Optimal. Leaf size=126 \[ -\frac {\sqrt {\pi } \left (c (d+e x)^n\right )^{2/n} e^{-\frac {1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \text {erfi}\left (\frac {-a b f \log (F)+b^2 (-f) \log (F) \log \left (c (d+e x)^n\right )+\frac {1}{n}}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^3 n \sqrt {\log (F)} (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.41, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {12, 2278, 2274, 15, 2276, 2234, 2204} \[ -\frac {\sqrt {\pi } \left (c (d+e x)^n\right )^{2/n} e^{-\frac {1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \text {Erfi}\left (\frac {-a b f \log (F)+b^2 (-f) \log (F) \log \left (c (d+e x)^n\right )+\frac {1}{n}}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^3 n \sqrt {\log (F)} (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 15
Rule 2204
Rule 2234
Rule 2274
Rule 2276
Rule 2278
Rubi steps
\begin {align*} \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{(d g+e g x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {F^{f \left (a+b \log \left (c x^n\right )\right )^2}}{g^3 x^3} \, dx,x,d+e x\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {F^{f \left (a+b \log \left (c x^n\right )\right )^2}}{x^3} \, dx,x,d+e x\right )}{e g^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {F^{a^2 f+2 a b f \log \left (c x^n\right )+b^2 f \log ^2\left (c x^n\right )}}{x^3} \, dx,x,d+e x\right )}{e g^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} \left (c x^n\right )^{2 a b f \log (F)}}{x^3} \, dx,x,d+e x\right )}{e g^3}\\ &=\frac {\left ((d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \operatorname {Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{-3+2 a b f n \log (F)} \, dx,x,d+e x\right )}{e g^3}\\ &=\frac {\left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {-2+2 a b f n \log (F)}{n}} \operatorname {Subst}\left (\int \exp \left (a^2 f \log (F)+b^2 f x^2 \log (F)+\frac {x (-2+2 a b f n \log (F))}{n}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g^3 n (d+e x)^2}\\ &=\frac {\left (\exp \left (a^2 f \log (F)-\frac {(-2+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {-2+2 a b f n \log (F)}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (\frac {\left (2 b^2 f x \log (F)+\frac {-2+2 a b f n \log (F)}{n}\right )^2}{4 b^2 f \log (F)}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g^3 n (d+e x)^2}\\ &=-\frac {e^{-\frac {1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \sqrt {\pi } \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {\frac {1}{n}-a b f \log (F)-b^2 f \log (F) \log \left (c (d+e x)^n\right )}{b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^3 n (d+e x)^2 \sqrt {\log (F)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.32, size = 121, normalized size = 0.96 \[ \frac {\sqrt {\pi } \left (c (d+e x)^n\right )^{2/n} e^{\frac {2 a b f n \log (F)-1}{b^2 f n^2 \log (F)}} \text {erfi}\left (\frac {b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )-1}{b \sqrt {f} n \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} g^3 n \sqrt {\log (F)} (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 129, normalized size = 1.02 \[ -\frac {\sqrt {\pi } \sqrt {-b^{2} f n^{2} \log \relax (F)} \operatorname {erf}\left (\frac {{\left (b^{2} f n^{2} \log \left (e x + d\right ) \log \relax (F) + b^{2} f n \log \relax (F) \log \relax (c) + a b f n \log \relax (F) - 1\right )} \sqrt {-b^{2} f n^{2} \log \relax (F)}}{b^{2} f n^{2} \log \relax (F)}\right ) e^{\left (\frac {2 \, b^{2} f n \log \relax (F) \log \relax (c) + 2 \, a b f n \log \relax (F) - 1}{b^{2} f n^{2} \log \relax (F)}\right )}}{2 \, b e g^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}}{{\left (e g x + d g\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.99, size = 0, normalized size = 0.00 \[ \int \frac {F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2} f}}{\left (e g x +d g \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}}{{\left (e g x + d g\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{f\,\ln \relax (F)\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}}{{\left (d\,g+e\,g\,x\right )}^3} \,d x \]
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]
________________________________________________________________________________________