3.84 \(\int F^{c (a+b x)} x^2 \log ^n(d x) (e+e n+e (3+b c x \log (F)) \log (d x)) \, dx\)

Optimal. Leaf size=22 \[ e x^3 \log ^{n+1}(d x) F^{c (a+b x)} \]

[Out]

e*F^(c*(a + b*x))*x^3*Log[d*x]^(1 + n)

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Rubi [A]  time = 0.192834, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026 \[ e x^3 \log ^{n+1}(d x) F^{c (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))*x^2*Log[d*x]^n*(e + e*n + e*(3 + b*c*x*Log[F])*Log[d*x]),x]

[Out]

e*F^(c*(a + b*x))*x^3*Log[d*x]^(1 + n)

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Rubi in Sympy [A]  time = 11.1257, size = 20, normalized size = 0.91 \[ F^{c \left (a + b x\right )} e x^{3} \log{\left (d x \right )}^{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*x**2*ln(d*x)**n*(e+e*n+e*(3+b*c*x*ln(F))*ln(d*x)),x)

[Out]

F**(c*(a + b*x))*e*x**3*log(d*x)**(n + 1)

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Mathematica [A]  time = 0.0498399, size = 23, normalized size = 1.05 \[ e x^3 \log ^{n+1}(d x) F^{a c+b c x} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))*x^2*Log[d*x]^n*(e + e*n + e*(3 + b*c*x*Log[F])*Log[d*x]),x]

[Out]

e*F^(a*c + b*c*x)*x^3*Log[d*x]^(1 + n)

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Maple [C]  time = 0.158, size = 198, normalized size = 9. \[ \left ( -{\frac{i}{2}}\pi \,e{x}^{3}{\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( idx \right ){F}^{c \left ( bx+a \right ) }+{\frac{i}{2}}\pi \,e{x}^{3}{\it csgn} \left ( id \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }+{\frac{i}{2}}\pi \,e{x}^{3}{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }-{\frac{i}{2}}\pi \,e{x}^{3} \left ({\it csgn} \left ( idx \right ) \right ) ^{3}{F}^{c \left ( bx+a \right ) }+\ln \left ( d \right ) e{x}^{3}{F}^{c \left ( bx+a \right ) }+e{x}^{3}{F}^{c \left ( bx+a \right ) }\ln \left ( x \right ) \right ) \left ( \ln \left ( d \right ) +\ln \left ( x \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( idx \right ) \left ( -{\it csgn} \left ( idx \right ) +{\it csgn} \left ( id \right ) \right ) \left ( -{\it csgn} \left ( idx \right ) +{\it csgn} \left ( ix \right ) \right ) \right ) ^{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*x^2*ln(d*x)^n*(e+e*n+e*(3+b*c*x*ln(F))*ln(d*x)),x)

[Out]

(-1/2*I*Pi*e*x^3*csgn(I*d)*csgn(I*x)*csgn(I*d*x)*F^(c*(b*x+a))+1/2*I*Pi*e*x^3*cs
gn(I*d)*csgn(I*d*x)^2*F^(c*(b*x+a))+1/2*I*Pi*e*x^3*csgn(I*x)*csgn(I*d*x)^2*F^(c*
(b*x+a))-1/2*I*Pi*e*x^3*csgn(I*d*x)^3*F^(c*(b*x+a))+ln(d)*e*x^3*F^(c*(b*x+a))+e*
x^3*F^(c*(b*x+a))*ln(x))*(ln(d)+ln(x)-1/2*I*Pi*csgn(I*d*x)*(-csgn(I*d*x)+csgn(I*
d))*(-csgn(I*d*x)+csgn(I*x)))^n

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Maxima [A]  time = 0.9331, size = 57, normalized size = 2.59 \[{\left (F^{a c} e x^{3} \log \left (d\right ) + F^{a c} e x^{3} \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + n \log \left (\log \left (d\right ) + \log \left (x\right )\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*c*x*log(F) + 3)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x^2*log(d*x)^n,x, algorithm="maxima")

[Out]

(F^(a*c)*e*x^3*log(d) + F^(a*c)*e*x^3*log(x))*e^(b*c*x*log(F) + n*log(log(d) + l
og(x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*c*x*log(F) + 3)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x^2*log(d*x)^n,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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**2*ln(d*x)**n*(e+e*n+e*(3+b*c*x*ln(F))*ln(d*x)),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*c*x*log(F) + 3)*e*log(d*x) + e*n + e)*F^((b*x + a)*c)*x^2*log(d*x)^n,x, algorithm="giac")

[Out]

Exception raised: RuntimeError