Optimal. Leaf size=154 \[ \frac{6 a^2 b \text{Ei}\left (\frac{c \sqrt{d+e x} \log (F)}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{2 a^3 \log \left (\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{6 a b^2 \text{Ei}\left (\frac{2 c \sqrt{d+e x} \log (F)}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{2 b^3 \text{Ei}\left (\frac{3 c \sqrt{d+e x} \log (F)}{\sqrt{f+g x}}\right )}{e f-d g} \]
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Rubi [A] time = 0.264809, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.06, Rules used = {2290, 2183, 2178} \[ \frac{6 a^2 b \text{Ei}\left (\frac{c \sqrt{d+e x} \log (F)}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{2 a^3 \log \left (\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{6 a b^2 \text{Ei}\left (\frac{2 c \sqrt{d+e x} \log (F)}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{2 b^3 \text{Ei}\left (\frac{3 c \sqrt{d+e x} \log (F)}{\sqrt{f+g x}}\right )}{e f-d g} \]
Antiderivative was successfully verified.
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Rule 2290
Rule 2183
Rule 2178
Rubi steps
\begin{align*} \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right )^3}{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 )^3}{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^3}{x}+\frac{3 a^2 b F^{c x}}{x}+\frac{3 a b^2 F^{2 c x}}{x}+\frac{b^3 F^{3 c x}}{x}\right ) \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{e f-d g}\\ &=\frac{2 a^3 \log \left (\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{\left (6 a^2 b\right ) \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 (6 a 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{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{F^{3 c x}}{x} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{e f-d g}\\ &=\frac{6 a^2 b \text{Ei}\left (\frac{c \sqrt{d+e x} \log (F)}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{6 a 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 b^3 \text{Ei}\left (\frac{3 c \sqrt{d+e x} \log (F)}{\sqrt{f+g x}}\right )}{e f-d g}+\frac{2 a^3 \log \left (\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{e f-d g}\\ \end{align*}
Mathematica [F] time = 1.51019, size = 0, normalized size = 0. \[ \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right )^3}{d f+(e f+d g) x+e g x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{df+ \left ( dg+fe \right ) x+eg{x}^{2}} \left ( a+b{F}^{{c\sqrt{ex+d}{\frac{1}{\sqrt{gx+f}}}}} \right ) ^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3}{\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^{3} \int \frac{F^{\frac{3 \, \sqrt{e x + d} c}{\sqrt{g x + f}}}}{e g x^{2} + d f +{\left (e f + d g\right )} x}\,{d x} + 3 \, a 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} + 3 \, a^{2} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (F^{\frac{\sqrt{e x + d} c}{\sqrt{g x + f}}} b + a\right )}^{3}}{e g x^{2} + d f +{\left (e f + d g\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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