Optimal. Leaf size=112 \[ \frac {2 a^2 \log \left (\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {4 a b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {2 b^2 \text {Ei}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g} \]
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Rubi [A] time = 0.23, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {2290, 2183, 2178} \[ \frac {2 a^2 \log \left (\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {4 a b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {2 b^2 \text {Ei}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2183
Rule 2290
Rubi steps
\begin {align*} \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^2}{d f+(e f+d g) x+e g x^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\left (a+b F^{c x}\right )^2}{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^2}{x}+\frac {2 a b F^{c x}}{x}+\frac {b^2 F^{2 c x}}{x}\right ) \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}\\ &=\frac {2 a^2 \log \left (\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {(4 a 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 {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {F^{2 c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g}\\ &=\frac {4 a b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {2 b^2 \text {Ei}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {2 a^2 \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 = 1.39, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^2}{d f+(e f+d g) x+e g x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 130.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} a b + F^{\frac {2 \, \sqrt {e x + d} c}{\sqrt {g x + f}}} b^{2} + a^{2}}{e g x^{2} + d f + {\left (e f + d g\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} b + a\right )}^{2}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,F^{\frac {\sqrt {e x +d}\, c}{\sqrt {g x +f}}}+a \right )^{2}}{e g \,x^{2}+d f +\left (d g +e f \right ) x}\, dx \]
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 \[ a^{2} {\left (\frac {\log \left (e x + d\right )}{e f - d g} - \frac {\log \left (g x + f\right )}{e f - d g}\right )} + b^{2} \int \frac {F^{\frac {2 \, \sqrt {e x + d} c}{\sqrt {g x + f}}}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\,{d x} + 2 \, a b \int \frac {F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\,{d x} \]
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 \[ \int \frac {{\left (a+F^{\frac {c\,\sqrt {d+e\,x}}{\sqrt {f+g\,x}}}\,b\right )}^2}{e\,g\,x^2+\left (d\,g+e\,f\right )\,x+d\,f} \,d x \]
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 \[ \int \frac {\left (F^{\frac {c \sqrt {d + e x}}{\sqrt {f + g x}}} b + a\right )^{2}}{\left (d + e x\right ) \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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