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

Optimal. Leaf size=242 \[ -\frac {6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {e^2 x F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^3 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.36, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2196, 2176, 2194} \[ -\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {3 f^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {6 f^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {e^2 x F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^3 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.15, size = 91, normalized size = 0.38 \[ \frac {F^{a+b (c+d x)} \left (b^3 d^3 x \log ^3(F) (e+f x)^2-b^2 d^2 \log ^2(F) \left (e^2+4 e f x+3 f^2 x^2\right )+2 b d f \log (F) (2 e+3 f x)-6 f^2\right )}{b^4 d^4 \log ^4(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 132, normalized size = 0.55 \[ \frac {{\left ({\left (b^{3} d^{3} f^{2} x^{3} + 2 \, b^{3} d^{3} e f x^{2} + b^{3} d^{3} e^{2} x\right )} \log \relax (F)^{3} - {\left (3 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \relax (F)^{2} - 6 \, f^{2} + 2 \, {\left (3 \, b d f^{2} x + 2 \, b d e f\right )} \log \relax (F)\right )} F^{b d x + b c + a}}{b^{4} d^{4} \log \relax (F)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [C]  time = 1.06, size = 4696, normalized size = 19.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 144, normalized size = 0.60 \[ \frac {\left (b^{3} d^{3} f^{2} x^{3} \ln \relax (F )^{3}+2 b^{3} d^{3} e f \,x^{2} \ln \relax (F )^{3}+b^{3} d^{3} e^{2} x \ln \relax (F )^{3}-3 b^{2} d^{2} f^{2} x^{2} \ln \relax (F )^{2}-4 b^{2} d^{2} e f x \ln \relax (F )^{2}-b^{2} d^{2} e^{2} \ln \relax (F )^{2}+6 b d \,f^{2} x \ln \relax (F )+4 b d e f \ln \relax (F )-6 f^{2}\right ) F^{b d x +b c +a}}{b^{4} d^{4} \ln \relax (F )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 196, normalized size = 0.81 \[ \frac {{\left (F^{b c + a} b d x \log \relax (F) - F^{b c + a}\right )} F^{b d x} e^{2}}{b^{2} d^{2} \log \relax (F)^{2}} + \frac {2 \, {\left (F^{b c + a} b^{2} d^{2} x^{2} \log \relax (F)^{2} - 2 \, F^{b c + a} b d x \log \relax (F) + 2 \, F^{b c + a}\right )} F^{b d x} e f}{b^{3} d^{3} \log \relax (F)^{3}} + \frac {{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \relax (F)^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \relax (F)^{2} + 6 \, F^{b c + a} b d x \log \relax (F) - 6 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{4} d^{4} \log \relax (F)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.52, size = 143, normalized size = 0.59 \[ \frac {F^{a+b\,c+b\,d\,x}\,\left (b^3\,d^3\,e^2\,x\,{\ln \relax (F)}^3+2\,b^3\,d^3\,e\,f\,x^2\,{\ln \relax (F)}^3+b^3\,d^3\,f^2\,x^3\,{\ln \relax (F)}^3-b^2\,d^2\,e^2\,{\ln \relax (F)}^2-4\,b^2\,d^2\,e\,f\,x\,{\ln \relax (F)}^2-3\,b^2\,d^2\,f^2\,x^2\,{\ln \relax (F)}^2+4\,b\,d\,e\,f\,\ln \relax (F)+6\,b\,d\,f^2\,x\,\ln \relax (F)-6\,f^2\right )}{b^4\,d^4\,{\ln \relax (F)}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.23, size = 199, normalized size = 0.82 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )} \left (b^{3} d^{3} e^{2} x \log {\relax (F )}^{3} + 2 b^{3} d^{3} e f x^{2} \log {\relax (F )}^{3} + b^{3} d^{3} f^{2} x^{3} \log {\relax (F )}^{3} - b^{2} d^{2} e^{2} \log {\relax (F )}^{2} - 4 b^{2} d^{2} e f x \log {\relax (F )}^{2} - 3 b^{2} d^{2} f^{2} x^{2} \log {\relax (F )}^{2} + 4 b d e f \log {\relax (F )} + 6 b d f^{2} x \log {\relax (F )} - 6 f^{2}\right )}{b^{4} d^{4} \log {\relax (F )}^{4}} & \text {for}\: b^{4} d^{4} \log {\relax (F )}^{4} \neq 0 \\\frac {e^{2} x^{2}}{2} + \frac {2 e f x^{3}}{3} + \frac {f^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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