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

Optimal. Leaf size=79 \[ \frac{2 e^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{2 e (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{(d+e x)^2 F^{c (a+b x)}}{b c \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.0825601, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 e^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{2 e (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{(d+e x)^2 F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.7132, size = 75, normalized size = 0.95 \[ \frac{F^{c \left (a + b x\right )} \left (d + e x\right )^{2}}{b c \log{\left (F \right )}} - \frac{2 F^{c \left (a + b x\right )} e \left (d + e x\right )}{b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{2 F^{c \left (a + b x\right )} e^{2}}{b^{3} c^{3} \log{\left (F \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0147851, size = 56, normalized size = 0.71 \[ \frac{F^{c (a+b x)} \left (b^2 c^2 \log ^2(F) (d+e x)^2-2 b c e \log (F) (d+e x)+2 e^2\right )}{b^3 c^3 \log ^3(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 91, normalized size = 1.2 \[{\frac{ \left ({e}^{2}{x}^{2}{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}dex+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}-2\,\ln \left ( F \right ) bc{e}^{2}x-2\,\ln \left ( F \right ) bced+2\,{e}^{2} \right ){F}^{c \left ( bx+a \right ) }}{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.734342, size = 166, normalized size = 2.1 \[ \frac{F^{b c x + a c} d^{2}}{b c \log \left (F\right )} + \frac{2 \,{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac{{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.228519, size = 113, normalized size = 1.43 \[ \frac{{\left ({\left (b^{2} c^{2} e^{2} x^{2} + 2 \, b^{2} c^{2} d e x + b^{2} c^{2} d^{2}\right )} \log \left (F\right )^{2} + 2 \, e^{2} - 2 \,{\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{3} c^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.394748, size = 133, normalized size = 1.68 \[ \begin{cases} \frac{F^{c \left (a + b x\right )} \left (b^{2} c^{2} d^{2} \log{\left (F \right )}^{2} + 2 b^{2} c^{2} d e x \log{\left (F \right )}^{2} + b^{2} c^{2} e^{2} x^{2} \log{\left (F \right )}^{2} - 2 b c d e \log{\left (F \right )} - 2 b c e^{2} x \log{\left (F \right )} + 2 e^{2}\right )}{b^{3} c^{3} \log{\left (F \right )}^{3}} & \text{for}\: b^{3} c^{3} \log{\left (F \right )}^{3} \neq 0 \\d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273253, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done