3.302 \(\int F^{a+\frac{b}{c+d x}} (c+d x)^4 \, dx\)

Optimal. Leaf size=29 \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{c+d x}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0734012, antiderivative size = 29, 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 \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x))*(c + d*x)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.90656, size = 29, normalized size = 1. \[ - \frac{F^{a} b^{5} \Gamma{\left (-5,- \frac{b \log{\left (F \right )}}{c + d x} \right )} \log{\left (F \right )}^{5}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b/(d*x+c))*(d*x+c)**4,x)

[Out]

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

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Mathematica [B]  time = 0.124079, size = 108, normalized size = 3.72 \[ \frac{F^a \left ((c+d x) F^{\frac{b}{c+d x}} \left (b^4 \log ^4(F)+b^3 \log ^3(F) (c+d x)+2 b^2 \log ^2(F) (c+d x)^2+6 b \log (F) (c+d x)^3+24 (c+d x)^4\right )-b^5 \log ^5(F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{c+d x}\right )\right )}{120 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b/(c + d*x))*(c + d*x)^4,x]

[Out]

(F^a*(-(b^5*ExpIntegralEi[(b*Log[F])/(c + d*x)]*Log[F]^5) + F^(b/(c + d*x))*(c +
d*x)*(24*(c + d*x)^4 + 6*b*(c + d*x)^3*Log[F] + 2*b^2*(c + d*x)^2*Log[F]^2 + b^
3*(c + d*x)*Log[F]^3 + b^4*Log[F]^4)))/(120*d)

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Maple [B]  time = 0.049, size = 634, normalized size = 21.9 \[{\frac{{d}^{4}{x}^{5}}{5}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{d}^{3}{F}^{{\frac{xda+ac+b}{dx+c}}}c{x}^{4}+2\,{d}^{2}{F}^{{\frac{xda+ac+b}{dx+c}}}{c}^{2}{x}^{3}+2\,d{F}^{{\frac{xda+ac+b}{dx+c}}}{c}^{3}{x}^{2}+{F}^{{\frac{xda+ac+b}{dx+c}}}{c}^{4}x+{\frac{{c}^{5}}{5\,d}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{\ln \left ( F \right ) b{d}^{3}{x}^{4}}{20}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{\ln \left ( F \right ) bc{d}^{2}{x}^{3}}{5}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{3\,\ln \left ( F \right ) b{c}^{2}d{x}^{2}}{10}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{\ln \left ( F \right ) b{c}^{3}x}{5}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{\ln \left ( F \right ) b{c}^{4}}{20\,d}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{x}^{3}}{60}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{d{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}c{x}^{2}}{20}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}x}{20}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{3}}{60\,d}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{d{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{x}^{2}}{120}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}cx}{60}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{2}}{120\,d}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}x}{120}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}c}{120\,d}{F}^{{\frac{xda+ac+b}{dx+c}}}}+{\frac{{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}{F}^{a}}{120\,d}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c))*(d*x+c)^4,x)

[Out]

1/5*d^4*F^((a*d*x+a*c+b)/(d*x+c))*x^5+d^3*F^((a*d*x+a*c+b)/(d*x+c))*c*x^4+2*d^2*
F^((a*d*x+a*c+b)/(d*x+c))*c^2*x^3+2*d*F^((a*d*x+a*c+b)/(d*x+c))*c^3*x^2+F^((a*d*
x+a*c+b)/(d*x+c))*c^4*x+1/5/d*F^((a*d*x+a*c+b)/(d*x+c))*c^5+1/20*d^3*b*ln(F)*F^(
(a*d*x+a*c+b)/(d*x+c))*x^4+1/5*d^2*b*ln(F)*F^((a*d*x+a*c+b)/(d*x+c))*c*x^3+3/10*
d*b*ln(F)*F^((a*d*x+a*c+b)/(d*x+c))*c^2*x^2+1/5*b*ln(F)*F^((a*d*x+a*c+b)/(d*x+c)
)*c^3*x+1/20/d*b*ln(F)*F^((a*d*x+a*c+b)/(d*x+c))*c^4+1/60*d^2*b^2*ln(F)^2*F^((a*
d*x+a*c+b)/(d*x+c))*x^3+1/20*d*b^2*ln(F)^2*F^((a*d*x+a*c+b)/(d*x+c))*c*x^2+1/20*
b^2*ln(F)^2*F^((a*d*x+a*c+b)/(d*x+c))*c^2*x+1/60/d*b^2*ln(F)^2*F^((a*d*x+a*c+b)/
(d*x+c))*c^3+1/120*d*b^3*ln(F)^3*F^((a*d*x+a*c+b)/(d*x+c))*x^2+1/60*b^3*ln(F)^3*
F^((a*d*x+a*c+b)/(d*x+c))*c*x+1/120/d*b^3*ln(F)^3*F^((a*d*x+a*c+b)/(d*x+c))*c^2+
1/120*b^4*ln(F)^4*F^((a*d*x+a*c+b)/(d*x+c))*x+1/120/d*b^4*ln(F)^4*F^((a*d*x+a*c+
b)/(d*x+c))*c+1/120/d*b^5*ln(F)^5*F^a*Ei(1,-b*ln(F)/(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{120} \,{\left (24 \, F^{a} d^{4} x^{5} + 6 \,{\left (F^{a} b d^{3} \log \left (F\right ) + 20 \, F^{a} c d^{3}\right )} x^{4} + 2 \,{\left (F^{a} b^{2} d^{2} \log \left (F\right )^{2} + 12 \, F^{a} b c d^{2} \log \left (F\right ) + 120 \, F^{a} c^{2} d^{2}\right )} x^{3} +{\left (F^{a} b^{3} d \log \left (F\right )^{3} + 6 \, F^{a} b^{2} c d \log \left (F\right )^{2} + 36 \, F^{a} b c^{2} d \log \left (F\right ) + 240 \, F^{a} c^{3} d\right )} x^{2} +{\left (F^{a} b^{4} \log \left (F\right )^{4} + 2 \, F^{a} b^{3} c \log \left (F\right )^{3} + 6 \, F^{a} b^{2} c^{2} \log \left (F\right )^{2} + 24 \, F^{a} b c^{3} \log \left (F\right ) + 120 \, F^{a} c^{4}\right )} x\right )} F^{\frac{b}{d x + c}} + \int \frac{{\left (F^{a} b^{5} d x \log \left (F\right )^{5} - F^{a} b^{4} c^{2} \log \left (F\right )^{4} - 2 \, F^{a} b^{3} c^{3} \log \left (F\right )^{3} - 6 \, F^{a} b^{2} c^{4} \log \left (F\right )^{2} - 24 \, F^{a} b c^{5} \log \left (F\right )\right )} F^{\frac{b}{d x + c}}}{120 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4*F^(a + b/(d*x + c)),x, algorithm="maxima")

[Out]

1/120*(24*F^a*d^4*x^5 + 6*(F^a*b*d^3*log(F) + 20*F^a*c*d^3)*x^4 + 2*(F^a*b^2*d^2
*log(F)^2 + 12*F^a*b*c*d^2*log(F) + 120*F^a*c^2*d^2)*x^3 + (F^a*b^3*d*log(F)^3 +
6*F^a*b^2*c*d*log(F)^2 + 36*F^a*b*c^2*d*log(F) + 240*F^a*c^3*d)*x^2 + (F^a*b^4*
log(F)^4 + 2*F^a*b^3*c*log(F)^3 + 6*F^a*b^2*c^2*log(F)^2 + 24*F^a*b*c^3*log(F) +
120*F^a*c^4)*x)*F^(b/(d*x + c)) + integrate(1/120*(F^a*b^5*d*x*log(F)^5 - F^a*b
^4*c^2*log(F)^4 - 2*F^a*b^3*c^3*log(F)^3 - 6*F^a*b^2*c^4*log(F)^2 - 24*F^a*b*c^5
*log(F))*F^(b/(d*x + c))/(d^2*x^2 + 2*c*d*x + c^2), x)

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Fricas [A]  time = 0.27941, size = 329, normalized size = 11.34 \[ -\frac{F^{a} b^{5}{\rm Ei}\left (\frac{b \log \left (F\right )}{d x + c}\right ) \log \left (F\right )^{5} -{\left (24 \, d^{5} x^{5} + 120 \, c d^{4} x^{4} + 240 \, c^{2} d^{3} x^{3} + 240 \, c^{3} d^{2} x^{2} + 120 \, c^{4} d x + 24 \, c^{5} +{\left (b^{4} d x + b^{4} c\right )} \log \left (F\right )^{4} +{\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} + 2 \,{\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (F\right )^{2} + 6 \,{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \left (F\right )\right )} F^{\frac{a d x + a c + b}{d x + c}}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4*F^(a + b/(d*x + c)),x, algorithm="fricas")

[Out]

-1/120*(F^a*b^5*Ei(b*log(F)/(d*x + c))*log(F)^5 - (24*d^5*x^5 + 120*c*d^4*x^4 +
240*c^2*d^3*x^3 + 240*c^3*d^2*x^2 + 120*c^4*d*x + 24*c^5 + (b^4*d*x + b^4*c)*log
(F)^4 + (b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*log(F)^3 + 2*(b^2*d^3*x^3 + 3*b^2*
c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*log(F)^2 + 6*(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6
*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4)*log(F))*F^((a*d*x + a*c + b)/(d*x + c)))/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))*(d*x+c)**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{4} F^{a + \frac{b}{d x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4*F^(a + b/(d*x + c)),x, algorithm="giac")

[Out]

integrate((d*x + c)^4*F^(a + b/(d*x + c)), x)