Optimal. Leaf size=57 \[ \frac {F^{a+\frac {b}{c+d x}}}{b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d \log (F) (c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac {F^{a+\frac {b}{c+d x}}}{b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d \log (F) (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^3} \, dx &=-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x) \log (F)}-\frac {\int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{b \log (F)}\\ &=\frac {F^{a+\frac {b}{c+d x}}}{b^2 d \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x) \log (F)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 41, normalized size = 0.72 \[ \frac {F^{a+\frac {b}{c+d x}} (-b \log (F)+c+d x)}{b^2 d \log ^2(F) (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 51, normalized size = 0.89 \[ \frac {{\left (d x - b \log \relax (F) + c\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{2} d^{2} x + b^{2} c d\right )} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 106, normalized size = 1.86 \[ \frac {\frac {d \,x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{2} \ln \relax (F )^{2}}-\frac {\left (b \ln \relax (F )-2 c \right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{2} \ln \relax (F )^{2}}-\frac {\left (b \ln \relax (F )-c \right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \relax (F )}}{b^{2} d \ln \relax (F )^{2}}}{\left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.23, size = 41, normalized size = 0.72 \[ \frac {F^{a+\frac {b}{c+d\,x}}\,\left (c+d\,x-b\,\ln \relax (F)\right )}{b^2\,d\,{\ln \relax (F)}^2\,\left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.22, size = 44, normalized size = 0.77 \[ \frac {F^{a + \frac {b}{c + d x}} \left (- b \log {\relax (F )} + c + d x\right )}{b^{2} c d \log {\relax (F )}^{2} + b^{2} d^{2} x \log {\relax (F )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________