Optimal. Leaf size=66 \[ -\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (-\frac {1}{n},-b \log (F);-\frac {1-n}{n};-\frac {d x^n}{c}\right )}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2274, 12, 365, 364} \[ -\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (-\frac {1}{n},-b \log (F);-\frac {1-n}{n};-\frac {d x^n}{c}\right )}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 364
Rule 365
Rule 2274
Rubi steps
\begin {align*} \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx &=\int \frac {F^a \left (c+d x^n\right )^{b \log (F)}}{x^2} \, dx\\ &=F^a \int \frac {\left (c+d x^n\right )^{b \log (F)}}{x^2} \, dx\\ &=\left (F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)}\right ) \int \frac {\left (1+\frac {d x^n}{c}\right )^{b \log (F)}}{x^2} \, dx\\ &=-\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (-\frac {1}{n},-b \log (F);-\frac {1-n}{n};-\frac {d x^n}{c}\right )}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 81, normalized size = 1.23 \[ -\frac {\left (-\frac {d x^n}{c}\right )^{\frac {1}{n}} \left (c+d x^n\right ) F^{a+b \log \left (c+d x^n\right )} \, _2F_1\left (1+\frac {1}{n},b \log (F)+1;b \log (F)+2;\frac {d x^n}{c}+1\right )}{c n x (b \log (F)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \ln \left (d \,x^{n}+c \right )+a}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {F^{a+b\,\ln \left (c+d\,x^n\right )}}{x^2} \,d x \]
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]
________________________________________________________________________________________