3.54 \(\int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx\)

Optimal. Leaf size=348 \[ \frac{24 h F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac{6 g F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac{24 h x F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac{2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac{6 g x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac{12 h x^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{3 g x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{4 h x^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{d F^{c (a+b x)}}{b c \log (F)}+\frac{e x F^{c (a+b x)}}{b c \log (F)}+\frac{f x^2 F^{c (a+b x)}}{b c \log (F)}+\frac{g x^3 F^{c (a+b x)}}{b c \log (F)}+\frac{h x^4 F^{c (a+b x)}}{b c \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.570169, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{24 h F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac{6 g F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac{24 h x F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac{2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac{6 g x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac{12 h x^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{3 g x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{4 h x^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{d F^{c (a+b x)}}{b c \log (F)}+\frac{e x F^{c (a+b x)}}{b c \log (F)}+\frac{f x^2 F^{c (a+b x)}}{b c \log (F)}+\frac{g x^3 F^{c (a+b x)}}{b c \log (F)}+\frac{h x^4 F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))*(d + e*x + f*x^2 + g*x^3 + h*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 57.0893, size = 347, normalized size = 1. \[ \frac{F^{c \left (a + b x\right )} d}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} e x}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} f x^{2}}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} g x^{3}}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} h x^{4}}{b c \log{\left (F \right )}} - \frac{F^{c \left (a + b x\right )} e}{b^{2} c^{2} \log{\left (F \right )}^{2}} - \frac{2 F^{c \left (a + b x\right )} f x}{b^{2} c^{2} \log{\left (F \right )}^{2}} - \frac{3 F^{c \left (a + b x\right )} g x^{2}}{b^{2} c^{2} \log{\left (F \right )}^{2}} - \frac{4 F^{c \left (a + b x\right )} h x^{3}}{b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{2 F^{c \left (a + b x\right )} f}{b^{3} c^{3} \log{\left (F \right )}^{3}} + \frac{6 F^{c \left (a + b x\right )} g x}{b^{3} c^{3} \log{\left (F \right )}^{3}} + \frac{12 F^{c \left (a + b x\right )} h x^{2}}{b^{3} c^{3} \log{\left (F \right )}^{3}} - \frac{6 F^{c \left (a + b x\right )} g}{b^{4} c^{4} \log{\left (F \right )}^{4}} - \frac{24 F^{c \left (a + b x\right )} h x}{b^{4} c^{4} \log{\left (F \right )}^{4}} + \frac{24 F^{c \left (a + b x\right )} h}{b^{5} c^{5} \log{\left (F \right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(h*x**4+g*x**3+f*x**2+e*x+d),x)

[Out]

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

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Mathematica [A]  time = 0.0972457, size = 117, normalized size = 0.34 \[ \frac{F^{c (a+b x)} \left (b^4 c^4 \log ^4(F) (d+x (e+x (f+x (g+h x))))-b^3 c^3 \log ^3(F) \left (e+x \left (2 f+3 g x+4 h x^2\right )\right )+2 b^2 c^2 \log ^2(F) (f+3 x (g+2 h x))-6 b c \log (F) (g+4 h x)+24 h\right )}{b^5 c^5 \log ^5(F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))*(d + e*x + f*x^2 + g*x^3 + h*x^4),x]

[Out]

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

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Maple [A]  time = 0.009, size = 212, normalized size = 0.6 \[{\frac{ \left ( h{x}^{4}{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}+ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}g{x}^{3}+ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}f{x}^{2}+ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}ex+ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}d-4\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}h{x}^{3}-3\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}g{x}^{2}-2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}fx- \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}e+12\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}h{x}^{2}+6\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}gx+2\,{c}^{2}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}f-24\,\ln \left ( F \right ) bchx-6\,gcb\ln \left ( F \right ) +24\,h \right ){F}^{c \left ( bx+a \right ) }}{{b}^{5}{c}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*(h*x^4+g*x^3+f*x^2+e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.835463, size = 393, normalized size = 1.13 \[ \frac{F^{b c x + a c} d}{b c \log \left (F\right )} + \frac{{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac{{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} f}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac{{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} g}{b^{4} c^{4} \log \left (F\right )^{4}} + \frac{{\left (F^{a c} b^{4} c^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{a c} b c x \log \left (F\right ) + 24 \, F^{a c}\right )} F^{b c x} h}{b^{5} c^{5} \log \left (F\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)*F^((b*x + a)*c),x, algorithm="maxima")

[Out]

F^(b*c*x + a*c)*d/(b*c*log(F)) + (F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b*c*x)*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)*f/(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)*g/(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 + 1
2*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)
*h/(b^5*c^5*log(F)^5)

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Fricas [A]  time = 0.271756, size = 246, normalized size = 0.71 \[ \frac{{\left ({\left (b^{4} c^{4} h x^{4} + b^{4} c^{4} g x^{3} + b^{4} c^{4} f x^{2} + b^{4} c^{4} e x + b^{4} c^{4} d\right )} \log \left (F\right )^{4} -{\left (4 \, b^{3} c^{3} h x^{3} + 3 \, b^{3} c^{3} g x^{2} + 2 \, b^{3} c^{3} f x + b^{3} c^{3} e\right )} \log \left (F\right )^{3} + 2 \,{\left (6 \, b^{2} c^{2} h x^{2} + 3 \, b^{2} c^{2} g x + b^{2} c^{2} f\right )} \log \left (F\right )^{2} - 6 \,{\left (4 \, b c h x + b c g\right )} \log \left (F\right ) + 24 \, h\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)*F^((b*x + a)*c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.605992, size = 284, normalized size = 0.82 \[ \begin{cases} \frac{F^{c \left (a + b x\right )} \left (b^{4} c^{4} d \log{\left (F \right )}^{4} + b^{4} c^{4} e x \log{\left (F \right )}^{4} + b^{4} c^{4} f x^{2} \log{\left (F \right )}^{4} + b^{4} c^{4} g x^{3} \log{\left (F \right )}^{4} + b^{4} c^{4} h x^{4} \log{\left (F \right )}^{4} - b^{3} c^{3} e \log{\left (F \right )}^{3} - 2 b^{3} c^{3} f x \log{\left (F \right )}^{3} - 3 b^{3} c^{3} g x^{2} \log{\left (F \right )}^{3} - 4 b^{3} c^{3} h x^{3} \log{\left (F \right )}^{3} + 2 b^{2} c^{2} f \log{\left (F \right )}^{2} + 6 b^{2} c^{2} g x \log{\left (F \right )}^{2} + 12 b^{2} c^{2} h x^{2} \log{\left (F \right )}^{2} - 6 b c g \log{\left (F \right )} - 24 b c h x \log{\left (F \right )} + 24 h\right )}{b^{5} c^{5} \log{\left (F \right )}^{5}} & \text{for}\: b^{5} c^{5} \log{\left (F \right )}^{5} \neq 0 \\d x + \frac{e x^{2}}{2} + \frac{f x^{3}}{3} + \frac{g x^{4}}{4} + \frac{h x^{5}}{5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(c*(b*x+a))*(h*x**4+g*x**3+f*x**2+e*x+d),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.32889, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)*F^((b*x + a)*c),x, algorithm="giac")

[Out]

Done