3.68 \(\int F^{a+b (c+d x)} (e+f x)^2 \, dx\)
Optimal. Leaf size=85 \[ -\frac{2 f (e+f x) F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{2 f^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac{(e+f x)^2 F^{a+b c+b d x}}{b d \log (F)} \]
[Out]
(2*f^2*F^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) - (2*f*F^(a + b*c + b*d*x)*(e + f*x))/(b^2*d^2*Log[F]^2) + (F^(
a + b*c + b*d*x)*(e + f*x)^2)/(b*d*Log[F])
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Rubi [A] time = 0.120364, antiderivative size = 85, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used =
{2187, 2176, 2194} \[ -\frac{2 f (e+f x) F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{2 f^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac{(e+f x)^2 F^{a+b c+b d x}}{b d \log (F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(a + b*(c + d*x))*(e + f*x)^2,x]
[Out]
(2*f^2*F^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) - (2*f*F^(a + b*c + b*d*x)*(e + f*x))/(b^2*d^2*Log[F]^2) + (F^(
a + b*c + b*d*x)*(e + f*x)^2)/(b*d*Log[F])
Rule 2187
Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*(u_)^(m_.), x_Symbol] :> Int[NormalizePowerOfLinear[u, x]^
m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ
[u, x] && !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) && IntegerQ[m]
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)} (e+f x)^2 \, dx &=\int F^{a+b c+b d x} (e+f x)^2 \, dx\\ &=\frac{F^{a+b c+b d x} (e+f x)^2}{b d \log (F)}-\frac{(2 f) \int F^{a+b c+b d x} (e+f x) \, dx}{b d \log (F)}\\ &=-\frac{2 f F^{a+b c+b d x} (e+f x)}{b^2 d^2 \log ^2(F)}+\frac{F^{a+b c+b d x} (e+f x)^2}{b d \log (F)}+\frac{\left (2 f^2\right ) \int F^{a+b c+b d x} \, dx}{b^2 d^2 \log ^2(F)}\\ &=\frac{2 f^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{2 f F^{a+b c+b d x} (e+f x)}{b^2 d^2 \log ^2(F)}+\frac{F^{a+b c+b d x} (e+f x)^2}{b d \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0689011, size = 58, normalized size = 0.68 \[ \frac{F^{a+b (c+d x)} \left (b^2 d^2 \log ^2(F) (e+f x)^2-2 b d f \log (F) (e+f x)+2 f^2\right )}{b^3 d^3 \log ^3(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(a + b*(c + d*x))*(e + f*x)^2,x]
[Out]
(F^(a + b*(c + d*x))*(2*f^2 - 2*b*d*f*(e + f*x)*Log[F] + b^2*d^2*(e + f*x)^2*Log[F]^2))/(b^3*d^3*Log[F]^3)
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Maple [A] time = 0.007, size = 93, normalized size = 1.1 \begin{align*}{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{f}^{2}{x}^{2}+2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}efx+ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}-2\,\ln \left ( F \right ) bd{f}^{2}x-2\,fe\ln \left ( F \right ) bd+2\,{f}^{2} \right ){F}^{bdx+bc+a}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(a+b*(d*x+c))*(f*x+e)^2,x)
[Out]
(ln(F)^2*b^2*d^2*f^2*x^2+2*ln(F)^2*b^2*d^2*e*f*x+ln(F)^2*b^2*d^2*e^2-2*ln(F)*b*d*f^2*x-2*f*e*ln(F)*b*d+2*f^2)*
F^(b*d*x+b*c+a)/b^3/d^3/ln(F)^3
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Maxima [A] time = 1.02178, size = 181, normalized size = 2.13 \begin{align*} \frac{F^{b d x + b c + a} e^{2}}{b d \log \left (F\right )} + \frac{2 \,{\left (F^{b c + a} b d x \log \left (F\right ) - F^{b c + a}\right )} F^{b d x} e f}{b^{2} d^{2} \log \left (F\right )^{2}} + \frac{{\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{3} d^{3} \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c))*(f*x+e)^2,x, algorithm="maxima")
[Out]
F^(b*d*x + b*c + a)*e^2/(b*d*log(F)) + 2*(F^(b*c + a)*b*d*x*log(F) - F^(b*c + a))*F^(b*d*x)*e*f/(b^2*d^2*log(F
)^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)*f^2/(b^3*d^3*
log(F)^3)
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Fricas [A] time = 1.48697, size = 192, normalized size = 2.26 \begin{align*} \frac{{\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} + 2 \, f^{2} - 2 \,{\left (b d f^{2} x + b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{3} d^{3} \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c))*(f*x+e)^2,x, algorithm="fricas")
[Out]
((b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(F)^2 + 2*f^2 - 2*(b*d*f^2*x + b*d*e*f)*log(F))*F^(b*d*x
+ b*c + a)/(b^3*d^3*log(F)^3)
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Sympy [A] time = 0.155241, size = 134, normalized size = 1.58 \begin{align*} \begin{cases} \frac{F^{a + b \left (c + d x\right )} \left (b^{2} d^{2} e^{2} \log{\left (F \right )}^{2} + 2 b^{2} d^{2} e f x \log{\left (F \right )}^{2} + b^{2} d^{2} f^{2} x^{2} \log{\left (F \right )}^{2} - 2 b d e f \log{\left (F \right )} - 2 b d f^{2} x \log{\left (F \right )} + 2 f^{2}\right )}{b^{3} d^{3} \log{\left (F \right )}^{3}} & \text{for}\: b^{3} d^{3} \log{\left (F \right )}^{3} \neq 0 \\e^{2} x + e f x^{2} + \frac{f^{2} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(a+b*(d*x+c))*(f*x+e)**2,x)
[Out]
Piecewise((F**(a + b*(c + d*x))*(b**2*d**2*e**2*log(F)**2 + 2*b**2*d**2*e*f*x*log(F)**2 + b**2*d**2*f**2*x**2*
log(F)**2 - 2*b*d*e*f*log(F) - 2*b*d*f**2*x*log(F) + 2*f**2)/(b**3*d**3*log(F)**3), Ne(b**3*d**3*log(F)**3, 0)
), (e**2*x + e*f*x**2 + f**2*x**3/3, True))
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Giac [C] time = 1.38049, size = 3708, normalized size = 43.62 \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))*(f*x+e)^2,x, algorithm="giac")
[Out]
2*(2*b*d*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
)*log(abs(F))/(4*b^2*d^2*log(abs(F))^2 + (pi*b*d*sgn(F) - pi*b*d)^2) - (pi*b*d*sgn(F) - pi*b*d)*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)/(4*b^2*d^2*log(abs(F)
)^2 + (pi*b*d*sgn(F) - pi*b*d)^2))*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)) + 2) - 1/2*I*(-2*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
)/(I*pi*b*d*sgn(F) - I*pi*b*d + 2*b*d*log(abs(F))) + 2*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)/(-I*pi*b*d*sgn(F) + I*pi*b*d + 2*b*d*log(abs(F)))
)*e^(b*d*x*log(abs(F)) + b*c*log(abs(F)) + a*log(abs(F)) + 2) + 2*(2*((pi*b^2*d^2*log(abs(F))*sgn(F) - pi*b^2*
d^2*log(abs(F)))*(pi*b*d*f*x*sgn(F) - pi*b*d*f*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*f*x*log(abs(F)) - f)/((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*f*x*sgn(F) - pi*b*d*f*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*f*x*log(abs(F)) - f)/((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*l
og(abs(F)) + 1) - 1/2*I*((4*pi*b*d*f*x*sgn(F) - 4*pi*b*d*f*x - 8*I*b*d*f*x*log(abs(F)) + 8*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)/(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) + (4*pi*b*d*f*x*sgn(F) - 4*pi*b*d*f*x + 8*I*b*d*f*x*log(abs(F)) - 8*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)/(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)) + 1) - ((2*(pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) - pi
*b^2*d^2*f^2*x^2*log(abs(F)) - pi*b*d*f^2*x*sgn(F) + pi*b*d*f^2*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)/((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*lo
g(abs(F)) + 2*b^3*d^3*log(abs(F))^3)^2) - (pi^2*b^2*d^2*f^2*x^2*sgn(F) - pi^2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*f^2*
x^2*log(abs(F))^2 - 4*b*d*f^2*x*log(abs(F)) + 4*f^2)*(3*pi^2*b^3*d^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*log(a
bs(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))*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*f^2*x^2*sgn(F) - pi^2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*f^2*x^2*log(abs(F))^2 - 4*b*d*f^2*
x*log(abs(F)) + 4*f^2)*(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^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*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) - pi*b^2*d^2*f^2*x^2*log(abs(F)) - pi*b*d*f^2*x*sgn(F) + pi*b*d*f^2*x)*
(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(ab
s(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))) + 1/2*I*((4*I*pi^2*b^2*d^2*f^2*x^2*sgn(F) - 8*pi*b^2*d^2*f^2*x^2*log(abs(F))*sgn(F) - 4*I*pi^2*b^
2*d^2*f^2*x^2 + 8*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 8*I*b^2*d^2*f^2*x^2*log(abs(F))^2 + 8*pi*b*d*f^2*x*sgn(F) -
8*pi*b*d*f^2*x - 16*I*b*d*f^2*x*log(abs(F)) + 16*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)/(-4*I*pi^3*b^3*d^3*sgn(F) + 12*pi^2*b^3*d^3*log(ab
s(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*f^2*x^2*sgn(F) + 8*pi*b^2*d^2*f^2*x^2*lo
g(abs(F))*sgn(F) - 4*I*pi^2*b^2*d^2*f^2*x^2 - 8*pi*b^2*d^2*f^2*x^2*log(abs(F)) + 8*I*b^2*d^2*f^2*x^2*log(abs(F
))^2 - 8*pi*b*d*f^2*x*sgn(F) + 8*pi*b*d*f^2*x - 16*I*b*d*f^2*x*log(abs(F)) + 16*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)/(4*I*pi^3*b^3*d^3*s
gn(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*lo
g(abs(F)) + a*log(abs(F)))