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

Optimal. Leaf size=229 \[ -\frac {6 g F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {6 g x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {3 g x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e x F^{c (a+b x)}}{b c \log (F)}+\frac {f x^2 F^{c (a+b x)}}{b c \log (F)}+\frac {g x^3 F^{c (a+b x)}}{b c \log (F)} \]

[Out]

-6*F^(c*(b*x+a))*g/b^4/c^4/ln(F)^4+2*f*F^(c*(b*x+a))/b^3/c^3/ln(F)^3+6*F^(c*(b*x+a))*g*x/b^3/c^3/ln(F)^3-e*F^(
c*(b*x+a))/b^2/c^2/ln(F)^2-2*f*F^(c*(b*x+a))*x/b^2/c^2/ln(F)^2-3*F^(c*(b*x+a))*g*x^2/b^2/c^2/ln(F)^2+d*F^(c*(b
*x+a))/b/c/ln(F)+e*F^(c*(b*x+a))*x/b/c/ln(F)+f*F^(c*(b*x+a))*x^2/b/c/ln(F)+F^(c*(b*x+a))*g*x^3/b/c/ln(F)

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Rubi [A]  time = 0.19, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2196, 2194, 2176} \[ -\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {3 g x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {6 g x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {6 g F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e x F^{c (a+b x)}}{b c \log (F)}+\frac {f x^2 F^{c (a+b x)}}{b c \log (F)}+\frac {g x^3 F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*F^(c*(a + b*x))*g)/(b^4*c^4*Log[F]^4) + (2*f*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) + (6*F^(c*(a + b*x))*g*x)
/(b^3*c^3*Log[F]^3) - (e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2) - (2*f*F^(c*(a + b*x))*x)/(b^2*c^2*Log[F]^2) - (3
*F^(c*(a + b*x))*g*x^2)/(b^2*c^2*Log[F]^2) + (d*F^(c*(a + b*x)))/(b*c*Log[F]) + (e*F^(c*(a + b*x))*x)/(b*c*Log
[F]) + (f*F^(c*(a + b*x))*x^2)/(b*c*Log[F]) + (F^(c*(a + b*x))*g*x^3)/(b*c*Log[F])

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]

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 84, normalized size = 0.37 \[ \frac {F^{c (a+b x)} \left (b^3 c^3 \log ^3(F) (d+x (e+x (f+g x)))-b^2 c^2 \log ^2(F) (e+x (2 f+3 g x))+2 b c \log (F) (f+3 g x)-6 g\right )}{b^4 c^4 \log ^4(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(c*(a + b*x))*(-6*g + 2*b*c*(f + 3*g*x)*Log[F] - b^2*c^2*(e + x*(2*f + 3*g*x))*Log[F]^2 + b^3*c^3*(d + x*(e
+ x*(f + g*x)))*Log[F]^3))/(b^4*c^4*Log[F]^4)

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fricas [A]  time = 0.43, size = 122, normalized size = 0.53 \[ \frac {{\left ({\left (b^{3} c^{3} g x^{3} + b^{3} c^{3} f x^{2} + b^{3} c^{3} e x + b^{3} c^{3} d\right )} \log \relax (F)^{3} - {\left (3 \, b^{2} c^{2} g x^{2} + 2 \, b^{2} c^{2} f x + b^{2} c^{2} e\right )} \log \relax (F)^{2} + 2 \, {\left (3 \, b c g x + b c f\right )} \log \relax (F) - 6 \, g\right )} F^{b c x + a c}}{b^{4} c^{4} \log \relax (F)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^3*c^3*g*x^3 + b^3*c^3*f*x^2 + b^3*c^3*e*x + b^3*c^3*d)*log(F)^3 - (3*b^2*c^2*g*x^2 + 2*b^2*c^2*f*x + b^2*c
^2*e)*log(F)^2 + 2*(3*b*c*g*x + b*c*f)*log(F) - 6*g)*F^(b*c*x + a*c)/(b^4*c^4*log(F)^4)

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giac [C]  time = 0.94, size = 4287, normalized size = 18.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*((pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))*(pi*b*c*x*sgn(F) - pi*b*c*x)/((pi^2*b^2*c^2*sgn(F
) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))^2)
+ (pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)*(b*c*x*log(abs(F)) - 1)/((pi^2*b^2*c^2*sgn(F)
- pi^2*b^2*c^2 + 2*b^2*c^2*log(abs(F))^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))^2))*
cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c) + ((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c
^2 + 2*b^2*c^2*log(abs(F))^2)*(pi*b*c*x*sgn(F) - pi*b*c*x)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*lo
g(abs(F))^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))^2) - 4*(pi*b^2*c^2*log(abs(F))*sgn
(F) - pi*b^2*c^2*log(abs(F)))*(b*c*x*log(abs(F)) - 1)/((pi^2*b^2*c^2*sgn(F) - pi^2*b^2*c^2 + 2*b^2*c^2*log(abs
(F))^2)^2 + 4*(pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))^2))*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b
*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)) + 1) - 1/2*I*((2*pi*b*c*x*sgn(F
) - 2*pi*b*c*x - 4*I*b*c*x*log(abs(F)) + 4*I)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F)
- 1/2*I*pi*a*c)/(2*pi^2*b^2*c^2*sgn(F) + 4*I*pi*b^2*c^2*log(abs(F))*sgn(F) - 2*pi^2*b^2*c^2 - 4*I*pi*b^2*c^2*l
og(abs(F)) + 4*b^2*c^2*log(abs(F))^2) + (2*pi*b*c*x*sgn(F) - 2*pi*b*c*x + 4*I*b*c*x*log(abs(F)) - 4*I)*e^(-1/2
*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(2*pi^2*b^2*c^2*sgn(F) - 4*I*pi*b^2*
c^2*log(abs(F))*sgn(F) - 2*pi^2*b^2*c^2 + 4*I*pi*b^2*c^2*log(abs(F)) + 4*b^2*c^2*log(abs(F))^2))*e^(b*c*x*log(
abs(F)) + a*c*log(abs(F)) + 1) - (((3*pi^2*b^3*c^3*g*x^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*g*x^3*log(abs(F))
+ 2*b^3*c^3*g*x^3*log(abs(F))^3 + 3*pi^2*b^3*c^3*f*x^2*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*f*x^2*log(abs(F))
+ 2*b^3*c^3*f*x^2*log(abs(F))^3 + 3*pi^2*b^3*c^3*d*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*d*log(abs(F)) + 2*b^3*c
^3*d*log(abs(F))^3 - 3*pi^2*b^2*c^2*g*x^2*sgn(F) + 3*pi^2*b^2*c^2*g*x^2 - 6*b^2*c^2*g*x^2*log(abs(F))^2 - 2*pi
^2*b^2*c^2*f*x*sgn(F) + 2*pi^2*b^2*c^2*f*x - 4*b^2*c^2*f*x*log(abs(F))^2 + 12*b*c*g*x*log(abs(F)) + 4*b*c*f*lo
g(abs(F)) - 12*g)*(pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*l
og(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4
*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^
4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)^2) - 4*(pi^3*b^3*c^3*g*x^3*s
gn(F) - 3*pi*b^3*c^3*g*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*g*x^3 + 3*pi*b^3*c^3*g*x^3*log(abs(F))^2 + pi^3
*b^3*c^3*f*x^2*sgn(F) - 3*pi*b^3*c^3*f*x^2*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*f*x^2 + 3*pi*b^3*c^3*f*x^2*log(
abs(F))^2 + pi^3*b^3*c^3*d*sgn(F) - 3*pi*b^3*c^3*d*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*d + 3*pi*b^3*c^3*d*log(
abs(F))^2 + 6*pi*b^2*c^2*g*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*c^2*g*x^2*log(abs(F)) + 4*pi*b^2*c^2*f*x*log(abs(
F))*sgn(F) - 4*pi*b^2*c^2*f*x*log(abs(F)) - 6*pi*b*c*g*x*sgn(F) + 6*pi*b*c*g*x - 2*pi*b*c*f*sgn(F) + 2*pi*b*c*
f)*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*
log(abs(F))^3)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log
(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F
) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)^2))*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*
a*c*sgn(F) + 1/2*pi*a*c) - ((pi^3*b^3*c^3*g*x^3*sgn(F) - 3*pi*b^3*c^3*g*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^
3*g*x^3 + 3*pi*b^3*c^3*g*x^3*log(abs(F))^2 + pi^3*b^3*c^3*f*x^2*sgn(F) - 3*pi*b^3*c^3*f*x^2*log(abs(F))^2*sgn(
F) - pi^3*b^3*c^3*f*x^2 + 3*pi*b^3*c^3*f*x^2*log(abs(F))^2 + pi^3*b^3*c^3*d*sgn(F) - 3*pi*b^3*c^3*d*log(abs(F)
)^2*sgn(F) - pi^3*b^3*c^3*d + 3*pi*b^3*c^3*d*log(abs(F))^2 + 6*pi*b^2*c^2*g*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*
c^2*g*x^2*log(abs(F)) + 4*pi*b^2*c^2*f*x*log(abs(F))*sgn(F) - 4*pi*b^2*c^2*f*x*log(abs(F)) - 6*pi*b*c*g*x*sgn(
F) + 6*pi*b*c*g*x - 2*pi*b*c*f*sgn(F) + 2*pi*b*c*f)*(pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F)
- pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c
^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^4*log(abs(F))^4)^2 + 16*(pi^3*
b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F
))^3)^2) + 4*(3*pi^2*b^3*c^3*g*x^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*g*x^3*log(abs(F)) + 2*b^3*c^3*g*x^3*log
(abs(F))^3 + 3*pi^2*b^3*c^3*f*x^2*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*f*x^2*log(abs(F)) + 2*b^3*c^3*f*x^2*log(
abs(F))^3 + 3*pi^2*b^3*c^3*d*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*d*log(abs(F)) + 2*b^3*c^3*d*log(abs(F))^3 - 3
*pi^2*b^2*c^2*g*x^2*sgn(F) + 3*pi^2*b^2*c^2*g*x^2 - 6*b^2*c^2*g*x^2*log(abs(F))^2 - 2*pi^2*b^2*c^2*f*x*sgn(F)
+ 2*pi^2*b^2*c^2*f*x - 4*b^2*c^2*f*x*log(abs(F))^2 + 12*b*c*g*x*log(abs(F)) + 4*b*c*f*log(abs(F)) - 12*g)*(pi^
3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs
(F))^3)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F)
)^2 - 2*b^4*c^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^
3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)^2))*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn
(F) + 1/2*pi*a*c))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 1/2*I*((8*pi^3*b^3*c^3*g*x^3*sgn(F) + 24*I*pi^2*b
^3*c^3*g*x^3*log(abs(F))*sgn(F) - 24*pi*b^3*c^3*g*x^3*log(abs(F))^2*sgn(F) - 8*pi^3*b^3*c^3*g*x^3 - 24*I*pi^2*
b^3*c^3*g*x^3*log(abs(F)) + 24*pi*b^3*c^3*g*x^3*log(abs(F))^2 + 16*I*b^3*c^3*g*x^3*log(abs(F))^3 + 8*pi^3*b^3*
c^3*f*x^2*sgn(F) + 24*I*pi^2*b^3*c^3*f*x^2*log(abs(F))*sgn(F) - 24*pi*b^3*c^3*f*x^2*log(abs(F))^2*sgn(F) - 8*p
i^3*b^3*c^3*f*x^2 - 24*I*pi^2*b^3*c^3*f*x^2*log(abs(F)) + 24*pi*b^3*c^3*f*x^2*log(abs(F))^2 + 16*I*b^3*c^3*f*x
^2*log(abs(F))^3 + 8*pi^3*b^3*c^3*d*sgn(F) + 24*I*pi^2*b^3*c^3*d*log(abs(F))*sgn(F) - 24*pi*b^3*c^3*d*log(abs(
F))^2*sgn(F) - 8*pi^3*b^3*c^3*d - 24*I*pi^2*b^3*c^3*d*log(abs(F)) + 24*pi*b^3*c^3*d*log(abs(F))^2 + 16*I*b^3*c
^3*d*log(abs(F))^3 - 24*I*pi^2*b^2*c^2*g*x^2*sgn(F) + 48*pi*b^2*c^2*g*x^2*log(abs(F))*sgn(F) + 24*I*pi^2*b^2*c
^2*g*x^2 - 48*pi*b^2*c^2*g*x^2*log(abs(F)) - 48*I*b^2*c^2*g*x^2*log(abs(F))^2 - 16*I*pi^2*b^2*c^2*f*x*sgn(F) +
32*pi*b^2*c^2*f*x*log(abs(F))*sgn(F) + 16*I*pi^2*b^2*c^2*f*x - 32*pi*b^2*c^2*f*x*log(abs(F)) - 32*I*b^2*c^2*f
*x*log(abs(F))^2 - 48*pi*b*c*g*x*sgn(F) + 48*pi*b*c*g*x + 96*I*b*c*g*x*log(abs(F)) - 16*pi*b*c*f*sgn(F) + 16*p
i*b*c*f + 32*I*b*c*f*log(abs(F)) - 96*I*g)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1
/2*I*pi*a*c)/(8*pi^4*b^4*c^4*sgn(F) + 32*I*pi^3*b^4*c^4*log(abs(F))*sgn(F) - 48*pi^2*b^4*c^4*log(abs(F))^2*sgn
(F) - 32*I*pi*b^4*c^4*log(abs(F))^3*sgn(F) - 8*pi^4*b^4*c^4 - 32*I*pi^3*b^4*c^4*log(abs(F)) + 48*pi^2*b^4*c^4*
log(abs(F))^2 + 32*I*pi*b^4*c^4*log(abs(F))^3 - 16*b^4*c^4*log(abs(F))^4) + (8*pi^3*b^3*c^3*g*x^3*sgn(F) - 24*
I*pi^2*b^3*c^3*g*x^3*log(abs(F))*sgn(F) - 24*pi*b^3*c^3*g*x^3*log(abs(F))^2*sgn(F) - 8*pi^3*b^3*c^3*g*x^3 + 24
*I*pi^2*b^3*c^3*g*x^3*log(abs(F)) + 24*pi*b^3*c^3*g*x^3*log(abs(F))^2 - 16*I*b^3*c^3*g*x^3*log(abs(F))^3 + 8*p
i^3*b^3*c^3*f*x^2*sgn(F) - 24*I*pi^2*b^3*c^3*f*x^2*log(abs(F))*sgn(F) - 24*pi*b^3*c^3*f*x^2*log(abs(F))^2*sgn(
F) - 8*pi^3*b^3*c^3*f*x^2 + 24*I*pi^2*b^3*c^3*f*x^2*log(abs(F)) + 24*pi*b^3*c^3*f*x^2*log(abs(F))^2 - 16*I*b^3
*c^3*f*x^2*log(abs(F))^3 + 8*pi^3*b^3*c^3*d*sgn(F) - 24*I*pi^2*b^3*c^3*d*log(abs(F))*sgn(F) - 24*pi*b^3*c^3*d*
log(abs(F))^2*sgn(F) - 8*pi^3*b^3*c^3*d + 24*I*pi^2*b^3*c^3*d*log(abs(F)) + 24*pi*b^3*c^3*d*log(abs(F))^2 - 16
*I*b^3*c^3*d*log(abs(F))^3 + 24*I*pi^2*b^2*c^2*g*x^2*sgn(F) + 48*pi*b^2*c^2*g*x^2*log(abs(F))*sgn(F) - 24*I*pi
^2*b^2*c^2*g*x^2 - 48*pi*b^2*c^2*g*x^2*log(abs(F)) + 48*I*b^2*c^2*g*x^2*log(abs(F))^2 + 16*I*pi^2*b^2*c^2*f*x*
sgn(F) + 32*pi*b^2*c^2*f*x*log(abs(F))*sgn(F) - 16*I*pi^2*b^2*c^2*f*x - 32*pi*b^2*c^2*f*x*log(abs(F)) + 32*I*b
^2*c^2*f*x*log(abs(F))^2 - 48*pi*b*c*g*x*sgn(F) + 48*pi*b*c*g*x - 96*I*b*c*g*x*log(abs(F)) - 16*pi*b*c*f*sgn(F
) + 16*pi*b*c*f - 32*I*b*c*f*log(abs(F)) + 96*I*g)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*s
gn(F) + 1/2*I*pi*a*c)/(8*pi^4*b^4*c^4*sgn(F) - 32*I*pi^3*b^4*c^4*log(abs(F))*sgn(F) - 48*pi^2*b^4*c^4*log(abs(
F))^2*sgn(F) + 32*I*pi*b^4*c^4*log(abs(F))^3*sgn(F) - 8*pi^4*b^4*c^4 + 32*I*pi^3*b^4*c^4*log(abs(F)) + 48*pi^2
*b^4*c^4*log(abs(F))^2 - 32*I*pi*b^4*c^4*log(abs(F))^3 - 16*b^4*c^4*log(abs(F))^4))*e^(b*c*x*log(abs(F)) + a*c
*log(abs(F)))

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maple [A]  time = 0.01, size = 138, normalized size = 0.60 \[ \frac {\left (b^{3} c^{3} g \,x^{3} \ln \relax (F )^{3}+b^{3} c^{3} f \,x^{2} \ln \relax (F )^{3}+b^{3} c^{3} e x \ln \relax (F )^{3}+b^{3} c^{3} d \ln \relax (F )^{3}-3 b^{2} c^{2} g \,x^{2} \ln \relax (F )^{2}-2 b^{2} c^{2} f x \ln \relax (F )^{2}-b^{2} c^{2} e \ln \relax (F )^{2}+6 b c g x \ln \relax (F )+2 b c f \ln \relax (F )-6 g \right ) F^{\left (b x +a \right ) c}}{b^{4} c^{4} \ln \relax (F )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(g*x^3*b^3*c^3*ln(F)^3+ln(F)^3*b^3*c^3*f*x^2+ln(F)^3*b^3*c^3*e*x+b^3*c^3*ln(F)^3*d-3*ln(F)^2*b^2*c^2*g*x^2-2*l
n(F)^2*b^2*c^2*f*x-b^2*c^2*ln(F)^2*e+6*ln(F)*b*c*g*x+2*f*b*c*ln(F)-6*g)*F^((b*x+a)*c)/b^4/c^4/ln(F)^4

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maxima [A]  time = 0.85, size = 194, normalized size = 0.85 \[ \frac {F^{b c x + a c} d}{b c \log \relax (F)} + \frac {{\left (F^{a c} b c x \log \relax (F) - F^{a c}\right )} F^{b c x} e}{b^{2} c^{2} \log \relax (F)^{2}} + \frac {{\left (F^{a c} b^{2} c^{2} x^{2} \log \relax (F)^{2} - 2 \, F^{a c} b c x \log \relax (F) + 2 \, F^{a c}\right )} F^{b c x} f}{b^{3} c^{3} \log \relax (F)^{3}} + \frac {{\left (F^{a c} b^{3} c^{3} x^{3} \log \relax (F)^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \relax (F)^{2} + 6 \, F^{a c} b c x \log \relax (F) - 6 \, F^{a c}\right )} F^{b c x} g}{b^{4} c^{4} \log \relax (F)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(b*c*x + a*c)*d/(b*c*log(F)) + (F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b*c*x)*e/(b^2*c^2*log(F)^2) + (F^(a*c)*b^
2*c^2*x^2*log(F)^2 - 2*F^(a*c)*b*c*x*log(F) + 2*F^(a*c))*F^(b*c*x)*f/(b^3*c^3*log(F)^3) + (F^(a*c)*b^3*c^3*x^3
*log(F)^3 - 3*F^(a*c)*b^2*c^2*x^2*log(F)^2 + 6*F^(a*c)*b*c*x*log(F) - 6*F^(a*c))*F^(b*c*x)*g/(b^4*c^4*log(F)^4
)

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mupad [B]  time = 3.54, size = 138, normalized size = 0.60 \[ \frac {F^{a\,c+b\,c\,x}\,\left (g\,b^3\,c^3\,x^3\,{\ln \relax (F)}^3+f\,b^3\,c^3\,x^2\,{\ln \relax (F)}^3+e\,b^3\,c^3\,x\,{\ln \relax (F)}^3+d\,b^3\,c^3\,{\ln \relax (F)}^3-3\,g\,b^2\,c^2\,x^2\,{\ln \relax (F)}^2-2\,f\,b^2\,c^2\,x\,{\ln \relax (F)}^2-e\,b^2\,c^2\,{\ln \relax (F)}^2+6\,g\,b\,c\,x\,\ln \relax (F)+2\,f\,b\,c\,\ln \relax (F)-6\,g\right )}{b^4\,c^4\,{\ln \relax (F)}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(a*c + b*c*x)*(2*b*c*f*log(F) - 6*g + b^3*c^3*d*log(F)^3 - b^2*c^2*e*log(F)^2 + b^3*c^3*f*x^2*log(F)^3 - 3*
b^2*c^2*g*x^2*log(F)^2 + b^3*c^3*g*x^3*log(F)^3 + 6*b*c*g*x*log(F) + b^3*c^3*e*x*log(F)^3 - 2*b^2*c^2*f*x*log(
F)^2))/(b^4*c^4*log(F)^4)

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sympy [A]  time = 0.22, size = 190, normalized size = 0.83 \[ \begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{3} c^{3} d \log {\relax (F )}^{3} + b^{3} c^{3} e x \log {\relax (F )}^{3} + b^{3} c^{3} f x^{2} \log {\relax (F )}^{3} + b^{3} c^{3} g x^{3} \log {\relax (F )}^{3} - b^{2} c^{2} e \log {\relax (F )}^{2} - 2 b^{2} c^{2} f x \log {\relax (F )}^{2} - 3 b^{2} c^{2} g x^{2} \log {\relax (F )}^{2} + 2 b c f \log {\relax (F )} + 6 b c g x \log {\relax (F )} - 6 g\right )}{b^{4} c^{4} \log {\relax (F )}^{4}} & \text {for}\: b^{4} c^{4} \log {\relax (F )}^{4} \neq 0 \\d x + \frac {e x^{2}}{2} + \frac {f x^{3}}{3} + \frac {g x^{4}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((F**(c*(a + b*x))*(b**3*c**3*d*log(F)**3 + b**3*c**3*e*x*log(F)**3 + b**3*c**3*f*x**2*log(F)**3 + b*
*3*c**3*g*x**3*log(F)**3 - b**2*c**2*e*log(F)**2 - 2*b**2*c**2*f*x*log(F)**2 - 3*b**2*c**2*g*x**2*log(F)**2 +
2*b*c*f*log(F) + 6*b*c*g*x*log(F) - 6*g)/(b**4*c**4*log(F)**4), Ne(b**4*c**4*log(F)**4, 0)), (d*x + e*x**2/2 +
f*x**3/3 + g*x**4/4, True))

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