Optimal. Leaf size=116 \[ -\frac {b d \log (F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^2}+\frac {d F^{a+\frac {b}{c+d x}}}{f (d e-c f)}-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.01, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2223, 6742, 2209, 2210, 2222, 2228, 2178} \[ -\frac {b d \log (F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^2}+\frac {d F^{a+\frac {b}{c+d x}}}{f (d e-c f)}-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 2209
Rule 2210
Rule 2222
Rule 2223
Rule 2228
Rule 6742
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^2} \, dx &=-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)}-\frac {(b d \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2 (e+f x)} \, dx}{f}\\ &=-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)}-\frac {(b d \log (F)) \int \left (\frac {d F^{a+\frac {b}{c+d x}}}{(d e-c f) (c+d x)^2}-\frac {d f F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (c+d x)}+\frac {f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (e+f x)}\right ) \, dx}{f}\\ &=-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)}+\frac {\left (b d^2 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{(d e-c f)^2}-\frac {(b d f \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{e+f x} \, dx}{(d e-c f)^2}-\frac {\left (b d^2 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{f (d e-c f)}\\ &=\frac {d F^{a+\frac {b}{c+d x}}}{f (d e-c f)}-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)}-\frac {b d F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log (F)}{(d e-c f)^2}-\frac {\left (b d^2 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{(d e-c f)^2}+\frac {(b d \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{d e-c f}\\ &=\frac {d F^{a+\frac {b}{c+d x}}}{f (d e-c f)}-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)}-\frac {(b d \log (F)) \operatorname {Subst}\left (\int \frac {F^{a-\frac {b f}{d e-c f}+\frac {b d x}{d e-c f}}}{x} \, dx,x,\frac {e+f x}{c+d x}\right )}{(d e-c f)^2}\\ &=\frac {d F^{a+\frac {b}{c+d x}}}{f (d e-c f)}-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)}-\frac {b d F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.33, size = 116, normalized size = 1.00 \[ -\frac {b d \log (F) F^{a+\frac {b f}{c f-d e}} \text {Ei}\left (\frac {b \log (F)}{c+d x}-\frac {b f \log (F)}{c f-d e}\right )}{(d e-c f)^2}+\frac {d F^{a+\frac {b}{c+d x}}}{f (d e-c f)}-\frac {F^{a+\frac {b}{c+d x}}}{f (e+f x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 179, normalized size = 1.54 \[ -\frac {{\left (b d f x + b d e\right )} F^{\frac {a d e - {\left (a c + b\right )} f}{d e - c f}} {\rm Ei}\left (\frac {{\left (b d f x + b d e\right )} \log \relax (F)}{c d e - c^{2} f + {\left (d^{2} e - c d f\right )} x}\right ) \log \relax (F) - {\left (c d e - c^{2} f + {\left (d^{2} e - c d f\right )} x\right )} F^{\frac {a d x + a c + b}{d x + c}}}{d^{2} e^{3} - 2 \, c d e^{2} f + c^{2} e f^{2} + {\left (d^{2} e^{2} f - 2 \, c d e f^{2} + c^{2} f^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.17, size = 191, normalized size = 1.65 \[ \frac {b d \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{\left (c f -d e \right )^{2} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )}+\frac {b d \,F^{\frac {a c f -a d e +b f}{c f -d e}} \Ei \left (1, -a \ln \relax (F )-\frac {b \ln \relax (F )}{d x +c}-\frac {-a c f \ln \relax (F )+a d e \ln \relax (F )-b f \ln \relax (F )}{c f -d e}\right ) \ln \relax (F )}{\left (c f -d e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{a+\frac {b}{c+d\,x}}}{{\left (e+f\,x\right )}^2} \,d x \]
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]
________________________________________________________________________________________