3.292 \(\int \frac{F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx\)

Optimal. Leaf size=31 \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^3\right )}{3 d} \]

[Out]

(b^5*F^a*Gamma[-5, -(b*(c + d*x)^3*Log[F])]*Log[F]^5)/(3*d)

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Rubi [A]  time = 0.0987708, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-b \log (F) (c+d x)^3\right )}{3 d} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^3)/(c + d*x)^16,x]

[Out]

(b^5*F^a*Gamma[-5, -(b*(c + d*x)^3*Log[F])]*Log[F]^5)/(3*d)

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Rubi in Sympy [A]  time = 6.03549, size = 31, normalized size = 1. \[ \frac{F^{a} b^{5} \Gamma{\left (-5,- b \left (c + d x\right )^{3} \log{\left (F \right )} \right )} \log{\left (F \right )}^{5}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**16,x)

[Out]

F**a*b**5*Gamma(-5, -b*(c + d*x)**3*log(F))*log(F)**5/(3*d)

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Mathematica [B]  time = 0.13009, size = 111, normalized size = 3.58 \[ \frac{F^a \left (b^5 \log ^5(F) \text{ExpIntegralEi}\left (b \log (F) (c+d x)^3\right )-\frac{F^{b (c+d x)^3} \left (b^4 \log ^4(F) (c+d x)^{12}+b^3 \log ^3(F) (c+d x)^9+2 b^2 \log ^2(F) (c+d x)^6+6 b \log (F) (c+d x)^3+24\right )}{(c+d x)^{15}}\right )}{360 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^3)/(c + d*x)^16,x]

[Out]

(F^a*(b^5*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*Log[F]^5 - (F^(b*(c + d*x)^3)*(24
+ 6*b*(c + d*x)^3*Log[F] + 2*b^2*(c + d*x)^6*Log[F]^2 + b^3*(c + d*x)^9*Log[F]^3
+ b^4*(c + d*x)^12*Log[F]^4))/(c + d*x)^15))/(360*d)

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Maple [F]  time = 0.287, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{ \left ( dx+c \right ) ^{16}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^3)/(d*x+c)^16,x)

[Out]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^16,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^16,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^16, x)

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Fricas [A]  time = 0.283486, size = 1192, normalized size = 38.45 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^16,x, algorithm="fricas")

[Out]

1/360*((b^5*d^15*x^15 + 15*b^5*c*d^14*x^14 + 105*b^5*c^2*d^13*x^13 + 455*b^5*c^3
*d^12*x^12 + 1365*b^5*c^4*d^11*x^11 + 3003*b^5*c^5*d^10*x^10 + 5005*b^5*c^6*d^9*
x^9 + 6435*b^5*c^7*d^8*x^8 + 6435*b^5*c^8*d^7*x^7 + 5005*b^5*c^9*d^6*x^6 + 3003*
b^5*c^10*d^5*x^5 + 1365*b^5*c^11*d^4*x^4 + 455*b^5*c^12*d^3*x^3 + 105*b^5*c^13*d
^2*x^2 + 15*b^5*c^14*d*x + b^5*c^15)*F^a*Ei((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2
*d*x + b*c^3)*log(F))*log(F)^5 - ((b^4*d^12*x^12 + 12*b^4*c*d^11*x^11 + 66*b^4*c
^2*d^10*x^10 + 220*b^4*c^3*d^9*x^9 + 495*b^4*c^4*d^8*x^8 + 792*b^4*c^5*d^7*x^7 +
924*b^4*c^6*d^6*x^6 + 792*b^4*c^7*d^5*x^5 + 495*b^4*c^8*d^4*x^4 + 220*b^4*c^9*d
^3*x^3 + 66*b^4*c^10*d^2*x^2 + 12*b^4*c^11*d*x + b^4*c^12)*log(F)^4 + (b^3*d^9*x
^9 + 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^7*x^7 + 84*b^3*c^3*d^6*x^6 + 126*b^3*c^4*d^5
*x^5 + 126*b^3*c^5*d^4*x^4 + 84*b^3*c^6*d^3*x^3 + 36*b^3*c^7*d^2*x^2 + 9*b^3*c^8
*d*x + b^3*c^9)*log(F)^3 + 2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4
+ 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4*d^2*x^2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 +
6*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F) + 24)*F^(b*d^3*x^3 +
3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/(d^16*x^15 + 15*c*d^15*x^14 + 105*c^2
*d^14*x^13 + 455*c^3*d^13*x^12 + 1365*c^4*d^12*x^11 + 3003*c^5*d^11*x^10 + 5005*
c^6*d^10*x^9 + 6435*c^7*d^9*x^8 + 6435*c^8*d^8*x^7 + 5005*c^9*d^7*x^6 + 3003*c^1
0*d^6*x^5 + 1365*c^11*d^5*x^4 + 455*c^12*d^4*x^3 + 105*c^13*d^3*x^2 + 15*c^14*d^
2*x + c^15*d)

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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**(a+b*(d*x+c)**3)/(d*x+c)**16,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^3*b + a)/(d*x + c)^16,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^16, x)