a f^2}{d \left (f \left (-\sqrt {a+\frac {e^2 x^2}{f^2}}\right )-e x\right )}}{2 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 187, normalized size = 1.60 \[ \frac {2 \, d e x - 2 \, d f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + {\left (a f^{2} + d^{2}\right )} \log \left (a f^{2} - d e x + d f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + {\left (a f^{2} + d^{2}\right )} \log \left (-a f^{2} + 2 \, d e x + d^{2}\right ) - {\left (a f^{2} + d^{2}\right )} \log \left (-e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} - d\right ) + {\left (a f^{2} - d^{2}\right )} \log \left (-e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right )}{4 \, d^{2} e} \]
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 \[ \left [\mathit {undef}, +\infty , 1\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1325, normalized size = 11.32 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} f + d}\,{d x} \]
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 \[ \int \frac {1}{d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}} \,d x \]
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 \[ \int \frac {1}{d + e x + f \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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