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

Optimal. Leaf size=135 \[ \frac{2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{d F^{c (a+b x)}}{b c \log (F)}+\frac{e x F^{c (a+b x)}}{b c \log (F)}+\frac{f x^2 F^{c (a+b x)}}{b c \log (F)} \]

[Out]

(2*f*F^(c*(a + b*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) + (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])

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Rubi [A]  time = 0.193089, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac{2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{d F^{c (a+b x)}}{b c \log (F)}+\frac{e x F^{c (a+b x)}}{b c \log (F)}+\frac{f x^2 F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

(2*f*F^(c*(a + b*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) + (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])

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Rubi in Sympy [A]  time = 23.661, size = 126, normalized size = 0.93 \[ \frac{F^{c \left (a + b x\right )} d}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} e x}{b c \log{\left (F \right )}} + \frac{F^{c \left (a + b x\right )} f x^{2}}{b c \log{\left (F \right )}} - \frac{F^{c \left (a + b x\right )} e}{b^{2} c^{2} \log{\left (F \right )}^{2}} - \frac{2 F^{c \left (a + b x\right )} f x}{b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{2 F^{c \left (a + b x\right )} f}{b^{3} c^{3} \log{\left (F \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(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))*e/(b**2*c**2*log(F)**2) - 2*F**(c*
(a + b*x))*f*x/(b**2*c**2*log(F)**2) + 2*F**(c*(a + b*x))*f/(b**3*c**3*log(F)**3
)

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Mathematica [A]  time = 0.0492556, size = 56, normalized size = 0.41 \[ \frac{F^{c (a+b x)} \left (b^2 c^2 \log ^2(F) (d+x (e+f x))-b c \log (F) (e+2 f x)+2 f\right )}{b^3 c^3 \log ^3(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 80, normalized size = 0.6 \[{\frac{ \left ( f{x}^{2}{c}^{2}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}ex+{c}^{2}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}d-2\,\ln \left ( F \right ) bcfx-\ln \left ( F \right ) bce+2\,f \right ){F}^{c \left ( bx+a \right ) }}{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.858383, size = 158, normalized size = 1.17 \[ \frac{F^{b c x + a c} d}{b c \log \left (F\right )} + \frac{{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac{{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} f}{b^{3} c^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)

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Fricas [A]  time = 0.307395, size = 100, normalized size = 0.74 \[ \frac{{\left ({\left (b^{2} c^{2} f x^{2} + b^{2} c^{2} e x + b^{2} c^{2} d\right )} \log \left (F\right )^{2} -{\left (2 \, b c f x + b c e\right )} \log \left (F\right ) + 2 \, f\right )} F^{b c x + a c}}{b^{3} c^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.38363, size = 116, normalized size = 0.86 \[ \begin{cases} \frac{F^{c \left (a + b x\right )} \left (b^{2} c^{2} d \log{\left (F \right )}^{2} + b^{2} c^{2} e x \log{\left (F \right )}^{2} + b^{2} c^{2} f x^{2} \log{\left (F \right )}^{2} - b c e \log{\left (F \right )} - 2 b c f x \log{\left (F \right )} + 2 f\right )}{b^{3} c^{3} \log{\left (F \right )}^{3}} & \text{for}\: b^{3} c^{3} \log{\left (F \right )}^{3} \neq 0 \\d x + \frac{e x^{2}}{2} + \frac{f x^{3}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.25308, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done