3.3 \(\int F^{c (a+b x)} (d+e x)^3 \, dx\)
Optimal. Leaf size=110 \[ -\frac {6 e^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {6 e^2 (d+e x) F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {3 e (d+e x)^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)} \]
[Out]
-6*e^3*F^(c*(b*x+a))/b^4/c^4/ln(F)^4+6*e^2*F^(c*(b*x+a))*(e*x+d)/b^3/c^3/ln(F)^3-3*e*F^(c*(b*x+a))*(e*x+d)^2/b
^2/c^2/ln(F)^2+F^(c*(b*x+a))*(e*x+d)^3/b/c/ln(F)
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Rubi [A] time = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used =
{2176, 2194} \[ \frac {6 e^2 (d+e x) F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {3 e (d+e x)^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {6 e^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(c*(a + b*x))*(d + e*x)^3,x]
[Out]
(-6*e^3*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4) + (6*e^2*F^(c*(a + b*x))*(d + e*x))/(b^3*c^3*Log[F]^3) - (3*e*F^(c
*(a + b*x))*(d + e*x)^2)/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*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]
Rubi steps
\begin {align*} \int F^{c (a+b x)} (d+e x)^3 \, dx &=\frac {F^{c (a+b x)} (d+e x)^3}{b c \log (F)}-\frac {(3 e) \int F^{c (a+b x)} (d+e x)^2 \, dx}{b c \log (F)}\\ &=-\frac {3 e F^{c (a+b x)} (d+e x)^2}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^3}{b c \log (F)}+\frac {\left (6 e^2\right ) \int F^{c (a+b x)} (d+e x) \, dx}{b^2 c^2 \log ^2(F)}\\ &=\frac {6 e^2 F^{c (a+b x)} (d+e x)}{b^3 c^3 \log ^3(F)}-\frac {3 e F^{c (a+b x)} (d+e x)^2}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^3}{b c \log (F)}-\frac {\left (6 e^3\right ) \int F^{c (a+b x)} \, dx}{b^3 c^3 \log ^3(F)}\\ &=-\frac {6 e^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {6 e^2 F^{c (a+b x)} (d+e x)}{b^3 c^3 \log ^3(F)}-\frac {3 e F^{c (a+b x)} (d+e x)^2}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^3}{b c \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 78, normalized size = 0.71 \[ \frac {F^{c (a+b x)} \left (b^3 c^3 \log ^3(F) (d+e x)^3-3 b^2 c^2 e \log ^2(F) (d+e x)^2+6 b c e^2 \log (F) (d+e x)-6 e^3\right )}{b^4 c^4 \log ^4(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(c*(a + b*x))*(d + e*x)^3,x]
[Out]
(F^(c*(a + b*x))*(-6*e^3 + 6*b*c*e^2*(d + e*x)*Log[F] - 3*b^2*c^2*e*(d + e*x)^2*Log[F]^2 + b^3*c^3*(d + e*x)^3
*Log[F]^3))/(b^4*c^4*Log[F]^4)
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fricas [A] time = 0.59, size = 147, normalized size = 1.34 \[ \frac {{\left ({\left (b^{3} c^{3} e^{3} x^{3} + 3 \, b^{3} c^{3} d e^{2} x^{2} + 3 \, b^{3} c^{3} d^{2} e x + b^{3} c^{3} d^{3}\right )} \log \relax (F)^{3} - 6 \, e^{3} - 3 \, {\left (b^{2} c^{2} e^{3} x^{2} + 2 \, b^{2} c^{2} d e^{2} x + b^{2} c^{2} d^{2} e\right )} \log \relax (F)^{2} + 6 \, {\left (b c e^{3} x + b c d e^{2}\right )} \log \relax (F)\right )} F^{b c x + a c}}{b^{4} c^{4} \log \relax (F)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^3,x, algorithm="fricas")
[Out]
((b^3*c^3*e^3*x^3 + 3*b^3*c^3*d*e^2*x^2 + 3*b^3*c^3*d^2*e*x + b^3*c^3*d^3)*log(F)^3 - 6*e^3 - 3*(b^2*c^2*e^3*x
^2 + 2*b^2*c^2*d*e^2*x + b^2*c^2*d^2*e)*log(F)^2 + 6*(b*c*e^3*x + b*c*d*e^2)*log(F))*F^(b*c*x + a*c)/(b^4*c^4*
log(F)^4)
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giac [C] time = 1.57, size = 4693, normalized size = 42.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^3,x, algorithm="giac")
[Out]
((4*(pi^3*b^3*c^3*x^3*sgn(F) - 3*pi*b^3*c^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*x^3 + 3*pi*b^3*c^3*x^3*log
(abs(F))^2 + 6*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*c^2*x^2*log(abs(F)) - 6*pi*b*c*x*sgn(F) + 6*pi*b*c
*x)*(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*lo
g(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) - (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)*(3*pi^2*b^3*c^3*x^3*log(a
bs(F))*sgn(F) - 3*pi^2*b^3*c^3*x^3*log(abs(F)) + 2*b^3*c^3*x^3*log(abs(F))^3 - 3*pi^2*b^2*c^2*x^2*sgn(F) + 3*p
i^2*b^2*c^2*x^2 - 6*b^2*c^2*x^2*log(abs(F))^2 + 12*b*c*x*log(abs(F)) - 12)/((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^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^3*b^3*c^3*x^3*sgn(F) - 3*pi*b^3*c^3*x^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*x^3 + 3*pi*b^3*c^3*x^3*log(abs
(F))^2 + 6*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - 6*pi*b^2*c^2*x^2*log(abs(F)) - 6*pi*b*c*x*sgn(F) + 6*pi*b*c*x)/
((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^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log(abs(F))^3*s
gn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)*(3*pi^2*b^3*c^3*x^3*log(abs(F))*sgn(F) - 3*pi^2*b
^3*c^3*x^3*log(abs(F)) + 2*b^3*c^3*x^3*log(abs(F))^3 - 3*pi^2*b^2*c^2*x^2*sgn(F) + 3*pi^2*b^2*c^2*x^2 - 6*b^2*
c^2*x^2*log(abs(F))^2 + 12*b*c*x*log(abs(F)) - 12)/((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)) + 3) - 1/
2*I*((8*pi^3*b^3*c^3*x^3*sgn(F) + 24*I*pi^2*b^3*c^3*x^3*log(abs(F))*sgn(F) - 24*pi*b^3*c^3*x^3*log(abs(F))^2*s
gn(F) - 8*pi^3*b^3*c^3*x^3 - 24*I*pi^2*b^3*c^3*x^3*log(abs(F)) + 24*pi*b^3*c^3*x^3*log(abs(F))^2 + 16*I*b^3*c^
3*x^3*log(abs(F))^3 - 24*I*pi^2*b^2*c^2*x^2*sgn(F) + 48*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) + 24*I*pi^2*b^2*c^2*
x^2 - 48*pi*b^2*c^2*x^2*log(abs(F)) - 48*I*b^2*c^2*x^2*log(abs(F))^2 - 48*pi*b*c*x*sgn(F) + 48*pi*b*c*x + 96*I
*b*c*x*log(abs(F)) - 96*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)/(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*x^3*sgn(F) - 24*I*pi^2*b^3*c^3*x^3
*log(abs(F))*sgn(F) - 24*pi*b^3*c^3*x^3*log(abs(F))^2*sgn(F) - 8*pi^3*b^3*c^3*x^3 + 24*I*pi^2*b^3*c^3*x^3*log(
abs(F)) + 24*pi*b^3*c^3*x^3*log(abs(F))^2 - 16*I*b^3*c^3*x^3*log(abs(F))^3 + 24*I*pi^2*b^2*c^2*x^2*sgn(F) + 48
*pi*b^2*c^2*x^2*log(abs(F))*sgn(F) - 24*I*pi^2*b^2*c^2*x^2 - 48*pi*b^2*c^2*x^2*log(abs(F)) + 48*I*b^2*c^2*x^2*
log(abs(F))^2 - 48*pi*b*c*x*sgn(F) + 48*pi*b*c*x - 96*I*b*c*x*log(abs(F)) + 96*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)/(8*pi^4*b^4*c^4*sgn(F) - 32*I*pi^3*b^4*c^4*log(abs(F))*sg
n(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)) + 3) + 3*(((pi^2*b^2*c^2*d*x^2*sgn(F) - pi^2*b^2*c^2*d*x^2 + 2*b^2
*c^2*d*x^2*log(abs(F))^2 - 4*b*c*d*x*log(abs(F)) + 4*d)*(3*pi^2*b^3*c^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*lo
g(abs(F)) + 2*b^3*c^3*log(abs(F))^3)/((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*(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*l
og(abs(F))^2)*(pi*b^2*c^2*d*x^2*log(abs(F))*sgn(F) - pi*b^2*c^2*d*x^2*log(abs(F)) - pi*b*c*d*x*sgn(F) + pi*b*c
*d*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*d*x^2*sgn(F) - pi^2*b^2*c^2*d*x^2 + 2*b
^2*c^2*d*x^2*log(abs(F))^2 - 4*b*c*d*x*log(abs(F)) + 4*d)/((pi^3*b^3*c^3*sgn(F) - 3*pi*b^3*c^3*log(abs(F))^2*s
gn(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*d*x^2*log(abs(F))*sgn(F) - pi*b^2*c^2*d*x^2*log(abs(F)) - pi*b*c*d*x*sgn(F
) + pi*b*c*d*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(ab
s(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))*si
n(-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*((12*I*pi^2*b^2*c^2*d*x^2*sgn(F) - 24*pi*b^2*c^2*d*x^2*log(abs(F))*sgn(F) - 12*I*pi^2*b^2*c^2*d
*x^2 + 24*pi*b^2*c^2*d*x^2*log(abs(F)) + 24*I*b^2*c^2*d*x^2*log(abs(F))^2 + 24*pi*b*c*d*x*sgn(F) - 24*pi*b*c*d
*x - 48*I*b*c*d*x*log(abs(F)) + 48*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)/(-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)
- (12*I*pi^2*b^2*c^2*d*x^2*sgn(F) + 24*pi*b^2*c^2*d*x^2*log(abs(F))*sgn(F) - 12*I*pi^2*b^2*c^2*d*x^2 - 24*pi*b
^2*c^2*d*x^2*log(abs(F)) + 24*I*b^2*c^2*d*x^2*log(abs(F))^2 - 24*pi*b*c*d*x*sgn(F) + 24*pi*b*c*d*x - 48*I*b*c*
d*x*log(abs(F)) + 48*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)/(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) + 3*(2*((pi*b*c*d^2*x*sgn(F) - pi*b*c*d^2*x)*(pi*b^2*c^2*log(abs(F))*sgn(F) - p
i*b^2*c^2*log(abs(F)))/((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(a
bs(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*d^2*x*log(abs(F)) - d^2)/((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*d^2*x*sgn(F) - pi*b*c*d
^2*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) - p
i*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^2*x*log(abs(F))
- d^2)/((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*((6*pi*b*c*d^2*x*sgn(F) - 6*pi*b*c*d^2*x - 12*I*b*c*d^2*x*log(abs
(F)) + 12*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)/(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(ab
s(F))^2) + (6*pi*b*c*d^2*x*sgn(F) - 6*pi*b*c*d^2*x + 12*I*b*c*d^2*x*log(abs(F)) - 12*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)/(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^3*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)*l
og(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^3*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^3*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^3*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)))
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maple [A] time = 0.01, size = 165, normalized size = 1.50 \[ \frac {\left (b^{3} c^{3} e^{3} x^{3} \ln \relax (F )^{3}+3 b^{3} c^{3} d \,e^{2} x^{2} \ln \relax (F )^{3}+3 b^{3} c^{3} d^{2} e x \ln \relax (F )^{3}+b^{3} c^{3} d^{3} \ln \relax (F )^{3}-3 b^{2} c^{2} e^{3} x^{2} \ln \relax (F )^{2}-6 b^{2} c^{2} d \,e^{2} x \ln \relax (F )^{2}-3 b^{2} c^{2} d^{2} e \ln \relax (F )^{2}+6 b c \,e^{3} x \ln \relax (F )+6 b c d \,e^{2} \ln \relax (F )-6 e^{3}\right ) F^{\left (b x +a \right ) c}}{b^{4} c^{4} \ln \relax (F )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^((b*x+a)*c)*(e*x+d)^3,x)
[Out]
(e^3*x^3*b^3*c^3*ln(F)^3+3*ln(F)^3*b^3*c^3*d*e^2*x^2+3*ln(F)^3*b^3*c^3*d^2*e*x+b^3*c^3*ln(F)^3*d^3-3*ln(F)^2*b
^2*c^2*e^3*x^2-6*ln(F)^2*b^2*c^2*d*e^2*x-3*b^2*c^2*ln(F)^2*e*d^2+6*ln(F)*b*c*e^3*x+6*d*e^2*b*c*ln(F)-6*e^3)*F^
((b*x+a)*c)/b^4/c^4/ln(F)^4
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maxima [A] time = 0.49, size = 206, normalized size = 1.87 \[ \frac {F^{b c x + a c} d^{3}}{b c \log \relax (F)} + \frac {3 \, {\left (F^{a c} b c x \log \relax (F) - F^{a c}\right )} F^{b c x} d^{2} e}{b^{2} c^{2} \log \relax (F)^{2}} + \frac {3 \, {\left (F^{a c} b^{2} c^{2} x^{2} \log \relax (F)^{2} - 2 \, F^{a c} b c x \log \relax (F) + 2 \, F^{a c}\right )} F^{b c x} d e^{2}}{b^{3} c^{3} \log \relax (F)^{3}} + \frac {{\left (F^{a c} b^{3} c^{3} x^{3} \log \relax (F)^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \relax (F)^{2} + 6 \, F^{a c} b c x \log \relax (F) - 6 \, F^{a c}\right )} F^{b c x} e^{3}}{b^{4} c^{4} \log \relax (F)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(e*x+d)^3,x, algorithm="maxima")
[Out]
F^(b*c*x + a*c)*d^3/(b*c*log(F)) + 3*(F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b*c*x)*d^2*e/(b^2*c^2*log(F)^2) + 3*(
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)*d*e^2/(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)*e^3/(
b^4*c^4*log(F)^4)
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mupad [B] time = 3.56, size = 165, normalized size = 1.50 \[ \frac {F^{a\,c+b\,c\,x}\,\left (b^3\,c^3\,d^3\,{\ln \relax (F)}^3+3\,b^3\,c^3\,d^2\,e\,x\,{\ln \relax (F)}^3+3\,b^3\,c^3\,d\,e^2\,x^2\,{\ln \relax (F)}^3+b^3\,c^3\,e^3\,x^3\,{\ln \relax (F)}^3-3\,b^2\,c^2\,d^2\,e\,{\ln \relax (F)}^2-6\,b^2\,c^2\,d\,e^2\,x\,{\ln \relax (F)}^2-3\,b^2\,c^2\,e^3\,x^2\,{\ln \relax (F)}^2+6\,b\,c\,d\,e^2\,\ln \relax (F)+6\,b\,c\,e^3\,x\,\ln \relax (F)-6\,e^3\right )}{b^4\,c^4\,{\ln \relax (F)}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(a + b*x))*(d + e*x)^3,x)
[Out]
(F^(a*c + b*c*x)*(b^3*c^3*d^3*log(F)^3 - 6*e^3 + 6*b*c*e^3*x*log(F) - 3*b^2*c^2*d^2*e*log(F)^2 - 3*b^2*c^2*e^3
*x^2*log(F)^2 + b^3*c^3*e^3*x^3*log(F)^3 + 6*b*c*d*e^2*log(F) - 6*b^2*c^2*d*e^2*x*log(F)^2 + 3*b^3*c^3*d^2*e*x
*log(F)^3 + 3*b^3*c^3*d*e^2*x^2*log(F)^3))/(b^4*c^4*log(F)^4)
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sympy [A] time = 0.22, size = 231, normalized size = 2.10 \[ \begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{3} c^{3} d^{3} \log {\relax (F )}^{3} + 3 b^{3} c^{3} d^{2} e x \log {\relax (F )}^{3} + 3 b^{3} c^{3} d e^{2} x^{2} \log {\relax (F )}^{3} + b^{3} c^{3} e^{3} x^{3} \log {\relax (F )}^{3} - 3 b^{2} c^{2} d^{2} e \log {\relax (F )}^{2} - 6 b^{2} c^{2} d e^{2} x \log {\relax (F )}^{2} - 3 b^{2} c^{2} e^{3} x^{2} \log {\relax (F )}^{2} + 6 b c d e^{2} \log {\relax (F )} + 6 b c e^{3} x \log {\relax (F )} - 6 e^{3}\right )}{b^{4} c^{4} \log {\relax (F )}^{4}} & \text {for}\: b^{4} c^{4} \log {\relax (F )}^{4} \neq 0 \\d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(c*(b*x+a))*(e*x+d)**3,x)
[Out]
Piecewise((F**(c*(a + b*x))*(b**3*c**3*d**3*log(F)**3 + 3*b**3*c**3*d**2*e*x*log(F)**3 + 3*b**3*c**3*d*e**2*x*
*2*log(F)**3 + b**3*c**3*e**3*x**3*log(F)**3 - 3*b**2*c**2*d**2*e*log(F)**2 - 6*b**2*c**2*d*e**2*x*log(F)**2 -
3*b**2*c**2*e**3*x**2*log(F)**2 + 6*b*c*d*e**2*log(F) + 6*b*c*e**3*x*log(F) - 6*e**3)/(b**4*c**4*log(F)**4),
Ne(b**4*c**4*log(F)**4, 0)), (d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4, True))
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