Optimal. Leaf size=118 \[ \frac {\sqrt {\pi } F^{a f} (d+e x) e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]
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Rubi [A] time = 0.09, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2275, 2234, 2204} \[ \frac {\sqrt {\pi } F^{a f} (d+e x) e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {Erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2275
Rubi steps
\begin {align*} \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} \, dx,x,d+e x\right )}{e}\\ &=\frac {\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{\frac {x}{n}+a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac {\left (e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{\frac {\left (\frac {1}{n}+2 b f x \log (F)\right )^2}{4 b f \log (F)}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 118, normalized size = 1.00 \[ \frac {\sqrt {\pi } F^{a f} (d+e x) e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 116, normalized size = 0.98 \[ -\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \relax (F)} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \relax (F) + 2 \, b f n \log \relax (F) \log \relax (c) + 1\right )} \sqrt {-b f n^{2} \log \relax (F)}}{2 \, b f n^{2} \log \relax (F)}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \relax (F)^{2} - 4 \, b f n \log \relax (F) \log \relax (c) - 1}{4 \, b f n^{2} \log \relax (F)}\right )}}{2 \, e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 101, normalized size = 0.86 \[ -\frac {\sqrt {\pi } F^{a f} \operatorname {erf}\left (-\sqrt {-b f \log \relax (F)} n \log \left (x e + d\right ) - \sqrt {-b f \log \relax (F)} \log \relax (c) - \frac {\sqrt {-b f \log \relax (F)}}{2 \, b f n \log \relax (F)}\right ) e^{\left (-\frac {1}{4 \, b f n^{2} \log \relax (F)} - 1\right )}}{2 \, \sqrt {-b f \log \relax (F)} c^{\left (\frac {1}{n}\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )^{2}+a \right ) f}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{b\,f\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2}\,F^{a\,f} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 29.05, size = 532, normalized size = 4.51 \[ \begin {cases} - \frac {2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f n^{2} \log {\relax (F )} \log {\left (d + e x \right )}}{e} - \frac {2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f n^{2} \log {\relax (F )}}{e} - \frac {2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f n \log {\relax (F )} \log {\relax (c )}}{e} - 2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b f n^{2} x \log {\relax (F )} \log {\left (d + e x \right )} + 2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b f n^{2} x \log {\relax (F )} - 2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b f n x \log {\relax (F )} \log {\relax (c )} + \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} d}{e} + F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} x & \text {for}\: e \neq 0 \\F^{f \left (a + b \log {\left (c d^{n} \right )}^{2}\right )} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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