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

Optimal. Leaf size=229 \[ -\frac{6 g F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac{2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac{6 g x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{3 g x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{d F^{c (a+b x)}}{b c \log (F)}+\frac{e x F^{c (a+b x)}}{b c \log (F)}+\frac{f x^2 F^{c (a+b x)}}{b c \log (F)}+\frac{g x^3 F^{c (a+b x)}}{b c \log (F)} \]

[Out]

(-6*F^(c*(a + b*x))*g)/(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) - (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) + (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])

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Rubi [A]  time = 0.349668, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{6 g F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac{2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac{6 g x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{3 g x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{d F^{c (a+b x)}}{b c \log (F)}+\frac{e x F^{c (a+b x)}}{b c \log (F)}+\frac{f x^2 F^{c (a+b x)}}{b c \log (F)}+\frac{g x^3 F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

(-6*F^(c*(a + b*x))*g)/(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) - (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) + (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])

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Rubi in Sympy [A]  time = 38.5173, size = 223, normalized size = 0.97 \[ \frac{F^{c \left (a + b x\right )} d}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} e x}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} f x^{2}}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} g x^{3}}{b c \log{\left (F \right )}} - \frac{F^{c \left (a + b x\right )} e}{b^{2} c^{2} \log{\left (F \right )}^{2}} - \frac{2 F^{c \left (a + b x\right )} f x}{b^{2} c^{2} \log{\left (F \right )}^{2}} - \frac{3 F^{c \left (a + b x\right )} g x^{2}}{b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{2 F^{c \left (a + b x\right )} f}{b^{3} c^{3} \log{\left (F \right )}^{3}} + \frac{6 F^{c \left (a + b x\right )} g x}{b^{3} c^{3} \log{\left (F \right )}^{3}} - \frac{6 F^{c \left (a + b x\right )} g}{b^{4} c^{4} \log{\left (F \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(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))*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) + 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) - 6*F**(c*(a + b*x))*g
/(b**4*c**4*log(F)**4)

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Mathematica [A]  time = 0.0777437, size = 84, normalized size = 0.37 \[ \frac{F^{c (a+b x)} \left (b^3 c^3 \log ^3(F) (d+x (e+x (f+g x)))-b^2 c^2 \log ^2(F) (e+x (2 f+3 g x))+2 b c \log (F) (f+3 g x)-6 g\right )}{b^4 c^4 \log ^4(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 138, normalized size = 0.6 \[{\frac{ \left ( g{x}^{3}{c}^{3}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}f{x}^{2}+ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}ex+{c}^{3}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}d-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}g{x}^{2}-2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}fx-{c}^{2}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}e+6\,\ln \left ( F \right ) bcgx+2\,fcb\ln \left ( F \right ) -6\,g \right ){F}^{c \left ( bx+a \right ) }}{{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.822496, size = 262, normalized size = 1.14 \[ \frac{F^{b c x + a c} d}{b c \log \left (F\right )} + \frac{{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac{{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} f}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac{{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} g}{b^{4} c^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)

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Fricas [A]  time = 0.268497, size = 165, normalized size = 0.72 \[ \frac{{\left ({\left (b^{3} c^{3} g x^{3} + b^{3} c^{3} f x^{2} + b^{3} c^{3} e x + b^{3} c^{3} d\right )} \log \left (F\right )^{3} -{\left (3 \, b^{2} c^{2} g x^{2} + 2 \, b^{2} c^{2} f x + b^{2} c^{2} e\right )} \log \left (F\right )^{2} + 2 \,{\left (3 \, b c g x + b c f\right )} \log \left (F\right ) - 6 \, g\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.486682, size = 190, normalized size = 0.83 \[ \begin{cases} \frac{F^{c \left (a + b x\right )} \left (b^{3} c^{3} d \log{\left (F \right )}^{3} + b^{3} c^{3} e x \log{\left (F \right )}^{3} + b^{3} c^{3} f x^{2} \log{\left (F \right )}^{3} + b^{3} c^{3} g x^{3} \log{\left (F \right )}^{3} - b^{2} c^{2} e \log{\left (F \right )}^{2} - 2 b^{2} c^{2} f x \log{\left (F \right )}^{2} - 3 b^{2} c^{2} g x^{2} \log{\left (F \right )}^{2} + 2 b c f \log{\left (F \right )} + 6 b c g x \log{\left (F \right )} - 6 g\right )}{b^{4} c^{4} \log{\left (F \right )}^{4}} & \text{for}\: b^{4} c^{4} \log{\left (F \right )}^{4} \neq 0 \\d x + \frac{e x^{2}}{2} + \frac{f x^{3}}{3} + \frac{g x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.289291, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done