3.4 \(\int F^{c (a+b x)} (d+e x)^2 \, dx\)
Optimal. Leaf size=79 \[ -\frac{2 e (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{2 e^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac{(d+e x)^2 F^{c (a+b x)}}{b c \log (F)} \]
[Out]
(2*e^2*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) - (2*e*F^(c*(a + b*x))*(d + e*x))/(b^2*c^2*Log[F]^2) + (F^(c*(a + b
*x))*(d + e*x)^2)/(b*c*Log[F])
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Rubi [A] time = 0.042787, antiderivative size = 79, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used =
{2176, 2194} \[ -\frac{2 e (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{2 e^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac{(d+e x)^2 F^{c (a+b x)}}{b c \log (F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(c*(a + b*x))*(d + e*x)^2,x]
[Out]
(2*e^2*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) - (2*e*F^(c*(a + b*x))*(d + e*x))/(b^2*c^2*Log[F]^2) + (F^(c*(a + b
*x))*(d + e*x)^2)/(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]
Rubi steps
\begin{align*} \int F^{c (a+b x)} (d+e x)^2 \, dx &=\frac{F^{c (a+b x)} (d+e x)^2}{b c \log (F)}-\frac{(2 e) \int F^{c (a+b x)} (d+e x) \, dx}{b c \log (F)}\\ &=-\frac{2 e F^{c (a+b x)} (d+e x)}{b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^2}{b c \log (F)}+\frac{\left (2 e^2\right ) \int F^{c (a+b x)} \, dx}{b^2 c^2 \log ^2(F)}\\ &=\frac{2 e^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{2 e F^{c (a+b x)} (d+e x)}{b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^2}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0660184, size = 56, normalized size = 0.71 \[ \frac{F^{c (a+b x)} \left (b^2 c^2 \log ^2(F) (d+e x)^2-2 b c e \log (F) (d+e x)+2 e^2\right )}{b^3 c^3 \log ^3(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(c*(a + b*x))*(d + e*x)^2,x]
[Out]
(F^(c*(a + b*x))*(2*e^2 - 2*b*c*e*(d + e*x)*Log[F] + b^2*c^2*(d + e*x)^2*Log[F]^2))/(b^3*c^3*Log[F]^3)
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Maple [A] time = 0.007, size = 91, normalized size = 1.2 \begin{align*}{\frac{ \left ({e}^{2}{x}^{2}{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}dex+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}-2\,\ln \left ( F \right ) bc{e}^{2}x-2\,\ln \left ( F \right ) bced+2\,{e}^{2} \right ){F}^{c \left ( bx+a \right ) }}{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(b*x+a))*(e*x+d)^2,x)
[Out]
(e^2*x^2*b^2*c^2*ln(F)^2+2*ln(F)^2*b^2*c^2*d*e*x+b^2*c^2*ln(F)^2*d^2-2*ln(F)*b*c*e^2*x-2*ln(F)*b*c*e*d+2*e^2)*
F^(c*(b*x+a))/b^3/c^3/ln(F)^3
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Maxima [A] time = 1.00751, size = 166, normalized size = 2.1 \begin{align*} \frac{F^{b c x + a c} d^{2}}{b c \log \left (F\right )} + \frac{2 \,{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac{{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^2,x, algorithm="maxima")
[Out]
F^(b*c*x + a*c)*d^2/(b*c*log(F)) + 2*(F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b*c*x)*d*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)*e^2/(b^3*c^3*log(F)^3)
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Fricas [A] time = 1.54573, size = 186, normalized size = 2.35 \begin{align*} \frac{{\left ({\left (b^{2} c^{2} e^{2} x^{2} + 2 \, b^{2} c^{2} d e x + b^{2} c^{2} d^{2}\right )} \log \left (F\right )^{2} + 2 \, e^{2} - 2 \,{\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{3} c^{3} \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^2,x, algorithm="fricas")
[Out]
((b^2*c^2*e^2*x^2 + 2*b^2*c^2*d*e*x + b^2*c^2*d^2)*log(F)^2 + 2*e^2 - 2*(b*c*e^2*x + b*c*d*e)*log(F))*F^(b*c*x
+ a*c)/(b^3*c^3*log(F)^3)
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Sympy [A] time = 0.235673, size = 133, normalized size = 1.68 \begin{align*} \begin{cases} \frac{F^{c \left (a + b x\right )} \left (b^{2} c^{2} d^{2} \log{\left (F \right )}^{2} + 2 b^{2} c^{2} d e x \log{\left (F \right )}^{2} + b^{2} c^{2} e^{2} x^{2} \log{\left (F \right )}^{2} - 2 b c d e \log{\left (F \right )} - 2 b c e^{2} x \log{\left (F \right )} + 2 e^{2}\right )}{b^{3} c^{3} \log{\left (F \right )}^{3}} & \text{for}\: b^{3} c^{3} \log{\left (F \right )}^{3} \neq 0 \\d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(c*(b*x+a))*(e*x+d)**2,x)
[Out]
Piecewise((F**(c*(a + b*x))*(b**2*c**2*d**2*log(F)**2 + 2*b**2*c**2*d*e*x*log(F)**2 + b**2*c**2*e**2*x**2*log(
F)**2 - 2*b*c*d*e*log(F) - 2*b*c*e**2*x*log(F) + 2*e**2)/(b**3*c**3*log(F)**3), Ne(b**3*c**3*log(F)**3, 0)), (
d**2*x + d*e*x**2 + e**2*x**3/3, True))
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Giac [C] time = 1.82994, size = 3367, normalized size = 42.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^2,x, algorithm="giac")
[Out]
(((3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*log(abs(F)) + 2*b^3*c^3*log(abs(F))^3)*(pi^2*b^2*c^2*x^2
*sgn(F) - pi^2*b^2*c^2*x^2 + 2*b^2*c^2*x^2*log(abs(F))^2 - 4*b*c*x*log(abs(F)) + 4)/((pi^3*b^3*c^3*sgn(F) - 3*
pi*b^3*c^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)^2 + (3*pi^2*b^3*c^3*log(abs(F))*s
gn(F) - 3*pi^2*b^3*c^3*log(abs(F)) + 2*b^3*c^3*log(abs(F))^3)^2) - 2*(pi^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(a
bs(F))^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)*(pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - pi*b^2*c^2*x
^2*log(abs(F)) - pi*b*c*x*sgn(F) + pi*b*c*x)/((pi^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(abs(F))^2*sgn(F) - pi^3*
b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)^2 + (3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*log(abs(F)) + 2*
b^3*c^3*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*sgn(F) - 3*pi*b^3*c^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)*(pi^2*b^2*c^2*
x^2*sgn(F) - pi^2*b^2*c^2*x^2 + 2*b^2*c^2*x^2*log(abs(F))^2 - 4*b*c*x*log(abs(F)) + 4)/((pi^3*b^3*c^3*sgn(F) -
3*pi*b^3*c^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3 + 3*pi*b^3*c^3*log(abs(F))^2)^2 + (3*pi^2*b^3*c^3*log(abs(F)
)*sgn(F) - 3*pi^2*b^3*c^3*log(abs(F)) + 2*b^3*c^3*log(abs(F))^3)^2) + 2*(3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3
*pi^2*b^3*c^3*log(abs(F)) + 2*b^3*c^3*log(abs(F))^3)*(pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - pi*b^2*c^2*x^2*log(a
bs(F)) - pi*b*c*x*sgn(F) + pi*b*c*x)/((pi^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3
+ 3*pi*b^3*c^3*log(abs(F))^2)^2 + (3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*log(abs(F)) + 2*b^3*c^3*
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)) + 2) + 1/2*I*((4*I*pi^2*b^2*c^2*x^2*sgn(F) - 8*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - 4*I*
pi^2*b^2*c^2*x^2 + 8*pi*b^2*c^2*x^2*log(abs(F)) + 8*I*b^2*c^2*x^2*log(abs(F))^2 + 8*pi*b*c*x*sgn(F) - 8*pi*b*c
*x - 16*I*b*c*x*log(abs(F)) + 16*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)/(-4*I*pi^3*b^3*c^3*sgn(F) + 12*pi^2*b^3*c^3*log(abs(F))*sgn(F) + 12*I*pi*b^3*c^3*log(abs(F))^2*sgn(F) +
4*I*pi^3*b^3*c^3 - 12*pi^2*b^3*c^3*log(abs(F)) - 12*I*pi*b^3*c^3*log(abs(F))^2 + 8*b^3*c^3*log(abs(F))^3) - (4
*I*pi^2*b^2*c^2*x^2*sgn(F) + 8*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - 4*I*pi^2*b^2*c^2*x^2 - 8*pi*b^2*c^2*x^2*log
(abs(F)) + 8*I*b^2*c^2*x^2*log(abs(F))^2 - 8*pi*b*c*x*sgn(F) + 8*pi*b*c*x - 16*I*b*c*x*log(abs(F)) + 16*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)/(4*I*pi^3*b^3*c^3*sgn(F) + 12*pi
^2*b^3*c^3*log(abs(F))*sgn(F) - 12*I*pi*b^3*c^3*log(abs(F))^2*sgn(F) - 4*I*pi^3*b^3*c^3 - 12*pi^2*b^3*c^3*log(
abs(F)) + 12*I*pi*b^3*c^3*log(abs(F))^2 + 8*b^3*c^3*log(abs(F))^3))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)) + 2
) + 2*(2*((pi*b^2*c^2*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))*(pi*b*c*d*x*sgn(F) - pi*b*c*d*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(ab
s(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*d*x*log(abs(F)) - d)/((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(a
bs(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*d*x*sgn(F) - pi*b*c*d*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) - 4*(pi*b^2*c^2
*log(abs(F))*sgn(F) - pi*b^2*c^2*log(abs(F)))*(b*c*d*x*log(abs(F)) - d)/((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*
((4*pi*b*c*d*x*sgn(F) - 4*pi*b*c*d*x - 8*I*b*c*d*x*log(abs(F)) + 8*I*d)*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) + (4*pi*b*c*d*x*sgn(F) - 4*pi*b*c*d*x + 8*I*b*
c*d*x*log(abs(F)) - 8*I*d)*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) + 2*(2*b*c*d^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)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2)
- (pi*b*c*sgn(F) - pi*b*c)*d^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)/(4*b
^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 1/2*I*(-2*I*d^2*
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)/(I*pi*b*c*sgn(F) - I*pi*b*c +
2*b*c*log(abs(F))) + 2*I*d^2*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)/
(-I*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F))))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)))