Optimal. Leaf size=68 \[ \frac {a \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e} \]
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Rubi [A] time = 0.18, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2291, 14, 2178} \[ \frac {a \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2178
Rule 2291
Rubi steps
\begin {align*} \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{d^2-e^2 x^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b F^{c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a}{x}+\frac {b F^{c x}}{x}\right ) \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}\\ &=\frac {a \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b \operatorname {Subst}\left (\int \frac {F^{c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}\\ &=\frac {b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {a \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}\\ \end {align*}
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Mathematica [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{d^2-e^2 x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 2.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a + \frac {b}{F^{\frac {\sqrt {-e f x + d f} \sqrt {e x + d} c}{e f x - d f}}}}{e^{2} x^{2} - d^{2}}, x\right ) \]
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 \[ \int -\frac {F^{\frac {\sqrt {e x + d} c}{\sqrt {-e f x + d f}}} b + a}{e^{2} x^{2} - d^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {b \,F^{\frac {\sqrt {e x +d}\, c}{\sqrt {-e f x +d f}}}+a}{-e^{2} x^{2}+d^{2}}\, dx \]
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 \[ \frac {1}{2} \, a {\left (\frac {\log \left (e x + d\right )}{d e} - \frac {\log \left (e x - d\right )}{d e}\right )} - b \int \frac {F^{\frac {\sqrt {e x + d} c}{\sqrt {-e x + d} \sqrt {f}}}}{e^{2} x^{2} - d^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,{\mathrm {e}}^{\frac {c\,\ln \relax (F)\,\sqrt {d+e\,x}}{\sqrt {d\,f-e\,f\,x}}}}{d^2-e^2\,x^2} \,d x \]
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 \[ - \int \frac {a}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac {F^{\frac {c \sqrt {d + e x}}{\sqrt {d f - e f x}}} b}{- d^{2} + e^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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