Optimal. Leaf size=68 \[ \frac {b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {a \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e} \]
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Rubi [A]
time = 0.12, 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 = {2329, 14, 2209}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2209
Rule 2329
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 {\text {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 {\text {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 \text {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 = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{d^2-e^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \,F^{\frac {c \sqrt {e x +d}}{\sqrt {-e f x +d f}}}}{-e^{2} x^{2}+d^{2}}\, dx\]
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 \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 {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 \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 \frac {a+b\,{\mathrm {e}}^{\frac {c\,\ln \left (F\right )\,\sqrt {d+e\,x}}{\sqrt {d\,f-e\,f\,x}}}}{d^2-e^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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