Optimal. Leaf size=121 \[ -\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} \sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {1-2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} g^2 n (d+e x) \sqrt {\log (F)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2308, 2266,
2235} \begin {gather*} -\frac {\sqrt {\pi } F^{a f} e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{\frac {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} g^2 n \sqrt {\log (F)} (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2308
Rubi steps
\begin {align*} \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{(d g+e g x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {F^{f \left (a+b \log ^2\left (c x^n\right )\right )}}{g^2 x^2} \, dx,x,d+e x\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {F^{f \left (a+b \log ^2\left (c x^n\right )\right )}}{x^2} \, dx,x,d+e x\right )}{e g^2}\\ &=\frac {\left (c (d+e x)^n\right )^{\frac {1}{n}} \text {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 g^2 n (d+e x)}\\ &=\frac {\left (e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} \left (c (d+e x)^n\right )^{\frac {1}{n}}\right ) \text {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 g^2 n (d+e x)}\\ &=-\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} \sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {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} g^2 n (d+e x) \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 121, normalized size = 1.00 \begin {gather*} \frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} \sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {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} g^2 n (d+e x) \sqrt {\log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {F^{f \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )^{2}\right )}}{\left (e g x +d g \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 119, normalized size = 0.98 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (x e + d\right ) \log \left (F\right ) + 2 \, b f n \log \left (F\right ) \log \left (c\right ) - 1\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{2 \, b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \left (F\right )^{2} + 4 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{4 \, b f n^{2} \log \left (F\right )} - 1\right )}}{2 \, g^{2} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 96.14, size = 211, normalized size = 1.74 \begin {gather*} \begin {cases} \frac {\tilde {\infty } F^{f \left (a + b \log {\left (0^{n} c \right )}^{2}\right )}}{g^{2} x} & \text {for}\: d = 0 \wedge e = 0 \\\tilde {\infty } F^{f \left (a + b \log {\left (0^{n} c \right )}^{2}\right )} x & \text {for}\: d = - e x \\\frac {F^{f \left (a + b \log {\left (c d^{n} \right )}^{2}\right )} x}{d^{2} g^{2}} & \text {for}\: e = 0 \\- \frac {2 F^{a f} F^{b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b f n^{2} \log {\left (F \right )}}{d e g^{2} + e^{2} g^{2} x} - \frac {2 F^{a f} F^{b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b f n \log {\left (F \right )} \log {\left (c \left (d + e x\right )^{n} \right )}}{d e g^{2} + e^{2} g^{2} x} - \frac {F^{a f} F^{b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}}{d e g^{2} + e^{2} g^{2} x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {e}}^{f\,\ln \left (F\right )\,\left (b\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2+a\right )}}{{\left (d\,g+e\,g\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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