Optimal. Leaf size=372 \[ \frac {\sqrt {\pi } h F^{a f} (d+e x)^2 (e g-d h) e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{\sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } F^{a f} (d+e x) (e g-d h)^2 e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } h^2 F^{a f} (d+e x)^3 e^{-\frac {9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}} \]
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Rubi [F] time = 0.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx \]
Verification is Not applicable to the result.
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\begin {align*} \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx &=\int \left (F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} g^2+2 F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} g h x+F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} h^2 x^2\right ) \, dx\\ &=g^2 \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} \, dx+(2 g h) \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x \, dx+h^2 \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x^2 \, dx\\ &=\frac {g^2 \operatorname {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} \, dx,x,d+e x\right )}{e}+(2 g h) \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x \, dx+h^2 \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x^2 \, dx\\ &=(2 g h) \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x \, dx+h^2 \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x^2 \, dx+\frac {\left (g^2 (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}\\ &=(2 g h) \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x \, dx+h^2 \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x^2 \, dx+\frac {\left (e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} g^2 (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} g^2 \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)}}+(2 g h) \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x \, dx+h^2 \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} x^2 \, dx\\ \end {align*}
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Mathematica [A] time = 0.81, size = 303, normalized size = 0.81 \[ \frac {\sqrt {\pi } F^{a f} (d+e x) e^{-\frac {9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \left ((e g-d h)^2 e^{\frac {2}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )-2 h (d+e x) (d h-e g) e^{\frac {5}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+h^2 (d+e x)^2 \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 367, normalized size = 0.99 \[ -\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \relax (F)} h^{2} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \relax (F) + 2 \, b f n \log \relax (F) \log \relax (c) + 3\right )} \sqrt {-b f n^{2} \log \relax (F)}}{2 \, b f n^{2} \log \relax (F)}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \relax (F)^{2} - 12 \, b f n \log \relax (F) \log \relax (c) - 9}{4 \, b f n^{2} \log \relax (F)}\right )} + \sqrt {\pi } \sqrt {-b f n^{2} \log \relax (F)} {\left (e^{2} g^{2} - 2 \, d e g h + d^{2} h^{2}\right )} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \relax (F) + 2 \, b f n \log \relax (F) \log \relax (c) + 1\right )} \sqrt {-b f n^{2} \log \relax (F)}}{2 \, b f n^{2} \log \relax (F)}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \relax (F)^{2} - 4 \, b f n \log \relax (F) \log \relax (c) - 1}{4 \, b f n^{2} \log \relax (F)}\right )} + 2 \, \sqrt {\pi } \sqrt {-b f n^{2} \log \relax (F)} {\left (e g h - d h^{2}\right )} \operatorname {erf}\left (\frac {{\left (b f n^{2} \log \left (e x + d\right ) \log \relax (F) + b f n \log \relax (F) \log \relax (c) + 1\right )} \sqrt {-b f n^{2} \log \relax (F)}}{b f n^{2} \log \relax (F)}\right ) e^{\left (\frac {a b f^{2} n^{2} \log \relax (F)^{2} - 2 \, b f n \log \relax (F) \log \relax (c) - 1}{b f n^{2} \log \relax (F)}\right )}}{2 \, e^{3} n} \]
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 \[ \int {\left (h x + g\right )}^{2} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \left (h x +g \right )^{2} F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )^{2}+a \right ) f}\, dx \]
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 \[ \int {\left (h x + g\right )}^{2} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {e}}^{f\,\ln \relax (F)\,\left (b\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2+a\right )}\,{\left (g+h\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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