∫ Fa+b(c+dx)x3(e + fx)2dx

3.65 $\int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx$

Optimal. Leaf size=414 \[ -\frac{3 e^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{6 e^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac{8 e f x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{24 e f x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{48 e f x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac{5 f^2 x^4 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{20 f^2 x^3 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{60 f^2 x^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{120 f^2 x F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac{120 f^2 F^{a+b c+b d x}}{b^6 d^6 \log ^6(F)}+\frac{e^2 x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac{2 e f x^4 F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x^5 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

(-120*f^2*F^(a + b*c + b*d*x))/(b^6*d^6*Log[F]^6) + (48*e*f*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) + (120*f^2

*F^(a + b*c + b*d*x)*x)/(b^5*d^5*Log[F]^5) - (6*e^2*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4) - (48*e*f*F^(a + b

*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) - (60*f^2*F^(a + b*c + b*d*x)*x^2)/(b^4*d^4*Log[F]^4) + (6*e^2*F^(a + b*c +

b*d*x)*x)/(b^3*d^3*Log[F]^3) + (24*e*f*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) + (20*f^2*F^(a + b*c + b*d*

x)*x^3)/(b^3*d^3*Log[F]^3) - (3*e^2*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) - (8*e*f*F^(a + b*c + b*d*x)*x

^3)/(b^2*d^2*Log[F]^2) - (5*f^2*F^(a + b*c + b*d*x)*x^4)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x^3)/(b

*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^4)/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x^5)/(b*d*Log[F])

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Rubi [A] time = 0.671541, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 3, integrand size = 22, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.136, Rules used = {2196, 2176, 2194} \[ -\frac{3 e^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{6 e^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac{8 e f x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{24 e f x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{48 e f x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac{5 f^2 x^4 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{20 f^2 x^3 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{60 f^2 x^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{120 f^2 x F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac{120 f^2 F^{a+b c+b d x}}{b^6 d^6 \log ^6(F)}+\frac{e^2 x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac{2 e f x^4 F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x^5 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[F^(a + b*(c + d*x))*x^3*(e + f*x)^2,x]

[Out]

(-120*f^2*F^(a + b*c + b*d*x))/(b^6*d^6*Log[F]^6) + (48*e*f*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) + (120*f^2

*F^(a + b*c + b*d*x)*x)/(b^5*d^5*Log[F]^5) - (6*e^2*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4) - (48*e*f*F^(a + b

*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) - (60*f^2*F^(a + b*c + b*d*x)*x^2)/(b^4*d^4*Log[F]^4) + (6*e^2*F^(a + b*c +

b*d*x)*x)/(b^3*d^3*Log[F]^3) + (24*e*f*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) + (20*f^2*F^(a + b*c + b*d*

x)*x^3)/(b^3*d^3*Log[F]^3) - (3*e^2*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) - (8*e*f*F^(a + b*c + b*d*x)*x

^3)/(b^2*d^2*Log[F]^2) - (5*f^2*F^(a + b*c + b*d*x)*x^4)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x^3)/(b

*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^4)/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x^5)/(b*d*Log[F])

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c

}, x] && PolynomialQ[u, x] && LinearQ[v, x] && !$UseGamma === True

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m

*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x

)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] && !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre

eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx &=\int \left (e^2 F^{a+b c+b d x} x^3+2 e f F^{a+b c+b d x} x^4+f^2 F^{a+b c+b d x} x^5\right ) \, dx\\ &=e^2 \int F^{a+b c+b d x} x^3 \, dx+(2 e f) \int F^{a+b c+b d x} x^4 \, dx+f^2 \int F^{a+b c+b d x} x^5 \, dx\\ &=\frac{e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac{2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac{f^2 F^{a+b c+b d x} x^5}{b d \log (F)}-\frac{\left (3 e^2\right ) \int F^{a+b c+b d x} x^2 \, dx}{b d \log (F)}-\frac{(8 e f) \int F^{a+b c+b d x} x^3 \, dx}{b d \log (F)}-\frac{\left (5 f^2\right ) \int F^{a+b c+b d x} x^4 \, dx}{b d \log (F)}\\ &=-\frac{3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac{8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac{5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac{e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac{2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac{f^2 F^{a+b c+b d x} x^5}{b d \log (F)}+\frac{\left (6 e^2\right ) \int F^{a+b c+b d x} x \, dx}{b^2 d^2 \log ^2(F)}+\frac{(24 e f) \int F^{a+b c+b d x} x^2 \, dx}{b^2 d^2 \log ^2(F)}+\frac{\left (20 f^2\right ) \int F^{a+b c+b d x} x^3 \, dx}{b^2 d^2 \log ^2(F)}\\ &=\frac{6 e^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac{24 e f F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}+\frac{20 f^2 F^{a+b c+b d x} x^3}{b^3 d^3 \log ^3(F)}-\frac{3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac{8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac{5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac{e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac{2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac{f^2 F^{a+b c+b d x} x^5}{b d \log (F)}-\frac{\left (6 e^2\right ) \int F^{a+b c+b d x} \, dx}{b^3 d^3 \log ^3(F)}-\frac{(48 e f) \int F^{a+b c+b d x} x \, dx}{b^3 d^3 \log ^3(F)}-\frac{\left (60 f^2\right ) \int F^{a+b c+b d x} x^2 \, dx}{b^3 d^3 \log ^3(F)}\\ &=-\frac{6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac{48 e f F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}-\frac{60 f^2 F^{a+b c+b d x} x^2}{b^4 d^4 \log ^4(F)}+\frac{6 e^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac{24 e f F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}+\frac{20 f^2 F^{a+b c+b d x} x^3}{b^3 d^3 \log ^3(F)}-\frac{3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac{8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac{5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac{e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac{2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac{f^2 F^{a+b c+b d x} x^5}{b d \log (F)}+\frac{(48 e f) \int F^{a+b c+b d x} \, dx}{b^4 d^4 \log ^4(F)}+\frac{\left (120 f^2\right ) \int F^{a+b c+b d x} x \, dx}{b^4 d^4 \log ^4(F)}\\ &=\frac{48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}+\frac{120 f^2 F^{a+b c+b d x} x}{b^5 d^5 \log ^5(F)}-\frac{6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac{48 e f F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}-\frac{60 f^2 F^{a+b c+b d x} x^2}{b^4 d^4 \log ^4(F)}+\frac{6 e^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac{24 e f F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}+\frac{20 f^2 F^{a+b c+b d x} x^3}{b^3 d^3 \log ^3(F)}-\frac{3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac{8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac{5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac{e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac{2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac{f^2 F^{a+b c+b d x} x^5}{b d \log (F)}-\frac{\left (120 f^2\right ) \int F^{a+b c+b d x} \, dx}{b^5 d^5 \log ^5(F)}\\ &=-\frac{120 f^2 F^{a+b c+b d x}}{b^6 d^6 \log ^6(F)}+\frac{48 e f F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}+\frac{120 f^2 F^{a+b c+b d x} x}{b^5 d^5 \log ^5(F)}-\frac{6 e^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac{48 e f F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}-\frac{60 f^2 F^{a+b c+b d x} x^2}{b^4 d^4 \log ^4(F)}+\frac{6 e^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac{24 e f F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}+\frac{20 f^2 F^{a+b c+b d x} x^3}{b^3 d^3 \log ^3(F)}-\frac{3 e^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac{8 e f F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}-\frac{5 f^2 F^{a+b c+b d x} x^4}{b^2 d^2 \log ^2(F)}+\frac{e^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac{2 e f F^{a+b c+b d x} x^4}{b d \log (F)}+\frac{f^2 F^{a+b c+b d x} x^5}{b d \log (F)}\\ \end{align*}

Mathematica [A] time = 0.119706, size = 159, normalized size = 0.38 \[ \frac{F^{a+b (c+d x)} \left (-b^4 d^4 x^2 \log ^4(F) \left (3 e^2+8 e f x+5 f^2 x^2\right )+2 b^3 d^3 x \log ^3(F) \left (3 e^2+12 e f x+10 f^2 x^2\right )-6 b^2 d^2 \log ^2(F) \left (e^2+8 e f x+10 f^2 x^2\right )+b^5 d^5 x^3 \log ^5(F) (e+f x)^2+24 b d f \log (F) (2 e+5 f x)-120 f^2\right )}{b^6 d^6 \log ^6(F)} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[F^(a + b*(c + d*x))*x^3*(e + f*x)^2,x]

[Out]

(F^(a + b*(c + d*x))*(-120*f^2 + 24*b*d*f*(2*e + 5*f*x)*Log[F] - 6*b^2*d^2*(e^2 + 8*e*f*x + 10*f^2*x^2)*Log[F]

^2 + 2*b^3*d^3*x*(3*e^2 + 12*e*f*x + 10*f^2*x^2)*Log[F]^3 - b^4*d^4*x^2*(3*e^2 + 8*e*f*x + 5*f^2*x^2)*Log[F]^4

+ b^5*d^5*x^3*(e + f*x)^2*Log[F]^5))/(b^6*d^6*Log[F]^6)

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Maple [A] time = 0.009, size = 250, normalized size = 0.6 \begin{align*}{\frac{ \left ({f}^{2}{x}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}{b}^{5}{d}^{5}+2\, \left ( \ln \left ( F \right ) \right ) ^{5}{b}^{5}{d}^{5}ef{x}^{4}+ \left ( \ln \left ( F \right ) \right ) ^{5}{b}^{5}{d}^{5}{e}^{2}{x}^{3}-5\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}{f}^{2}{x}^{4}-8\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}ef{x}^{3}-3\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}{e}^{2}{x}^{2}+20\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{f}^{2}{x}^{3}+24\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}ef{x}^{2}+6\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{e}^{2}x-60\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{f}^{2}{x}^{2}-48\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}efx-6\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}+120\,\ln \left ( F \right ) bd{f}^{2}x+48\,fe\ln \left ( F \right ) bd-120\,{f}^{2} \right ){F}^{bdx+bc+a}}{ \left ( \ln \left ( F \right ) \right ) ^{6}{b}^{6}{d}^{6}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x)

[Out]

(f^2*x^5*ln(F)^5*b^5*d^5+2*ln(F)^5*b^5*d^5*e*f*x^4+ln(F)^5*b^5*d^5*e^2*x^3-5*ln(F)^4*b^4*d^4*f^2*x^4-8*ln(F)^4

*b^4*d^4*e*f*x^3-3*ln(F)^4*b^4*d^4*e^2*x^2+20*ln(F)^3*b^3*d^3*f^2*x^3+24*ln(F)^3*b^3*d^3*e*f*x^2+6*ln(F)^3*b^3

*d^3*e^2*x-60*ln(F)^2*b^2*d^2*f^2*x^2-48*ln(F)^2*b^2*d^2*e*f*x-6*ln(F)^2*b^2*d^2*e^2+120*ln(F)*b*d*f^2*x+48*f*

e*ln(F)*b*d-120*f^2)*F^(b*d*x+b*c+a)/ln(F)^6/b^6/d^6

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Maxima [A] time = 1.06091, size = 443, normalized size = 1.07 \begin{align*} \frac{{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} e^{2}}{b^{4} d^{4} \log \left (F\right )^{4}} + \frac{2 \,{\left (F^{b c + a} b^{4} d^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{b c + a} b d x \log \left (F\right ) + 24 \, F^{b c + a}\right )} F^{b d x} e f}{b^{5} d^{5} \log \left (F\right )^{5}} + \frac{{\left (F^{b c + a} b^{5} d^{5} x^{5} \log \left (F\right )^{5} - 5 \, F^{b c + a} b^{4} d^{4} x^{4} \log \left (F\right )^{4} + 20 \, F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 60 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 120 \, F^{b c + a} b d x \log \left (F\right ) - 120 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{6} d^{6} \log \left (F\right )^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x, algorithm="maxima")

[Out]

(F^(b*c + a)*b^3*d^3*x^3*log(F)^3 - 3*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 + 6*F^(b*c + a)*b*d*x*log(F) - 6*F^(b*c

+ a))*F^(b*d*x)*e^2/(b^4*d^4*log(F)^4) + 2*(F^(b*c + a)*b^4*d^4*x^4*log(F)^4 - 4*F^(b*c + a)*b^3*d^3*x^3*log(

F)^3 + 12*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 - 24*F^(b*c + a)*b*d*x*log(F) + 24*F^(b*c + a))*F^(b*d*x)*e*f/(b^5*

d^5*log(F)^5) + (F^(b*c + a)*b^5*d^5*x^5*log(F)^5 - 5*F^(b*c + a)*b^4*d^4*x^4*log(F)^4 + 20*F^(b*c + a)*b^3*d^

3*x^3*log(F)^3 - 60*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 + 120*F^(b*c + a)*b*d*x*log(F) - 120*F^(b*c + a))*F^(b*d*

x)*f^2/(b^6*d^6*log(F)^6)

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Fricas [A] time = 1.55277, size = 493, normalized size = 1.19 \begin{align*} \frac{{\left ({\left (b^{5} d^{5} f^{2} x^{5} + 2 \, b^{5} d^{5} e f x^{4} + b^{5} d^{5} e^{2} x^{3}\right )} \log \left (F\right )^{5} -{\left (5 \, b^{4} d^{4} f^{2} x^{4} + 8 \, b^{4} d^{4} e f x^{3} + 3 \, b^{4} d^{4} e^{2} x^{2}\right )} \log \left (F\right )^{4} + 2 \,{\left (10 \, b^{3} d^{3} f^{2} x^{3} + 12 \, b^{3} d^{3} e f x^{2} + 3 \, b^{3} d^{3} e^{2} x\right )} \log \left (F\right )^{3} - 6 \,{\left (10 \, b^{2} d^{2} f^{2} x^{2} + 8 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 120 \, f^{2} + 24 \,{\left (5 \, b d f^{2} x + 2 \, b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{6} d^{6} \log \left (F\right )^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x, algorithm="fricas")

[Out]

((b^5*d^5*f^2*x^5 + 2*b^5*d^5*e*f*x^4 + b^5*d^5*e^2*x^3)*log(F)^5 - (5*b^4*d^4*f^2*x^4 + 8*b^4*d^4*e*f*x^3 + 3

*b^4*d^4*e^2*x^2)*log(F)^4 + 2*(10*b^3*d^3*f^2*x^3 + 12*b^3*d^3*e*f*x^2 + 3*b^3*d^3*e^2*x)*log(F)^3 - 6*(10*b^

2*d^2*f^2*x^2 + 8*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(F)^2 - 120*f^2 + 24*(5*b*d*f^2*x + 2*b*d*e*f)*log(F))*F^(b*

d*x + b*c + a)/(b^6*d^6*log(F)^6)

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Sympy [A] time = 0.209837, size = 323, normalized size = 0.78 \begin{align*} \begin{cases} \frac{F^{a + b \left (c + d x\right )} \left (b^{5} d^{5} e^{2} x^{3} \log{\left (F \right )}^{5} + 2 b^{5} d^{5} e f x^{4} \log{\left (F \right )}^{5} + b^{5} d^{5} f^{2} x^{5} \log{\left (F \right )}^{5} - 3 b^{4} d^{4} e^{2} x^{2} \log{\left (F \right )}^{4} - 8 b^{4} d^{4} e f x^{3} \log{\left (F \right )}^{4} - 5 b^{4} d^{4} f^{2} x^{4} \log{\left (F \right )}^{4} + 6 b^{3} d^{3} e^{2} x \log{\left (F \right )}^{3} + 24 b^{3} d^{3} e f x^{2} \log{\left (F \right )}^{3} + 20 b^{3} d^{3} f^{2} x^{3} \log{\left (F \right )}^{3} - 6 b^{2} d^{2} e^{2} \log{\left (F \right )}^{2} - 48 b^{2} d^{2} e f x \log{\left (F \right )}^{2} - 60 b^{2} d^{2} f^{2} x^{2} \log{\left (F \right )}^{2} + 48 b d e f \log{\left (F \right )} + 120 b d f^{2} x \log{\left (F \right )} - 120 f^{2}\right )}{b^{6} d^{6} \log{\left (F \right )}^{6}} & \text{for}\: b^{6} d^{6} \log{\left (F \right )}^{6} \neq 0 \\\frac{e^{2} x^{4}}{4} + \frac{2 e f x^{5}}{5} + \frac{f^{2} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c))*x**3*(f*x+e)**2,x)

[Out]

Piecewise((F**(a + b*(c + d*x))*(b**5*d**5*e**2*x**3*log(F)**5 + 2*b**5*d**5*e*f*x**4*log(F)**5 + b**5*d**5*f*

*2*x**5*log(F)**5 - 3*b**4*d**4*e**2*x**2*log(F)**4 - 8*b**4*d**4*e*f*x**3*log(F)**4 - 5*b**4*d**4*f**2*x**4*l

og(F)**4 + 6*b**3*d**3*e**2*x*log(F)**3 + 24*b**3*d**3*e*f*x**2*log(F)**3 + 20*b**3*d**3*f**2*x**3*log(F)**3 -

6*b**2*d**2*e**2*log(F)**2 - 48*b**2*d**2*e*f*x*log(F)**2 - 60*b**2*d**2*f**2*x**2*log(F)**2 + 48*b*d*e*f*log

(F) + 120*b*d*f**2*x*log(F) - 120*f**2)/(b**6*d**6*log(F)**6), Ne(b**6*d**6*log(F)**6, 0)), (e**2*x**4/4 + 2*e

*f*x**5/5 + f**2*x**6/6, True))

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Giac [C] time = 1.85998, size = 13437, normalized size = 32.46 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*x^3*(f*x+e)^2,x, algorithm="giac")

[Out]

((4*(pi^3*b^3*d^3*x^3*sgn(F) - 3*pi*b^3*d^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*x^3 + 3*pi*b^3*d^3*x^3*log

(abs(F))^2 + 6*pi*b^2*d^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*d^2*x^2*log(abs(F)) - 6*pi*b*d*x*sgn(F) + 6*pi*b*d

*x)*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4

*log(abs(F))^3)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*lo

g(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(

F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)^2) - (pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F

))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)*(3*pi^2*b^3*d^3*x^3*log(a

bs(F))*sgn(F) - 3*pi^2*b^3*d^3*x^3*log(abs(F)) + 2*b^3*d^3*x^3*log(abs(F))^3 - 3*pi^2*b^2*d^2*x^2*sgn(F) + 3*p

i^2*b^2*d^2*x^2 - 6*b^2*d^2*x^2*log(abs(F))^2 + 12*b*d*x*log(abs(F)) - 12)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*

d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3

*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(

F))^3)^2))*cos(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi

*a) + ((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^

2 - 2*b^4*d^4*log(abs(F))^4)*(pi^3*b^3*d^3*x^3*sgn(F) - 3*pi*b^3*d^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*x

^3 + 3*pi*b^3*d^3*x^3*log(abs(F))^2 + 6*pi*b^2*d^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*d^2*x^2*log(abs(F)) - 6*p

i*b*d*x*sgn(F) + 6*pi*b*d*x)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi

^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(a

bs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)^2) + 4*(pi^3*b^4*d^4*log(abs(F))*sgn(F)

- pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log(abs(F))^3)*(3*pi^2*b^3*d^3*x^3*

log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*x^3*log(abs(F)) + 2*b^3*d^3*x^3*log(abs(F))^3 - 3*pi^2*b^2*d^2*x^2*sgn(F)

+ 3*pi^2*b^2*d^2*x^2 - 6*b^2*d^2*x^2*log(abs(F))^2 + 12*b*d*x*log(abs(F)) - 12)/((pi^4*b^4*d^4*sgn(F) - 6*pi^2

*b^4*d^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4 + 6*pi^2*b^4*d^4*log(abs(F))^2 - 2*b^4*d^4*log(abs(F))^4)^2 + 16*

(pi^3*b^4*d^4*log(abs(F))*sgn(F) - pi*b^4*d^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*log(abs(F)) + pi*b^4*d^4*log

(abs(F))^3)^2))*sin(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1

/2*pi*a))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)) + 2) - 1/2*I*((8*pi^3*b^3*d^3*x^3*sgn(F) + 24

*I*pi^2*b^3*d^3*x^3*log(abs(F))*sgn(F) - 24*pi*b^3*d^3*x^3*log(abs(F))^2*sgn(F) - 8*pi^3*b^3*d^3*x^3 - 24*I*pi

^2*b^3*d^3*x^3*log(abs(F)) + 24*pi*b^3*d^3*x^3*log(abs(F))^2 + 16*I*b^3*d^3*x^3*log(abs(F))^3 - 24*I*pi^2*b^2*

d^2*x^2*sgn(F) + 48*pi*b^2*d^2*x^2*log(abs(F))*sgn(F) + 24*I*pi^2*b^2*d^2*x^2 - 48*pi*b^2*d^2*x^2*log(abs(F))

- 48*I*b^2*d^2*x^2*log(abs(F))^2 - 48*pi*b*d*x*sgn(F) + 48*pi*b*d*x + 96*I*b*d*x*log(abs(F)) - 96*I)*e^(1/2*I*

pi*b*d*x*sgn(F) - 1/2*I*pi*b*d*x + 1/2*I*pi*b*c*sgn(F) - 1/2*I*pi*b*c + 1/2*I*pi*a*sgn(F) - 1/2*I*pi*a)/(8*pi^

4*b^4*d^4*sgn(F) + 32*I*pi^3*b^4*d^4*log(abs(F))*sgn(F) - 48*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) - 32*I*pi*b^4*d

^4*log(abs(F))^3*sgn(F) - 8*pi^4*b^4*d^4 - 32*I*pi^3*b^4*d^4*log(abs(F)) + 48*pi^2*b^4*d^4*log(abs(F))^2 + 32*

I*pi*b^4*d^4*log(abs(F))^3 - 16*b^4*d^4*log(abs(F))^4) + (8*pi^3*b^3*d^3*x^3*sgn(F) - 24*I*pi^2*b^3*d^3*x^3*lo

g(abs(F))*sgn(F) - 24*pi*b^3*d^3*x^3*log(abs(F))^2*sgn(F) - 8*pi^3*b^3*d^3*x^3 + 24*I*pi^2*b^3*d^3*x^3*log(abs

(F)) + 24*pi*b^3*d^3*x^3*log(abs(F))^2 - 16*I*b^3*d^3*x^3*log(abs(F))^3 + 24*I*pi^2*b^2*d^2*x^2*sgn(F) + 48*pi

*b^2*d^2*x^2*log(abs(F))*sgn(F) - 24*I*pi^2*b^2*d^2*x^2 - 48*pi*b^2*d^2*x^2*log(abs(F)) + 48*I*b^2*d^2*x^2*log

(abs(F))^2 - 48*pi*b*d*x*sgn(F) + 48*pi*b*d*x - 96*I*b*d*x*log(abs(F)) + 96*I)*e^(-1/2*I*pi*b*d*x*sgn(F) + 1/2

*I*pi*b*d*x - 1/2*I*pi*b*c*sgn(F) + 1/2*I*pi*b*c - 1/2*I*pi*a*sgn(F) + 1/2*I*pi*a)/(8*pi^4*b^4*d^4*sgn(F) - 32

*I*pi^3*b^4*d^4*log(abs(F))*sgn(F) - 48*pi^2*b^4*d^4*log(abs(F))^2*sgn(F) + 32*I*pi*b^4*d^4*log(abs(F))^3*sgn(

F) - 8*pi^4*b^4*d^4 + 32*I*pi^3*b^4*d^4*log(abs(F)) + 48*pi^2*b^4*d^4*log(abs(F))^2 - 32*I*pi*b^4*d^4*log(abs(

F))^3 - 16*b^4*d^4*log(abs(F))^4))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)) + 2) - 2*((4*(pi^3*b

^4*d^4*f*x^4*log(abs(F))*sgn(F) - pi*b^4*d^4*f*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*f*x^4*log(abs(F)) + pi*

b^4*d^4*f*x^4*log(abs(F))^3 - pi^3*b^3*d^3*f*x^3*sgn(F) + 3*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) + pi^3*b^3*d

^3*f*x^3 - 3*pi*b^3*d^3*f*x^3*log(abs(F))^2 - 6*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) + 6*pi*b^2*d^2*f*x^2*log(a

bs(F)) + 6*pi*b*d*f*x*sgn(F) - 6*pi*b*d*f*x)*(pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*p

i*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)/((

pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5

+ 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)^2 + (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2

*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs

(F))^5)^2) - (pi^4*b^4*d^4*f*x^4*sgn(F) - 6*pi^2*b^4*d^4*f*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4*f*x^4 + 6*p

i^2*b^4*d^4*f*x^4*log(abs(F))^2 - 2*b^4*d^4*f*x^4*log(abs(F))^4 + 12*pi^2*b^3*d^3*f*x^3*log(abs(F))*sgn(F) - 1

2*pi^2*b^3*d^3*f*x^3*log(abs(F)) + 8*b^3*d^3*f*x^3*log(abs(F))^3 - 12*pi^2*b^2*d^2*f*x^2*sgn(F) + 12*pi^2*b^2*

d^2*f*x^2 - 24*b^2*d^2*f*x^2*log(abs(F))^2 + 48*b*d*f*x*log(abs(F)) - 48*f)*(5*pi^4*b^5*d^5*log(abs(F))*sgn(F)

- 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d

^5*log(abs(F))^5)/((pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sg

n(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)^2 + (5*pi^4*b^5*d^5*log(abs(

F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3

- 2*b^5*d^5*log(abs(F))^5)^2))*cos(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2

*pi*a*sgn(F) + 1/2*pi*a) - ((pi^4*b^4*d^4*f*x^4*sgn(F) - 6*pi^2*b^4*d^4*f*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*

d^4*f*x^4 + 6*pi^2*b^4*d^4*f*x^4*log(abs(F))^2 - 2*b^4*d^4*f*x^4*log(abs(F))^4 + 12*pi^2*b^3*d^3*f*x^3*log(abs

(F))*sgn(F) - 12*pi^2*b^3*d^3*f*x^3*log(abs(F)) + 8*b^3*d^3*f*x^3*log(abs(F))^3 - 12*pi^2*b^2*d^2*f*x^2*sgn(F)

+ 12*pi^2*b^2*d^2*f*x^2 - 24*b^2*d^2*f*x^2*log(abs(F))^2 + 48*b*d*f*x*log(abs(F)) - 48*f)*(pi^5*b^5*d^5*sgn(F

) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*

log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)/((pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*sgn(F) + 5*p

i*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(abs(F))^4)^2

+ (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10*

pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)^2) + 4*(pi^3*b^4*d^4*f*x^4*log(abs(F))*sgn(F) - pi*b^4*d

^4*f*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*f*x^4*log(abs(F)) + pi*b^4*d^4*f*x^4*log(abs(F))^3 - pi^3*b^3*d^3

*f*x^3*sgn(F) + 3*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) + pi^3*b^3*d^3*f*x^3 - 3*pi*b^3*d^3*f*x^3*log(abs(F))^

2 - 6*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) + 6*pi*b^2*d^2*f*x^2*log(abs(F)) + 6*pi*b*d*f*x*sgn(F) - 6*pi*b*d*f*

x)*(5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(abs(F)) + 10

*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)/((pi^5*b^5*d^5*sgn(F) - 10*pi^3*b^5*d^5*log(abs(F))^2*s

gn(F) + 5*pi*b^5*d^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5 + 10*pi^3*b^5*d^5*log(abs(F))^2 - 5*pi*b^5*d^5*log(ab

s(F))^4)^2 + (5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*log(ab

s(F)) + 10*pi^2*b^5*d^5*log(abs(F))^3 - 2*b^5*d^5*log(abs(F))^5)^2))*sin(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x -

1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi*a))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(

abs(F)) + 1) + 1/2*I*((-32*I*pi^4*b^4*d^4*f*x^4*sgn(F) + 128*pi^3*b^4*d^4*f*x^4*log(abs(F))*sgn(F) + 192*I*pi^

2*b^4*d^4*f*x^4*log(abs(F))^2*sgn(F) - 128*pi*b^4*d^4*f*x^4*log(abs(F))^3*sgn(F) + 32*I*pi^4*b^4*d^4*f*x^4 - 1

28*pi^3*b^4*d^4*f*x^4*log(abs(F)) - 192*I*pi^2*b^4*d^4*f*x^4*log(abs(F))^2 + 128*pi*b^4*d^4*f*x^4*log(abs(F))^

3 + 64*I*b^4*d^4*f*x^4*log(abs(F))^4 - 128*pi^3*b^3*d^3*f*x^3*sgn(F) - 384*I*pi^2*b^3*d^3*f*x^3*log(abs(F))*sg

n(F) + 384*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) + 128*pi^3*b^3*d^3*f*x^3 + 384*I*pi^2*b^3*d^3*f*x^3*log(abs(F

)) - 384*pi*b^3*d^3*f*x^3*log(abs(F))^2 - 256*I*b^3*d^3*f*x^3*log(abs(F))^3 + 384*I*pi^2*b^2*d^2*f*x^2*sgn(F)

- 768*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) - 384*I*pi^2*b^2*d^2*f*x^2 + 768*pi*b^2*d^2*f*x^2*log(abs(F)) + 768*

I*b^2*d^2*f*x^2*log(abs(F))^2 + 768*pi*b*d*f*x*sgn(F) - 768*pi*b*d*f*x - 1536*I*b*d*f*x*log(abs(F)) + 1536*I*f

)*e^(1/2*I*pi*b*d*x*sgn(F) - 1/2*I*pi*b*d*x + 1/2*I*pi*b*c*sgn(F) - 1/2*I*pi*b*c + 1/2*I*pi*a*sgn(F) - 1/2*I*p

i*a)/(16*I*pi^5*b^5*d^5*sgn(F) - 80*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 160*I*pi^3*b^5*d^5*log(abs(F))^2*sgn(F)

+ 160*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) + 80*I*pi*b^5*d^5*log(abs(F))^4*sgn(F) - 16*I*pi^5*b^5*d^5 + 80*pi^4*b

^5*d^5*log(abs(F)) + 160*I*pi^3*b^5*d^5*log(abs(F))^2 - 160*pi^2*b^5*d^5*log(abs(F))^3 - 80*I*pi*b^5*d^5*log(a

bs(F))^4 + 32*b^5*d^5*log(abs(F))^5) - (-32*I*pi^4*b^4*d^4*f*x^4*sgn(F) - 128*pi^3*b^4*d^4*f*x^4*log(abs(F))*s

gn(F) + 192*I*pi^2*b^4*d^4*f*x^4*log(abs(F))^2*sgn(F) + 128*pi*b^4*d^4*f*x^4*log(abs(F))^3*sgn(F) + 32*I*pi^4*

b^4*d^4*f*x^4 + 128*pi^3*b^4*d^4*f*x^4*log(abs(F)) - 192*I*pi^2*b^4*d^4*f*x^4*log(abs(F))^2 - 128*pi*b^4*d^4*f

*x^4*log(abs(F))^3 + 64*I*b^4*d^4*f*x^4*log(abs(F))^4 + 128*pi^3*b^3*d^3*f*x^3*sgn(F) - 384*I*pi^2*b^3*d^3*f*x

^3*log(abs(F))*sgn(F) - 384*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) - 128*pi^3*b^3*d^3*f*x^3 + 384*I*pi^2*b^3*d^

3*f*x^3*log(abs(F)) + 384*pi*b^3*d^3*f*x^3*log(abs(F))^2 - 256*I*b^3*d^3*f*x^3*log(abs(F))^3 + 384*I*pi^2*b^2*

d^2*f*x^2*sgn(F) + 768*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) - 384*I*pi^2*b^2*d^2*f*x^2 - 768*pi*b^2*d^2*f*x^2*l

og(abs(F)) + 768*I*b^2*d^2*f*x^2*log(abs(F))^2 - 768*pi*b*d*f*x*sgn(F) + 768*pi*b*d*f*x - 1536*I*b*d*f*x*log(a

bs(F)) + 1536*I*f)*e^(-1/2*I*pi*b*d*x*sgn(F) + 1/2*I*pi*b*d*x - 1/2*I*pi*b*c*sgn(F) + 1/2*I*pi*b*c - 1/2*I*pi*

a*sgn(F) + 1/2*I*pi*a)/(-16*I*pi^5*b^5*d^5*sgn(F) - 80*pi^4*b^5*d^5*log(abs(F))*sgn(F) + 160*I*pi^3*b^5*d^5*lo

g(abs(F))^2*sgn(F) + 160*pi^2*b^5*d^5*log(abs(F))^3*sgn(F) - 80*I*pi*b^5*d^5*log(abs(F))^4*sgn(F) + 16*I*pi^5*

b^5*d^5 + 80*pi^4*b^5*d^5*log(abs(F)) - 160*I*pi^3*b^5*d^5*log(abs(F))^2 - 160*pi^2*b^5*d^5*log(abs(F))^3 + 80

*I*pi*b^5*d^5*log(abs(F))^4 + 32*b^5*d^5*log(abs(F))^5))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)

) + 1) - (((5*pi^4*b^5*d^5*f^2*x^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3*sgn(F) - 5*pi^4*

b^5*d^5*f^2*x^5*log(abs(F)) + 10*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3 - 2*b^5*d^5*f^2*x^5*log(abs(F))^5 - 5*pi^4

*b^4*d^4*f^2*x^4*sgn(F) + 30*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) + 5*pi^4*b^4*d^4*f^2*x^4 - 30*pi^2*b^4*

d^4*f^2*x^4*log(abs(F))^2 + 10*b^4*d^4*f^2*x^4*log(abs(F))^4 - 60*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 60

*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 40*b^3*d^3*f^2*x^3*log(abs(F))^3 + 60*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 60*pi^

2*b^2*d^2*f^2*x^2 + 120*b^2*d^2*f^2*x^2*log(abs(F))^2 - 240*b*d*f^2*x*log(abs(F)) + 240*f^2)*(pi^6*b^6*d^6*sgn

(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^4*b^6

*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6*log(abs(F))^6)/((pi^6*b^6*d^6*sgn(F) - 15*pi^4*

b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^4*b^6*d^6*log(abs(F

))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6*log(abs(F))^6)^2 + 4*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*

pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 3*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b

^6*d^6*log(abs(F))^3 - 3*pi*b^6*d^6*log(abs(F))^5)^2) - 2*(pi^5*b^5*d^5*f^2*x^5*sgn(F) - 10*pi^3*b^5*d^5*f^2*x

^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*f^2*x^5*log(abs(F))^4*sgn(F) - pi^5*b^5*d^5*f^2*x^5 + 10*pi^3*b^5*d^5*f

^2*x^5*log(abs(F))^2 - 5*pi*b^5*d^5*f^2*x^5*log(abs(F))^4 + 20*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) - 20*pi

*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - 20*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) + 20*pi*b^4*d^4*f^2*x^4*log(abs(F)

)^3 - 20*pi^3*b^3*d^3*f^2*x^3*sgn(F) + 60*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) + 20*pi^3*b^3*d^3*f^2*x^3 -

60*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 - 120*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) + 120*pi*b^2*d^2*f^2*x^2*log(a

bs(F)) + 120*pi*b*d*f^2*x*sgn(F) - 120*pi*b*d*f^2*x)*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(

abs(F))^3*sgn(F) + 3*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b^6*d^6*log(abs(F)

)^3 - 3*pi*b^6*d^6*log(abs(F))^5)/((pi^6*b^6*d^6*sgn(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d

^6*log(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^4*b^6*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6

*d^6*log(abs(F))^6)^2 + 4*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 3*pi*b^6

*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b^6*d^6*log(abs(F))^3 - 3*pi*b^6*d^6*log(abs(

F))^5)^2))*cos(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi

*a) - ((pi^5*b^5*d^5*f^2*x^5*sgn(F) - 10*pi^3*b^5*d^5*f^2*x^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*d^5*f^2*x^5*log(

abs(F))^4*sgn(F) - pi^5*b^5*d^5*f^2*x^5 + 10*pi^3*b^5*d^5*f^2*x^5*log(abs(F))^2 - 5*pi*b^5*d^5*f^2*x^5*log(abs

(F))^4 + 20*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) - 20*pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - 20*pi^3*b^4

*d^4*f^2*x^4*log(abs(F)) + 20*pi*b^4*d^4*f^2*x^4*log(abs(F))^3 - 20*pi^3*b^3*d^3*f^2*x^3*sgn(F) + 60*pi*b^3*d^

3*f^2*x^3*log(abs(F))^2*sgn(F) + 20*pi^3*b^3*d^3*f^2*x^3 - 60*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 - 120*pi*b^2*d^

2*f^2*x^2*log(abs(F))*sgn(F) + 120*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 120*pi*b*d*f^2*x*sgn(F) - 120*pi*b*d*f^2*x

)*(pi^6*b^6*d^6*sgn(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - pi^6*b^

6*d^6 + 15*pi^4*b^6*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6*log(abs(F))^6)/((pi^6*b^6*d^

6*sgn(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^

4*b^6*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6*log(abs(F))^6)^2 + 4*(3*pi^5*b^6*d^6*log(a

bs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 3*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(

abs(F)) + 10*pi^3*b^6*d^6*log(abs(F))^3 - 3*pi*b^6*d^6*log(abs(F))^5)^2) + 2*(5*pi^4*b^5*d^5*f^2*x^5*log(abs(F

))*sgn(F) - 10*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3*sgn(F) - 5*pi^4*b^5*d^5*f^2*x^5*log(abs(F)) + 10*pi^2*b^5*d^

5*f^2*x^5*log(abs(F))^3 - 2*b^5*d^5*f^2*x^5*log(abs(F))^5 - 5*pi^4*b^4*d^4*f^2*x^4*sgn(F) + 30*pi^2*b^4*d^4*f^

2*x^4*log(abs(F))^2*sgn(F) + 5*pi^4*b^4*d^4*f^2*x^4 - 30*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 + 10*b^4*d^4*f^2*x

^4*log(abs(F))^4 - 60*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 60*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 40*b^3*d

^3*f^2*x^3*log(abs(F))^3 + 60*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 60*pi^2*b^2*d^2*f^2*x^2 + 120*b^2*d^2*f^2*x^2*log(

abs(F))^2 - 240*b*d*f^2*x*log(abs(F)) + 240*f^2)*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(abs(

F))^3*sgn(F) + 3*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b^6*d^6*log(abs(F))^3

- 3*pi*b^6*d^6*log(abs(F))^5)/((pi^6*b^6*d^6*sgn(F) - 15*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 15*pi^2*b^6*d^6*l

og(abs(F))^4*sgn(F) - pi^6*b^6*d^6 + 15*pi^4*b^6*d^6*log(abs(F))^2 - 15*pi^2*b^6*d^6*log(abs(F))^4 + 2*b^6*d^6

*log(abs(F))^6)^2 + 4*(3*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 3*pi*b^6*d^6

*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*d^6*log(abs(F)) + 10*pi^3*b^6*d^6*log(abs(F))^3 - 3*pi*b^6*d^6*log(abs(F))^

5)^2))*sin(-1/2*pi*b*d*x*sgn(F) + 1/2*pi*b*d*x - 1/2*pi*b*c*sgn(F) + 1/2*pi*b*c - 1/2*pi*a*sgn(F) + 1/2*pi*a))

*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F))) - 1/2*I*((32*pi^5*b^5*d^5*f^2*x^5*sgn(F) + 160*I*pi^4

*b^5*d^5*f^2*x^5*log(abs(F))*sgn(F) - 320*pi^3*b^5*d^5*f^2*x^5*log(abs(F))^2*sgn(F) - 320*I*pi^2*b^5*d^5*f^2*x

^5*log(abs(F))^3*sgn(F) + 160*pi*b^5*d^5*f^2*x^5*log(abs(F))^4*sgn(F) - 32*pi^5*b^5*d^5*f^2*x^5 - 160*I*pi^4*b

^5*d^5*f^2*x^5*log(abs(F)) + 320*pi^3*b^5*d^5*f^2*x^5*log(abs(F))^2 + 320*I*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3

- 160*pi*b^5*d^5*f^2*x^5*log(abs(F))^4 - 64*I*b^5*d^5*f^2*x^5*log(abs(F))^5 - 160*I*pi^4*b^4*d^4*f^2*x^4*sgn(

F) + 640*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) + 960*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) - 640*pi*b^

4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) + 160*I*pi^4*b^4*d^4*f^2*x^4 - 640*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) - 960*I

*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 + 640*pi*b^4*d^4*f^2*x^4*log(abs(F))^3 + 320*I*b^4*d^4*f^2*x^4*log(abs(F))

^4 - 640*pi^3*b^3*d^3*f^2*x^3*sgn(F) - 1920*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 1920*pi*b^3*d^3*f^2*x^

3*log(abs(F))^2*sgn(F) + 640*pi^3*b^3*d^3*f^2*x^3 + 1920*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 1920*pi*b^3*d^3*

f^2*x^3*log(abs(F))^2 - 1280*I*b^3*d^3*f^2*x^3*log(abs(F))^3 + 1920*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 3840*pi*b^

2*d^2*f^2*x^2*log(abs(F))*sgn(F) - 1920*I*pi^2*b^2*d^2*f^2*x^2 + 3840*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 3840*I*

b^2*d^2*f^2*x^2*log(abs(F))^2 + 3840*pi*b*d*f^2*x*sgn(F) - 3840*pi*b*d*f^2*x - 7680*I*b*d*f^2*x*log(abs(F)) +

7680*I*f^2)*e^(1/2*I*pi*b*d*x*sgn(F) - 1/2*I*pi*b*d*x + 1/2*I*pi*b*c*sgn(F) - 1/2*I*pi*b*c + 1/2*I*pi*a*sgn(F)

- 1/2*I*pi*a)/(32*pi^6*b^6*d^6*sgn(F) + 192*I*pi^5*b^6*d^6*log(abs(F))*sgn(F) - 480*pi^4*b^6*d^6*log(abs(F))^

2*sgn(F) - 640*I*pi^3*b^6*d^6*log(abs(F))^3*sgn(F) + 480*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) + 192*I*pi*b^6*d^6*

log(abs(F))^5*sgn(F) - 32*pi^6*b^6*d^6 - 192*I*pi^5*b^6*d^6*log(abs(F)) + 480*pi^4*b^6*d^6*log(abs(F))^2 + 640

*I*pi^3*b^6*d^6*log(abs(F))^3 - 480*pi^2*b^6*d^6*log(abs(F))^4 - 192*I*pi*b^6*d^6*log(abs(F))^5 + 64*b^6*d^6*l

og(abs(F))^6) + (32*pi^5*b^5*d^5*f^2*x^5*sgn(F) - 160*I*pi^4*b^5*d^5*f^2*x^5*log(abs(F))*sgn(F) - 320*pi^3*b^5

*d^5*f^2*x^5*log(abs(F))^2*sgn(F) + 320*I*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3*sgn(F) + 160*pi*b^5*d^5*f^2*x^5*l

og(abs(F))^4*sgn(F) - 32*pi^5*b^5*d^5*f^2*x^5 + 160*I*pi^4*b^5*d^5*f^2*x^5*log(abs(F)) + 320*pi^3*b^5*d^5*f^2*

x^5*log(abs(F))^2 - 320*I*pi^2*b^5*d^5*f^2*x^5*log(abs(F))^3 - 160*pi*b^5*d^5*f^2*x^5*log(abs(F))^4 + 64*I*b^5

*d^5*f^2*x^5*log(abs(F))^5 + 160*I*pi^4*b^4*d^4*f^2*x^4*sgn(F) + 640*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) -

960*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) - 640*pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - 160*I*pi^4*b^

4*d^4*f^2*x^4 - 640*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) + 960*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 + 640*pi*b^4*d

^4*f^2*x^4*log(abs(F))^3 - 320*I*b^4*d^4*f^2*x^4*log(abs(F))^4 - 640*pi^3*b^3*d^3*f^2*x^3*sgn(F) + 1920*I*pi^2

*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 1920*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) + 640*pi^3*b^3*d^3*f^2*x^3

- 1920*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 1920*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 + 1280*I*b^3*d^3*f^2*x^3*log

(abs(F))^3 - 1920*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 3840*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) + 1920*I*pi^2*b^2

*d^2*f^2*x^2 + 3840*pi*b^2*d^2*f^2*x^2*log(abs(F)) - 3840*I*b^2*d^2*f^2*x^2*log(abs(F))^2 + 3840*pi*b*d*f^2*x*

sgn(F) - 3840*pi*b*d*f^2*x + 7680*I*b*d*f^2*x*log(abs(F)) - 7680*I*f^2)*e^(-1/2*I*pi*b*d*x*sgn(F) + 1/2*I*pi*b

*d*x - 1/2*I*pi*b*c*sgn(F) + 1/2*I*pi*b*c - 1/2*I*pi*a*sgn(F) + 1/2*I*pi*a)/(32*pi^6*b^6*d^6*sgn(F) - 192*I*pi

^5*b^6*d^6*log(abs(F))*sgn(F) - 480*pi^4*b^6*d^6*log(abs(F))^2*sgn(F) + 640*I*pi^3*b^6*d^6*log(abs(F))^3*sgn(F

) + 480*pi^2*b^6*d^6*log(abs(F))^4*sgn(F) - 192*I*pi*b^6*d^6*log(abs(F))^5*sgn(F) - 32*pi^6*b^6*d^6 + 192*I*pi

^5*b^6*d^6*log(abs(F)) + 480*pi^4*b^6*d^6*log(abs(F))^2 - 640*I*pi^3*b^6*d^6*log(abs(F))^3 - 480*pi^2*b^6*d^6*

log(abs(F))^4 + 192*I*pi*b^6*d^6*log(abs(F))^5 + 64*b^6*d^6*log(abs(F))^6))*e^(b*d*x*log(abs(F)) + b*c*log(abs

(F)) + a*log(abs(F)))

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3.65 \(\int F^{a+b (c+d x)} x^3 (e+f x)^2 \, dx\)