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

Optimal. Leaf size=328 \[ \frac {24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {24 f^2 x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 e f x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {6 e f x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {e^2 x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^4 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

24*f^2*F^(b*d*x+b*c+a)/b^5/d^5/ln(F)^5-12*e*f*F^(b*d*x+b*c+a)/b^4/d^4/ln(F)^4-24*f^2*F^(b*d*x+b*c+a)*x/b^4/d^4
/ln(F)^4+2*e^2*F^(b*d*x+b*c+a)/b^3/d^3/ln(F)^3+12*e*f*F^(b*d*x+b*c+a)*x/b^3/d^3/ln(F)^3+12*f^2*F^(b*d*x+b*c+a)
*x^2/b^3/d^3/ln(F)^3-2*e^2*F^(b*d*x+b*c+a)*x/b^2/d^2/ln(F)^2-6*e*f*F^(b*d*x+b*c+a)*x^2/b^2/d^2/ln(F)^2-4*f^2*F
^(b*d*x+b*c+a)*x^3/b^2/d^2/ln(F)^2+e^2*F^(b*d*x+b*c+a)*x^2/b/d/ln(F)+2*e*f*F^(b*d*x+b*c+a)*x^3/b/d/ln(F)+f^2*F
^(b*d*x+b*c+a)*x^4/b/d/ln(F)

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Rubi [A]  time = 0.53, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2196, 2176, 2194} \[ -\frac {2 e^2 x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {6 e f x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {12 e f x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {4 f^2 x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {12 f^2 x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {24 f^2 x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}+\frac {e^2 x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^4 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*f^2*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) - (12*e*f*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4) - (24*f^2*F^
(a + b*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) + (2*e^2*F^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) + (12*e*f*F^(a + b*c
+ b*d*x)*x)/(b^3*d^3*Log[F]^3) + (12*f^2*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) - (2*e^2*F^(a + b*c + b*d
*x)*x)/(b^2*d^2*Log[F]^2) - (6*e*f*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) - (4*f^2*F^(a + b*c + b*d*x)*x^
3)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x^2)/(b*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^3)/(b*d*Log[
F]) + (f^2*F^(a + b*c + b*d*x)*x^4)/(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^2 (e+f x)^2 \, dx &=\int \left (e^2 F^{a+b c+b d x} x^2+2 e f F^{a+b c+b d x} x^3+f^2 F^{a+b c+b d x} x^4\right ) \, dx\\ &=e^2 \int F^{a+b c+b d x} x^2 \, dx+(2 e f) \int F^{a+b c+b d x} x^3 \, dx+f^2 \int F^{a+b c+b d x} x^4 \, dx\\ &=\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}-\frac {\left (2 e^2\right ) \int F^{a+b c+b d x} x \, dx}{b d \log (F)}-\frac {(6 e f) \int F^{a+b c+b d x} x^2 \, dx}{b d \log (F)}-\frac {\left (4 f^2\right ) \int F^{a+b c+b d x} x^3 \, dx}{b d \log (F)}\\ &=-\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {\left (2 e^2\right ) \int F^{a+b c+b d x} \, dx}{b^2 d^2 \log ^2(F)}+\frac {(12 e f) \int F^{a+b c+b d x} x \, dx}{b^2 d^2 \log ^2(F)}+\frac {\left (12 f^2\right ) \int F^{a+b c+b d x} x^2 \, dx}{b^2 d^2 \log ^2(F)}\\ &=\frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 e f F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}-\frac {(12 e f) \int F^{a+b c+b d x} \, dx}{b^3 d^3 \log ^3(F)}-\frac {\left (24 f^2\right ) \int F^{a+b c+b d x} x \, dx}{b^3 d^3 \log ^3(F)}\\ &=-\frac {12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {24 f^2 F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}+\frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 e f F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}+\frac {\left (24 f^2\right ) \int F^{a+b c+b d x} \, dx}{b^4 d^4 \log ^4(F)}\\ &=\frac {24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac {12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac {24 f^2 F^{a+b c+b d x} x}{b^4 d^4 \log ^4(F)}+\frac {2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {12 e f F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}+\frac {12 f^2 F^{a+b c+b d x} x^2}{b^3 d^3 \log ^3(F)}-\frac {2 e^2 F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {6 e f F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}-\frac {4 f^2 F^{a+b c+b d x} x^3}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^4}{b d \log (F)}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 121, normalized size = 0.37 \[ \frac {F^{a+b (c+d x)} \left (b^4 d^4 x^2 \log ^4(F) (e+f x)^2-2 b^3 d^3 x \log ^3(F) \left (e^2+3 e f x+2 f^2 x^2\right )+2 b^2 d^2 \log ^2(F) \left (e^2+6 e f x+6 f^2 x^2\right )-12 b d f \log (F) (e+2 f x)+24 f^2\right )}{b^5 d^5 \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(a + b*(c + d*x))*(24*f^2 - 12*b*d*f*(e + 2*f*x)*Log[F] + 2*b^2*d^2*(e^2 + 6*e*f*x + 6*f^2*x^2)*Log[F]^2 -
2*b^3*d^3*x*(e^2 + 3*e*f*x + 2*f^2*x^2)*Log[F]^3 + b^4*d^4*x^2*(e + f*x)^2*Log[F]^4))/(b^5*d^5*Log[F]^5)

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fricas [A]  time = 0.42, size = 178, normalized size = 0.54 \[ \frac {{\left ({\left (b^{4} d^{4} f^{2} x^{4} + 2 \, b^{4} d^{4} e f x^{3} + b^{4} d^{4} e^{2} x^{2}\right )} \log \relax (F)^{4} - 2 \, {\left (2 \, b^{3} d^{3} f^{2} x^{3} + 3 \, b^{3} d^{3} e f x^{2} + b^{3} d^{3} e^{2} x\right )} \log \relax (F)^{3} + 2 \, {\left (6 \, b^{2} d^{2} f^{2} x^{2} + 6 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \relax (F)^{2} + 24 \, f^{2} - 12 \, {\left (2 \, b d f^{2} x + b d e f\right )} \log \relax (F)\right )} F^{b d x + b c + a}}{b^{5} d^{5} \log \relax (F)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^4*d^4*f^2*x^4 + 2*b^4*d^4*e*f*x^3 + b^4*d^4*e^2*x^2)*log(F)^4 - 2*(2*b^3*d^3*f^2*x^3 + 3*b^3*d^3*e*f*x^2 +
b^3*d^3*e^2*x)*log(F)^3 + 2*(6*b^2*d^2*f^2*x^2 + 6*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(F)^2 + 24*f^2 - 12*(2*b*d
*f^2*x + b*d*e*f)*log(F))*F^(b*d*x + b*c + a)/(b^5*d^5*log(F)^5)

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giac [C]  time = 1.34, size = 7061, normalized size = 21.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((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^2*b^2*d^2*x^2
*sgn(F) - pi^2*b^2*d^2*x^2 + 2*b^2*d^2*x^2*log(abs(F))^2 - 4*b*d*x*log(abs(F)) + 4)/((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))*s
gn(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(a
bs(F))^2*sgn(F) - pi^3*b^3*d^3 + 3*pi*b^3*d^3*log(abs(F))^2)*(pi*b^2*d^2*x^2*log(abs(F))*sgn(F) - pi*b^2*d^2*x
^2*log(abs(F)) - pi*b*d*x*sgn(F) + pi*b*d*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*x^2*sgn(F) - pi^2*b^2*d^2*x^2 + 2*b^2*d^2*x^2*log(abs(F))^2 - 4*b*d*x*log(abs(F))
+ 4)/((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*x^2*log(abs(F))*
sgn(F) - pi*b^2*d^2*x^2*log(abs(F)) - pi*b*d*x*sgn(F) + pi*b*d*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))*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*
((4*I*pi^2*b^2*d^2*x^2*sgn(F) - 8*pi*b^2*d^2*x^2*log(abs(F))*sgn(F) - 4*I*pi^2*b^2*d^2*x^2 + 8*pi*b^2*d^2*x^2*
log(abs(F)) + 8*I*b^2*d^2*x^2*log(abs(F))^2 + 8*pi*b*d*x*sgn(F) - 8*pi*b*d*x - 16*I*b*d*x*log(abs(F)) + 16*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)/(-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) - (4*I*
pi^2*b^2*d^2*x^2*sgn(F) + 8*pi*b^2*d^2*x^2*log(abs(F))*sgn(F) - 4*I*pi^2*b^2*d^2*x^2 - 8*pi*b^2*d^2*x^2*log(ab
s(F)) + 8*I*b^2*d^2*x^2*log(abs(F))^2 - 8*pi*b*d*x*sgn(F) + 8*pi*b*d*x - 16*I*b*d*x*log(abs(F)) + 16*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)/(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))*e^(b*d*x*log
(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)) + 2) - 2*(((3*pi^2*b^3*d^3*f*x^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^
3*f*x^3*log(abs(F)) + 2*b^3*d^3*f*x^3*log(abs(F))^3 - 3*pi^2*b^2*d^2*f*x^2*sgn(F) + 3*pi^2*b^2*d^2*f*x^2 - 6*b
^2*d^2*f*x^2*log(abs(F))^2 + 12*b*d*f*x*log(abs(F)) - 12*f)*(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 + 1
6*(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*l
og(abs(F))^3)^2) - 4*(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)*(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*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)*(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*s
gn(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*x^3*log(abs(F))*sgn(F) -
3*pi^2*b^3*d^3*f*x^3*log(abs(F)) + 2*b^3*d^3*f*x^3*log(abs(F))^3 - 3*pi^2*b^2*d^2*f*x^2*sgn(F) + 3*pi^2*b^2*d
^2*f*x^2 - 6*b^2*d^2*f*x^2*log(abs(F))^2 + 12*b*d*f*x*log(abs(F)) - 12*f)*(pi^3*b^4*d^4*log(abs(F))*sgn(F) - p
i*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) - 1/2*I*((16*pi^3*b^3*d^3*f*x^3*sg
n(F) + 48*I*pi^2*b^3*d^3*f*x^3*log(abs(F))*sgn(F) - 48*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) - 16*pi^3*b^3*d^3
*f*x^3 - 48*I*pi^2*b^3*d^3*f*x^3*log(abs(F)) + 48*pi*b^3*d^3*f*x^3*log(abs(F))^2 + 32*I*b^3*d^3*f*x^3*log(abs(
F))^3 - 48*I*pi^2*b^2*d^2*f*x^2*sgn(F) + 96*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) + 48*I*pi^2*b^2*d^2*f*x^2 - 96
*pi*b^2*d^2*f*x^2*log(abs(F)) - 96*I*b^2*d^2*f*x^2*log(abs(F))^2 - 96*pi*b*d*f*x*sgn(F) + 96*pi*b*d*f*x + 192*
I*b*d*f*x*log(abs(F)) - 192*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)/(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) + (16*pi^3*b^
3*d^3*f*x^3*sgn(F) - 48*I*pi^2*b^3*d^3*f*x^3*log(abs(F))*sgn(F) - 48*pi*b^3*d^3*f*x^3*log(abs(F))^2*sgn(F) - 1
6*pi^3*b^3*d^3*f*x^3 + 48*I*pi^2*b^3*d^3*f*x^3*log(abs(F)) + 48*pi*b^3*d^3*f*x^3*log(abs(F))^2 - 32*I*b^3*d^3*
f*x^3*log(abs(F))^3 + 48*I*pi^2*b^2*d^2*f*x^2*sgn(F) + 96*pi*b^2*d^2*f*x^2*log(abs(F))*sgn(F) - 48*I*pi^2*b^2*
d^2*f*x^2 - 96*pi*b^2*d^2*f*x^2*log(abs(F)) + 96*I*b^2*d^2*f*x^2*log(abs(F))^2 - 96*pi*b*d*f*x*sgn(F) + 96*pi*
b*d*f*x - 192*I*b*d*f*x*log(abs(F)) + 192*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)/(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)) + 1) - ((4*(pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F)
- pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*f^2*x^4*log(abs(F)) + pi*b^4*d^4*f^2*x^4*log(abs(F))^
3 - 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^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*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*l
og(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^2*x^4*sgn(F) - 6*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) - pi^4*b^4*d^4*f^2*x^4 + 6*pi^2*
b^4*d^4*f^2*x^4*log(abs(F))^2 - 2*b^4*d^4*f^2*x^4*log(abs(F))^4 + 12*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) -
12*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) + 8*b^3*d^3*f^2*x^3*log(abs(F))^3 - 12*pi^2*b^2*d^2*f^2*x^2*sgn(F) + 12*p
i^2*b^2*d^2*f^2*x^2 - 24*b^2*d^2*f^2*x^2*log(abs(F))^2 + 48*b*d*f^2*x*log(abs(F)) - 48*f^2)*(5*pi^4*b^5*d^5*lo
g(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*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))*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^2*x^4*sgn(F) - 6*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^
2*sgn(F) - pi^4*b^4*d^4*f^2*x^4 + 6*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 - 2*b^4*d^4*f^2*x^4*log(abs(F))^4 + 12*
pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) - 12*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) + 8*b^3*d^3*f^2*x^3*log(abs(F))^
3 - 12*pi^2*b^2*d^2*f^2*x^2*sgn(F) + 12*pi^2*b^2*d^2*f^2*x^2 - 24*b^2*d^2*f^2*x^2*log(abs(F))^2 + 48*b*d*f^2*x
*log(abs(F)) - 48*f^2)*(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*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) + 4*(pi^3*b
^4*d^4*f^2*x^4*log(abs(F))*sgn(F) - pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) - pi^3*b^4*d^4*f^2*x^4*log(abs(F))
+ pi*b^4*d^4*f^2*x^4*log(abs(F))^3 - 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)*(5*pi^4*b^5*d^5*log(abs(F))*sgn(F) - 10*p
i^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*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))*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*s
gn(F) + 1/2*pi*a))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F))) + 1/2*I*((-16*I*pi^4*b^4*d^4*f^2*x^
4*sgn(F) + 64*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F) + 96*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) - 64*pi
*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) + 16*I*pi^4*b^4*d^4*f^2*x^4 - 64*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) - 96*I
*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 + 64*pi*b^4*d^4*f^2*x^4*log(abs(F))^3 + 32*I*b^4*d^4*f^2*x^4*log(abs(F))^4
- 64*pi^3*b^3*d^3*f^2*x^3*sgn(F) - 192*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F))*sgn(F) + 192*pi*b^3*d^3*f^2*x^3*log
(abs(F))^2*sgn(F) + 64*pi^3*b^3*d^3*f^2*x^3 + 192*I*pi^2*b^3*d^3*f^2*x^3*log(abs(F)) - 192*pi*b^3*d^3*f^2*x^3*
log(abs(F))^2 - 128*I*b^3*d^3*f^2*x^3*log(abs(F))^3 + 192*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 384*pi*b^2*d^2*f^2*x
^2*log(abs(F))*sgn(F) - 192*I*pi^2*b^2*d^2*f^2*x^2 + 384*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 384*I*b^2*d^2*f^2*x^
2*log(abs(F))^2 + 384*pi*b*d*f^2*x*sgn(F) - 384*pi*b*d*f^2*x - 768*I*b*d*f^2*x*log(abs(F)) + 768*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)/(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*l
og(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) - (-16*I*pi^4*b^4*d^4*f^2*x^4*sgn(F) - 64*pi^3*b^4*d^4*f^2*x^4*log(abs(F))*sgn(F)
+ 96*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2*sgn(F) + 64*pi*b^4*d^4*f^2*x^4*log(abs(F))^3*sgn(F) + 16*I*pi^4*b^4
*d^4*f^2*x^4 + 64*pi^3*b^4*d^4*f^2*x^4*log(abs(F)) - 96*I*pi^2*b^4*d^4*f^2*x^4*log(abs(F))^2 - 64*pi*b^4*d^4*f
^2*x^4*log(abs(F))^3 + 32*I*b^4*d^4*f^2*x^4*log(abs(F))^4 + 64*pi^3*b^3*d^3*f^2*x^3*sgn(F) - 192*I*pi^2*b^3*d^
3*f^2*x^3*log(abs(F))*sgn(F) - 192*pi*b^3*d^3*f^2*x^3*log(abs(F))^2*sgn(F) - 64*pi^3*b^3*d^3*f^2*x^3 + 192*I*p
i^2*b^3*d^3*f^2*x^3*log(abs(F)) + 192*pi*b^3*d^3*f^2*x^3*log(abs(F))^2 - 128*I*b^3*d^3*f^2*x^3*log(abs(F))^3 +
192*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) + 384*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) - 192*I*pi^2*b^2*d^2*f^2*x^2 -
384*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 384*I*b^2*d^2*f^2*x^2*log(abs(F))^2 - 384*pi*b*d*f^2*x*sgn(F) + 384*pi*b*
d*f^2*x - 768*I*b*d*f^2*x*log(abs(F)) + 768*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*s
gn(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*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*l
og(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)))

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maple [A]  time = 0.01, size = 197, normalized size = 0.60 \[ \frac {\left (b^{4} d^{4} f^{2} x^{4} \ln \relax (F )^{4}+2 b^{4} d^{4} e f \,x^{3} \ln \relax (F )^{4}+b^{4} d^{4} e^{2} x^{2} \ln \relax (F )^{4}-4 b^{3} d^{3} f^{2} x^{3} \ln \relax (F )^{3}-6 b^{3} d^{3} e f \,x^{2} \ln \relax (F )^{3}-2 b^{3} d^{3} e^{2} x \ln \relax (F )^{3}+12 b^{2} d^{2} f^{2} x^{2} \ln \relax (F )^{2}+12 b^{2} d^{2} e f x \ln \relax (F )^{2}+2 b^{2} d^{2} e^{2} \ln \relax (F )^{2}-24 b d \,f^{2} x \ln \relax (F )-12 b d e f \ln \relax (F )+24 f^{2}\right ) F^{b d x +b c +a}}{b^{5} d^{5} \ln \relax (F )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^4*d^4*f^2*x^4*ln(F)^4+2*b^4*d^4*e*f*x^3*ln(F)^4+b^4*d^4*e^2*x^2*ln(F)^4-4*b^3*d^3*f^2*x^3*ln(F)^3-6*b^3*d^3
*e*f*x^2*ln(F)^3-2*b^3*d^3*e^2*x*ln(F)^3+12*b^2*d^2*f^2*x^2*ln(F)^2+12*b^2*d^2*e*f*x*ln(F)^2+2*b^2*d^2*e^2*ln(
F)^2-24*b*d*f^2*x*ln(F)-12*b*d*e*f*ln(F)+24*f^2)*F^(b*d*x+b*c+a)/ln(F)^5/b^5/d^5

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maxima [A]  time = 0.72, size = 262, normalized size = 0.80 \[ \frac {{\left (F^{b c + a} b^{2} d^{2} x^{2} \log \relax (F)^{2} - 2 \, F^{b c + a} b d x \log \relax (F) + 2 \, F^{b c + a}\right )} F^{b d x} e^{2}}{b^{3} d^{3} \log \relax (F)^{3}} + \frac {2 \, {\left (F^{b c + a} b^{3} d^{3} x^{3} \log \relax (F)^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \relax (F)^{2} + 6 \, F^{b c + a} b d x \log \relax (F) - 6 \, F^{b c + a}\right )} F^{b d x} e f}{b^{4} d^{4} \log \relax (F)^{4}} + \frac {{\left (F^{b c + a} b^{4} d^{4} x^{4} \log \relax (F)^{4} - 4 \, F^{b c + a} b^{3} d^{3} x^{3} \log \relax (F)^{3} + 12 \, F^{b c + a} b^{2} d^{2} x^{2} \log \relax (F)^{2} - 24 \, F^{b c + a} b d x \log \relax (F) + 24 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{5} d^{5} \log \relax (F)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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^2/(b^3*d^3*log(F)^
3) + 2*(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*f/(b^4*d^4*log(F)^4) + (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)*f^2/
(b^5*d^5*log(F)^5)

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mupad [B]  time = 3.56, size = 196, normalized size = 0.60 \[ \frac {F^{a+b\,c+b\,d\,x}\,\left (b^4\,d^4\,e^2\,x^2\,{\ln \relax (F)}^4+2\,b^4\,d^4\,e\,f\,x^3\,{\ln \relax (F)}^4+b^4\,d^4\,f^2\,x^4\,{\ln \relax (F)}^4-2\,b^3\,d^3\,e^2\,x\,{\ln \relax (F)}^3-6\,b^3\,d^3\,e\,f\,x^2\,{\ln \relax (F)}^3-4\,b^3\,d^3\,f^2\,x^3\,{\ln \relax (F)}^3+2\,b^2\,d^2\,e^2\,{\ln \relax (F)}^2+12\,b^2\,d^2\,e\,f\,x\,{\ln \relax (F)}^2+12\,b^2\,d^2\,f^2\,x^2\,{\ln \relax (F)}^2-12\,b\,d\,e\,f\,\ln \relax (F)-24\,b\,d\,f^2\,x\,\ln \relax (F)+24\,f^2\right )}{b^5\,d^5\,{\ln \relax (F)}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(a + b*c + b*d*x)*(24*f^2 + 2*b^2*d^2*e^2*log(F)^2 - 24*b*d*f^2*x*log(F) - 2*b^3*d^3*e^2*x*log(F)^3 + b^4*d
^4*e^2*x^2*log(F)^4 + 12*b^2*d^2*f^2*x^2*log(F)^2 - 4*b^3*d^3*f^2*x^3*log(F)^3 + b^4*d^4*f^2*x^4*log(F)^4 - 12
*b*d*e*f*log(F) + 12*b^2*d^2*e*f*x*log(F)^2 - 6*b^3*d^3*e*f*x^2*log(F)^3 + 2*b^4*d^4*e*f*x^3*log(F)^4))/(b^5*d
^5*log(F)^5)

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sympy [A]  time = 0.28, size = 260, normalized size = 0.79 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )} \left (b^{4} d^{4} e^{2} x^{2} \log {\relax (F )}^{4} + 2 b^{4} d^{4} e f x^{3} \log {\relax (F )}^{4} + b^{4} d^{4} f^{2} x^{4} \log {\relax (F )}^{4} - 2 b^{3} d^{3} e^{2} x \log {\relax (F )}^{3} - 6 b^{3} d^{3} e f x^{2} \log {\relax (F )}^{3} - 4 b^{3} d^{3} f^{2} x^{3} \log {\relax (F )}^{3} + 2 b^{2} d^{2} e^{2} \log {\relax (F )}^{2} + 12 b^{2} d^{2} e f x \log {\relax (F )}^{2} + 12 b^{2} d^{2} f^{2} x^{2} \log {\relax (F )}^{2} - 12 b d e f \log {\relax (F )} - 24 b d f^{2} x \log {\relax (F )} + 24 f^{2}\right )}{b^{5} d^{5} \log {\relax (F )}^{5}} & \text {for}\: b^{5} d^{5} \log {\relax (F )}^{5} \neq 0 \\\frac {e^{2} x^{3}}{3} + \frac {e f x^{4}}{2} + \frac {f^{2} x^{5}}{5} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((F**(a + b*(c + d*x))*(b**4*d**4*e**2*x**2*log(F)**4 + 2*b**4*d**4*e*f*x**3*log(F)**4 + b**4*d**4*f*
*2*x**4*log(F)**4 - 2*b**3*d**3*e**2*x*log(F)**3 - 6*b**3*d**3*e*f*x**2*log(F)**3 - 4*b**3*d**3*f**2*x**3*log(
F)**3 + 2*b**2*d**2*e**2*log(F)**2 + 12*b**2*d**2*e*f*x*log(F)**2 + 12*b**2*d**2*f**2*x**2*log(F)**2 - 12*b*d*
e*f*log(F) - 24*b*d*f**2*x*log(F) + 24*f**2)/(b**5*d**5*log(F)**5), Ne(b**5*d**5*log(F)**5, 0)), (e**2*x**3/3
+ e*f*x**4/2 + f**2*x**5/5, True))

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