3.2 \(\int F^{c (a+b x)} (d+e x)^4 \, dx\)

Optimal. Leaf size=141 \[ \frac{24 e^4 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac{24 e^3 (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac{12 e^2 (d+e x)^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{4 e (d+e x)^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{(d+e x)^4 F^{c (a+b x)}}{b c \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.202553, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{24 e^4 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac{24 e^3 (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac{12 e^2 (d+e x)^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{4 e (d+e x)^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{(d+e x)^4 F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))*(d + e*x)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 39.7458, size = 139, normalized size = 0.99 \[ \frac{F^{c \left (a + b x\right )} \left (d + e x\right )^{4}}{b c \log{\left (F \right )}} - \frac{4 F^{c \left (a + b x\right )} e \left (d + e x\right )^{3}}{b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{12 F^{c \left (a + b x\right )} e^{2} \left (d + e x\right )^{2}}{b^{3} c^{3} \log{\left (F \right )}^{3}} - \frac{24 F^{c \left (a + b x\right )} e^{3} \left (d + e x\right )}{b^{4} c^{4} \log{\left (F \right )}^{4}} + \frac{24 F^{c \left (a + b x\right )} e^{4}}{b^{5} c^{5} \log{\left (F \right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(e*x+d)**4,x)

[Out]

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

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Mathematica [A]  time = 0.0744396, size = 100, normalized size = 0.71 \[ \frac{F^{c (a+b x)} \left (b^4 c^4 \log ^4(F) (d+e x)^4-4 b^3 c^3 e \log ^3(F) (d+e x)^3+12 b^2 c^2 e^2 \log ^2(F) (d+e x)^2-24 b c e^3 \log (F) (d+e x)+24 e^4\right )}{b^5 c^5 \log ^5(F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))*(d + e*x)^4,x]

[Out]

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

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Maple [A]  time = 0.017, size = 260, normalized size = 1.8 \[{\frac{ \left ({e}^{4}{x}^{4}{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}+4\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}d{e}^{3}{x}^{3}+6\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}{d}^{2}{e}^{2}{x}^{2}+4\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}{d}^{3}ex+ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}{d}^{4}-4\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{e}^{4}{x}^{3}-12\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}d{e}^{3}{x}^{2}-12\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{d}^{2}{e}^{2}x-4\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{d}^{3}e+12\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{e}^{4}{x}^{2}+24\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}d{e}^{3}x+12\,{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}{e}^{2}-24\,\ln \left ( F \right ) bc{e}^{4}x-24\,d{e}^{3}bc\ln \left ( F \right ) +24\,{e}^{4} \right ){F}^{c \left ( bx+a \right ) }}{{b}^{5}{c}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 0.715837, size = 417, normalized size = 2.96 \[ \frac{F^{b c x + a c} d^{4}}{b c \log \left (F\right )} + \frac{4 \,{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d^{3} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac{6 \,{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} d^{2} e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac{4 \,{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} d e^{3}}{b^{4} c^{4} \log \left (F\right )^{4}} + \frac{{\left (F^{a c} b^{4} c^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{a c} b c x \log \left (F\right ) + 24 \, F^{a c}\right )} F^{b c x} e^{4}}{b^{5} c^{5} \log \left (F\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4*F^((b*x + a)*c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.257058, size = 306, normalized size = 2.17 \[ \frac{{\left ({\left (b^{4} c^{4} e^{4} x^{4} + 4 \, b^{4} c^{4} d e^{3} x^{3} + 6 \, b^{4} c^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} c^{4} d^{3} e x + b^{4} c^{4} d^{4}\right )} \log \left (F\right )^{4} + 24 \, e^{4} - 4 \,{\left (b^{3} c^{3} e^{4} x^{3} + 3 \, b^{3} c^{3} d e^{3} x^{2} + 3 \, b^{3} c^{3} d^{2} e^{2} x + b^{3} c^{3} d^{3} e\right )} \log \left (F\right )^{3} + 12 \,{\left (b^{2} c^{2} e^{4} x^{2} + 2 \, b^{2} c^{2} d e^{3} x + b^{2} c^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 24 \,{\left (b c e^{4} x + b c d e^{3}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4*F^((b*x + a)*c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.604649, size = 350, normalized size = 2.48 \[ \begin{cases} \frac{F^{c \left (a + b x\right )} \left (b^{4} c^{4} d^{4} \log{\left (F \right )}^{4} + 4 b^{4} c^{4} d^{3} e x \log{\left (F \right )}^{4} + 6 b^{4} c^{4} d^{2} e^{2} x^{2} \log{\left (F \right )}^{4} + 4 b^{4} c^{4} d e^{3} x^{3} \log{\left (F \right )}^{4} + b^{4} c^{4} e^{4} x^{4} \log{\left (F \right )}^{4} - 4 b^{3} c^{3} d^{3} e \log{\left (F \right )}^{3} - 12 b^{3} c^{3} d^{2} e^{2} x \log{\left (F \right )}^{3} - 12 b^{3} c^{3} d e^{3} x^{2} \log{\left (F \right )}^{3} - 4 b^{3} c^{3} e^{4} x^{3} \log{\left (F \right )}^{3} + 12 b^{2} c^{2} d^{2} e^{2} \log{\left (F \right )}^{2} + 24 b^{2} c^{2} d e^{3} x \log{\left (F \right )}^{2} + 12 b^{2} c^{2} e^{4} x^{2} \log{\left (F \right )}^{2} - 24 b c d e^{3} \log{\left (F \right )} - 24 b c e^{4} x \log{\left (F \right )} + 24 e^{4}\right )}{b^{5} c^{5} \log{\left (F \right )}^{5}} & \text{for}\: b^{5} c^{5} \log{\left (F \right )}^{5} \neq 0 \\d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac{e^{4} x^{5}}{5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(c*(b*x+a))*(e*x+d)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.355179, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4*F^((b*x + a)*c),x, algorithm="giac")

[Out]

Done