Optimal. Leaf size=70 \[ \frac {2 a \log \left (\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {2 b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g} \]
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Rubi [A] time = 0.12, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2290, 14, 2178} \[ \frac {2 a \log \left (\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {2 b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2178
Rule 2290
Rubi steps
\begin {align*} \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{d f+(e f+d g) x+e g x^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {a+b F^{c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {a}{x}+\frac {b F^{c x}}{x}\right ) \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}\\ &=\frac {2 a \log \left (\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {F^{c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}\\ &=\frac {2 b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {2 a \log \left (\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}\\ \end {align*}
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Mathematica [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{d f+(e f+d g) x+e g x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 58.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} b + a}{e g x^{2} + d f + {\left (e f + d g\right )} x}, x\right ) \]
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 \[ \int \frac {F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} b + a}{e g x^{2} + d f + {\left (e f + d g\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {b \,F^{\frac {\sqrt {e x +d}\, c}{\sqrt {g x +f}}}+a}{e g \,x^{2}+d f +\left (d g +e f \right ) x}\, dx \]
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 \[ a {\left (\frac {\log \left (e x + d\right )}{e f - d g} - \frac {\log \left (g x + f\right )}{e f - d g}\right )} + b \int \frac {F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\,{d x} \]
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 \[ \int \frac {a+F^{\frac {c\,\sqrt {d+e\,x}}{\sqrt {f+g\,x}}}\,b}{e\,g\,x^2+\left (d\,g+e\,f\right )\,x+d\,f} \,d x \]
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 \[ \int \frac {F^{\frac {c \sqrt {d + e x}}{\sqrt {f + g x}}} b + a}{\left (d + e x\right ) \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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