3.16 \(\int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx\)

Optimal. Leaf size=41 \[ \frac{\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]

[Out]

(a + b*(F^(e*(c + d*x)))^n)^(1 + p)/(b*d*e*n*(1 + p)*Log[F])

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Rubi [A]  time = 0.115201, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*(F^(e*(c + d*x)))^n)^(1 + p)/(b*d*e*n*(1 + p)*Log[F])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.304654, size = 0, normalized size = 0. \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.013, size = 42, normalized size = 1. \[{\frac{ \left ( a+b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n} \right ) ^{1+p}}{bden \left ( 1+p \right ) \ln \left ( F \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a+b*(F^(e*(d*x+c)))^n)^(1+p)/b/d/e/n/(1+p)/ln(F)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252022, size = 72, normalized size = 1.76 \[ \frac{{\left (F^{d e n x + c e n} b + a\right )}{\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(F^(d*e*n*x + c*e*n)*b + a)*(F^(d*e*n*x + c*e*n)*b + a)^p/((b*d*e*n*p + b*d*e*n)
*log(F))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.245405, size = 58, normalized size = 1.41 \[ \frac{{\left (F^{d n x e + c n e} b + a\right )}^{p + 1} e^{\left (-1\right )}}{b d n{\left (p + 1\right )}{\rm ln}\left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(F^(d*n*x*e + c*n*e)*b + a)^(p + 1)*e^(-1)/(b*d*n*(p + 1)*ln(F))