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

Optimal. Leaf size=48 \[ \frac{(d+e x) F^{c (a+b x)}}{b c \log (F)}-\frac{e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)} \]

[Out]

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

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Rubi [A]  time = 0.0365302, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{(d+e x) F^{c (a+b x)}}{b c \log (F)}-\frac{e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.63371, size = 41, normalized size = 0.85 \[ \frac{F^{c \left (a + b x\right )} \left (d + e x\right )}{b c \log{\left (F \right )}} - \frac{F^{c \left (a + b x\right )} e}{b^{2} c^{2} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0260575, size = 34, normalized size = 0.71 \[ \frac{F^{c (a+b x)} (b c \log (F) (d+e x)-e)}{b^2 c^2 \log ^2(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \[{\frac{ \left ( \ln \left ( F \right ) bcex+\ln \left ( F \right ) bcd-e \right ){F}^{c \left ( bx+a \right ) }}{{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*(e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.691583, size = 81, normalized size = 1.69 \[ \frac{F^{b c x + a c} d}{b c \log \left (F\right )} + \frac{{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} e}{b^{2} c^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)

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Fricas [A]  time = 0.233659, size = 51, normalized size = 1.06 \[ \frac{{\left ({\left (b c e x + b c d\right )} \log \left (F\right ) - e\right )} F^{b c x + a c}}{b^{2} c^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.293582, size = 60, normalized size = 1.25 \[ \begin{cases} \frac{F^{c \left (a + b x\right )} \left (b c d \log{\left (F \right )} + b c e x \log{\left (F \right )} - e\right )}{b^{2} c^{2} \log{\left (F \right )}^{2}} & \text{for}\: b^{2} c^{2} \log{\left (F \right )}^{2} \neq 0 \\d x + \frac{e x^{2}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260921, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done