Optimal. Leaf size=104 \[ \frac {F^{\frac {f (b g-a h)}{d g-c h}+e} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{h}-\frac {F^{\frac {b f}{d}+e} \text {Ei}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right )}{h} \]
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Rubi [A] time = 1.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2231, 2230, 2210, 2233, 2178} \[ \frac {F^{\frac {f (b g-a h)}{d g-c h}+e} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{h}-\frac {F^{\frac {b f}{d}+e} \text {Ei}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right )}{h} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2210
Rule 2230
Rule 2231
Rule 2233
Rubi steps
\begin {align*} \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx &=\frac {d \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{h}-\frac {(d g-c h) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{h}\\ &=\frac {\operatorname {Subst}\left (\int \frac {F^{e+\frac {f (b g-a h)}{d g-c h}-\frac {(b c-a d) f x}{d g-c h}}}{x} \, dx,x,\frac {g+h x}{c+d x}\right )}{h}+\frac {d \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{h}\\ &=-\frac {F^{e+\frac {b f}{d}} \text {Ei}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right )}{h}+\frac {F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{h}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 103, normalized size = 0.99 \[ \frac {F^{\frac {b f}{d}+e} \left (F^{\frac {f h (b c-a d)}{d (d g-c h)}} \text {Ei}\left (\frac {(b c-a d) f (g+h x) \log (F)}{(c h-d g) (c+d x)}\right )-\text {Ei}\left (\frac {(a d f-b c f) \log (F)}{d (c+d x)}\right )\right )}{h} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 135, normalized size = 1.30 \[ -\frac {F^{\frac {d e + b f}{d}} {\rm Ei}\left (-\frac {{\left (b c - a d\right )} f \log \relax (F)}{d^{2} x + c d}\right ) - F^{\frac {{\left (d e + b f\right )} g - {\left (c e + a f\right )} h}{d g - c h}} {\rm Ei}\left (-\frac {{\left ({\left (b c - a d\right )} f h x + {\left (b c - a d\right )} f g\right )} \log \relax (F)}{c d g - c^{2} h + {\left (d^{2} g - c d h\right )} x}\right )}{h} \]
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 \[ \int \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 432, normalized size = 4.15 \[ \frac {a d \,F^{\frac {b f +d e}{d}} \Ei \left (1, -\frac {\left (a d -b c \right ) f \ln \relax (F )}{\left (d x +c \right ) d}-\frac {\left (b f +d e \right ) \ln \relax (F )}{d}-\frac {-b f \ln \relax (F )-d e \ln \relax (F )}{d}\right )}{\left (a d -b c \right ) h}-\frac {a d \,F^{\frac {a f h -b f g +c e h -d e g}{c h -d g}} \Ei \left (1, -\frac {\left (a d -b c \right ) f \ln \relax (F )}{\left (d x +c \right ) d}-\frac {\left (b f +d e \right ) \ln \relax (F )}{d}-\frac {-a f h \ln \relax (F )+b f g \ln \relax (F )-c e h \ln \relax (F )+d e g \ln \relax (F )}{c h -d g}\right )}{\left (a d -b c \right ) h}-\frac {b c \,F^{\frac {b f +d e}{d}} \Ei \left (1, -\frac {\left (a d -b c \right ) f \ln \relax (F )}{\left (d x +c \right ) d}-\frac {\left (b f +d e \right ) \ln \relax (F )}{d}-\frac {-b f \ln \relax (F )-d e \ln \relax (F )}{d}\right )}{\left (a d -b c \right ) h}+\frac {b c \,F^{\frac {a f h -b f g +c e h -d e g}{c h -d g}} \Ei \left (1, -\frac {\left (a d -b c \right ) f \ln \relax (F )}{\left (d x +c \right ) d}-\frac {\left (b f +d e \right ) \ln \relax (F )}{d}-\frac {-a f h \ln \relax (F )+b f g \ln \relax (F )-c e h \ln \relax (F )+d e g \ln \relax (F )}{c h -d g}\right )}{\left (a d -b c \right ) h} \]
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 \[ \int \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{h x + g}\,{d x} \]
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 \[ \int \frac {F^{e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}}}{g+h\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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