Optimal. Leaf size=159 \[ \frac {f \log (F) (b c-a d) F^{\frac {f (b g-a h)}{d g-c h}+e} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^2}+\frac {d F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{h (d g-c h)}-\frac {F^{\frac {f (a+b x)}{c+d x}+e}}{h (g+h x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.56, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2232, 6742, 2230, 2209, 2210, 2231, 2233, 2178} \[ \frac {f \log (F) (b c-a d) F^{\frac {f (b g-a h)}{d g-c h}+e} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^2}+\frac {d F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{h (d g-c h)}-\frac {F^{\frac {f (a+b x)}{c+d x}+e}}{h (g+h x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 2209
Rule 2210
Rule 2230
Rule 2231
Rule 2232
Rule 2233
Rule 6742
Rubi steps
\begin {align*} \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx &=-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {((b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)} \, dx}{h}\\ &=-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {((b c-a d) f \log (F)) \int \left (\frac {d F^{e+\frac {f (a+b x)}{c+d x}}}{(d g-c h) (c+d x)^2}-\frac {d F^{e+\frac {f (a+b x)}{c+d x}} h}{(d g-c h)^2 (c+d x)}+\frac {F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^2 (g+h x)}\right ) \, dx}{h}\\ &=-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}-\frac {(d (b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{(d g-c h)^2}+\frac {((b c-a d) f h \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx}{(d g-c h)^2}+\frac {(d (b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{h (d g-c h)}\\ &=-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}-\frac {(d (b c-a d) f \log (F)) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{(d g-c h)^2}+\frac {(d (b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{(d g-c h)^2}-\frac {((b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{d g-c h}+\frac {(d (b c-a d) f \log (F)) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{h (d g-c h)}\\ &=\frac {d F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{h (d g-c h)}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {(b c-a d) f F^{e+\frac {b f}{d}} \text {Ei}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right ) \log (F)}{(d g-c h)^2}+\frac {((b c-a d) f \log (F)) \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 )}{(d g-c h)^2}+\frac {(d (b c-a d) f \log (F)) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{(d g-c h)^2}\\ &=\frac {d F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{h (d g-c h)}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {(b c-a d) f 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 ) \log (F)}{(d g-c h)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 1.19, size = 0, normalized size = 0.00 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 220, normalized size = 1.38 \[ \frac {{\left ({\left (b c - a d\right )} f h x + {\left (b c - a d\right )} f g\right )} F^{\frac {{\left (d e + b f\right )} g - {\left (c e + a f\right )} h}{d g - c h}} {\rm Ei}\left (-\frac {{\left ({\left (b c - a d\right )} f h x + {\left (b c - a d\right )} f g\right )} \log \relax (F)}{c d g - c^{2} h + {\left (d^{2} g - c d h\right )} x}\right ) \log \relax (F) + {\left (c d g - c^{2} h + {\left (d^{2} g - c d h\right )} x\right )} F^{\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}}}{d^{2} g^{3} - 2 \, c d g^{2} h + c^{2} g h^{2} + {\left (d^{2} g^{2} h - 2 \, c d g h^{2} + c^{2} h^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.21, size = 580, normalized size = 3.65 \[ \frac {a d f \,F^{\frac {b f +d e}{d}} F^{\frac {\left (a d -b c \right ) f}{\left (d x +c \right ) d}} \ln \relax (F )}{\left (c h -d g \right )^{2} \left (-\frac {a f h \ln \relax (F )}{c h -d g}+\frac {b f g \ln \relax (F )}{c h -d g}-\frac {c e h \ln \relax (F )}{c h -d g}+\frac {d e g \ln \relax (F )}{c h -d g}+\frac {a f \ln \relax (F )}{d x +c}-\frac {b c f \ln \relax (F )}{\left (d x +c \right ) d}+\frac {b f \ln \relax (F )}{d}+e \ln \relax (F )\right )}+\frac {a d f \,F^{\frac {a f h -b f g +c e h -d e g}{c h -d g}} \Ei \left (1, -\frac {\left (a d -b c \right ) f \ln \relax (F )}{\left (d x +c \right ) d}-\frac {\left (b f +d e \right ) \ln \relax (F )}{d}-\frac {-a f h \ln \relax (F )+b f g \ln \relax (F )-c e h \ln \relax (F )+d e g \ln \relax (F )}{c h -d g}\right ) \ln \relax (F )}{\left (c h -d g \right )^{2}}-\frac {b c f \,F^{\frac {b f +d e}{d}} F^{\frac {\left (a d -b c \right ) f}{\left (d x +c \right ) d}} \ln \relax (F )}{\left (c h -d g \right )^{2} \left (-\frac {a f h \ln \relax (F )}{c h -d g}+\frac {b f g \ln \relax (F )}{c h -d g}-\frac {c e h \ln \relax (F )}{c h -d g}+\frac {d e g \ln \relax (F )}{c h -d g}+\frac {a f \ln \relax (F )}{d x +c}-\frac {b c f \ln \relax (F )}{\left (d x +c \right ) d}+\frac {b f \ln \relax (F )}{d}+e \ln \relax (F )\right )}-\frac {b c f \,F^{\frac {a f h -b f g +c e h -d e g}{c h -d g}} \Ei \left (1, -\frac {\left (a d -b c \right ) f \ln \relax (F )}{\left (d x +c \right ) d}-\frac {\left (b f +d e \right ) \ln \relax (F )}{d}-\frac {-a f h \ln \relax (F )+b f g \ln \relax (F )-c e h \ln \relax (F )+d e g \ln \relax (F )}{c h -d g}\right ) \ln \relax (F )}{\left (c h -d g \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}}}{{\left (g+h\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________