Optimal. Leaf size=152 \[ \frac {a^3 \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {3 a^2 b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {3 a b^2 \text {Ei}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^3 \text {Ei}\left (\frac {3 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e} \]
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Rubi [A] time = 0.33, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {2291, 2183, 2178} \[ \frac {3 a^2 b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {a^3 \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {3 a b^2 \text {Ei}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^3 \text {Ei}\left (\frac {3 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2183
Rule 2291
Rubi steps
\begin {align*} \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b F^{c x}\right )^3}{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^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 {d f-e f x}}\right )}{d e}\\ &=\frac {a^3 \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {\left (3 a^2 b\right ) \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 {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \frac {F^{2 c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^3 \operatorname {Subst}\left (\int \frac {F^{3 c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}\\ &=\frac {3 a^2 b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {3 a b^2 \text {Ei}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^3 \text {Ei}\left (\frac {3 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {a^3 \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 = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 5.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{3} + \frac {3 \, a^{2} b}{F^{\frac {\sqrt {-e f x + d f} \sqrt {e x + d} c}{e f x - d f}}} + \frac {3 \, a b^{2}}{F^{\frac {2 \, \sqrt {-e f x + d f} \sqrt {e x + d} c}{e f x - d f}}} + \frac {b^{3}}{F^{\frac {3 \, \sqrt {-e f x + d f} \sqrt {e x + d} c}{e f x - d f}}}}{e^{2} x^{2} - d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,F^{\frac {\sqrt {e x +d}\, c}{\sqrt {-e f x +d f}}}+a \right )^{3}}{-e^{2} x^{2}+d^{2}}\, dx \]
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 \[ \frac {1}{2} \, a^{3} {\left (\frac {\log \left (e x + d\right )}{d e} - \frac {\log \left (e x - d\right )}{d e}\right )} - b^{3} \int \frac {F^{\frac {3 \, \sqrt {e x + d} c}{\sqrt {-e x + d} \sqrt {f}}}}{e^{2} x^{2} - d^{2}}\,{d x} - 3 \, a b^{2} \int \frac {F^{\frac {2 \, \sqrt {e x + d} c}{\sqrt {-e x + d} \sqrt {f}}}}{e^{2} x^{2} - d^{2}}\,{d x} - 3 \, a^{2} b \int \frac {F^{\frac {\sqrt {e x + d} c}{\sqrt {-e x + d} \sqrt {f}}}}{e^{2} x^{2} - d^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,{\mathrm {e}}^{\frac {c\,\ln \relax (F)\,\sqrt {d+e\,x}}{\sqrt {d\,f-e\,f\,x}}}\right )}^3}{d^2-e^2\,x^2} \,d x \]
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 \[ - \int \frac {a^{3}}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac {F^{\frac {3 c \sqrt {d + e x}}{\sqrt {d f - e f x}}} b^{3}}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac {3 F^{\frac {c \sqrt {d + e x}}{\sqrt {d f - e f x}}} a^{2} b}{- d^{2} + e^{2} x^{2}}\, dx - \int \frac {3 F^{\frac {2 c \sqrt {d + e x}}{\sqrt {d f - e f x}}} a b^{2}}{- d^{2} + e^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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