3.83 \(\int F^{c (a+b x)} x^m \log ^n(d x) (e+e n+e (1+m+b c x \log (F)) \log (d x)) \, dx\)

Optimal. Leaf size=24 \[ e x^{m+1} \log ^{n+1}(d x) F^{c (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.221407, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026 \[ e x^{m+1} \log ^{n+1}(d x) F^{c (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))*x^m*Log[d*x]^n*(e + e*n + e*(1 + m + b*c*x*Log[F])*Log[d*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.1184, size = 22, normalized size = 0.92 \[ F^{c \left (a + b x\right )} e x^{m + 1} \log{\left (d x \right )}^{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*x**m*ln(d*x)**n*(e+e*n+e*(1+m+b*c*x*ln(F))*ln(d*x)),x)

[Out]

F**(c*(a + b*x))*e*x**(m + 1)*log(d*x)**(n + 1)

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Mathematica [A]  time = 0.116804, size = 24, normalized size = 1. \[ e x^{m+1} \log ^{n+1}(d x) F^{c (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))*x^m*Log[d*x]^n*(e + e*n + e*(1 + m + b*c*x*Log[F])*Log[d*x]),x]

[Out]

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

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{F}^{c \left ( bx+a \right ) }{x}^{m} \left ( \ln \left ( dx \right ) \right ) ^{n} \left ( e+en+e \left ( 1+m+bcx\ln \left ( F \right ) \right ) \ln \left ( dx \right ) \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*x^m*ln(d*x)^n*(e+e*n+e*(1+m+b*c*x*ln(F))*ln(d*x)),x)

[Out]

int(F^(c*(b*x+a))*x^m*ln(d*x)^n*(e+e*n+e*(1+m+b*c*x*ln(F))*ln(d*x)),x)

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Maxima [A]  time = 0.965532, size = 57, normalized size = 2.38 \[{\left (F^{a c} e x \log \left (d\right ) + F^{a c} e x \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + m \log \left (x\right ) + n \log \left (\log \left (d\right ) + \log \left (x\right )\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*c*x*log(F) + m + 1)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x^m*log(d*x)^n,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.260418, size = 43, normalized size = 1.79 \[{\left (e x \log \left (d\right ) + e x \log \left (x\right )\right )} F^{b c x + a c} x^{m}{\left (\log \left (d\right ) + \log \left (x\right )\right )}^{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*c*x*log(F) + m + 1)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x^m*log(d*x)^n,x, algorithm="fricas")

[Out]

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

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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**(c*(b*x+a))*x**m*ln(d*x)**n*(e+e*n+e*(1+m+b*c*x*ln(F))*ln(d*x)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left ({\left (b c x \log \left (F\right ) + m + 1\right )} e \log \left (d x\right ) + e n + e\right )} F^{{\left (b x + a\right )} c} x^{m} \log \left (d x\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*c*x*log(F) + m + 1)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x^m*log(d*x)^n,x, algorithm="giac")

[Out]

integrate(((b*c*x*log(F) + m + 1)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x^m*log(
d*x)^n, x)