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

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

[Out]

F^a*(d*x+c)^5*Ei(6,-b*ln(F)/(d*x+c))/d

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Rubi [A]  time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac {b^5 F^a \log ^5(F) \text {Gamma}\left (-5,-\frac {b \log (F)}{c+d x}\right )}{d} \]

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)

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin {align*} \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx &=-\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right ) \log ^5(F)}{d}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \[ -\frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[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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fricas [B]  time = 0.43, size = 244, normalized size = 8.41 \[ -\frac {F^{a} b^{5} {\rm Ei}\left (\frac {b \log \relax (F)}{d x + c}\right ) \log \relax (F)^{5} - {\left (24 \, d^{5} x^{5} + 120 \, c d^{4} x^{4} + 240 \, c^{2} d^{3} x^{3} + 240 \, c^{3} d^{2} x^{2} + 120 \, c^{4} d x + 24 \, c^{5} + {\left (b^{4} d x + b^{4} c\right )} \log \relax (F)^{4} + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \relax (F)^{3} + 2 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \relax (F)^{2} + 6 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \relax (F)\right )} F^{\frac {a d x + a c + b}{d x + c}}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c))*(d*x+c)^4,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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{4} F^{a + \frac {b}{d x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.14, size = 534, normalized size = 18.41 \[ \frac {b^{5} F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{d x +c}\right ) \ln \relax (F )^{5}}{120 d}+\frac {b^{4} x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{4}}{120}+\frac {b^{3} d \,x^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{120}+\frac {b^{2} d^{2} x^{3} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{60}+\frac {b \,d^{3} x^{4} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{20}+\frac {d^{4} x^{5} F^{a} F^{\frac {b}{d x +c}}}{5}+\frac {b^{4} c \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{4}}{120 d}+\frac {b^{3} c x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{60}+\frac {b^{2} c d \,x^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{20}+\frac {b c \,d^{2} x^{3} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{5}+c \,d^{3} x^{4} F^{a} F^{\frac {b}{d x +c}}+\frac {b^{3} c^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{120 d}+\frac {b^{2} c^{2} x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{20}+\frac {3 b \,c^{2} d \,x^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{10}+2 c^{2} d^{2} x^{3} F^{a} F^{\frac {b}{d x +c}}+\frac {b^{2} c^{3} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{60 d}+\frac {b \,c^{3} x \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{5}+2 c^{3} d \,x^{2} F^{a} F^{\frac {b}{d x +c}}+\frac {b \,c^{4} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{20 d}+c^{4} x \,F^{a} F^{\frac {b}{d x +c}}+\frac {c^{5} F^{a} F^{\frac {b}{d x +c}}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)*b)*(d*x+c)^4,x)

[Out]

1/5*d^4*F^a*F^(1/(d*x+c)*b)*x^5+d^3*F^a*F^(1/(d*x+c)*b)*c*x^4+2*d^2*F^a*F^(1/(d*x+c)*b)*c^2*x^3+2*d*F^a*F^(1/(
d*x+c)*b)*c^3*x^2+F^a*F^(1/(d*x+c)*b)*c^4*x+1/5/d*F^a*F^(1/(d*x+c)*b)*c^5+1/20*d^3*b*ln(F)*F^a*F^(1/(d*x+c)*b)
*x^4+1/5*d^2*b*ln(F)*F^a*F^(1/(d*x+c)*b)*c*x^3+3/10*d*b*ln(F)*F^a*F^(1/(d*x+c)*b)*c^2*x^2+1/5*b*ln(F)*F^a*F^(1
/(d*x+c)*b)*c^3*x+1/20/d*b*ln(F)*F^a*F^(1/(d*x+c)*b)*c^4+1/60*d^2*b^2*ln(F)^2*F^a*F^(1/(d*x+c)*b)*x^3+1/20*d*b
^2*ln(F)^2*F^a*F^(1/(d*x+c)*b)*c*x^2+1/20*b^2*ln(F)^2*F^a*F^(1/(d*x+c)*b)*c^2*x+1/60/d*b^2*ln(F)^2*F^a*F^(1/(d
*x+c)*b)*c^3+1/120*d*b^3*ln(F)^3*F^a*F^(1/(d*x+c)*b)*x^2+1/60*b^3*ln(F)^3*F^a*F^(1/(d*x+c)*b)*c*x+1/120/d*b^3*
ln(F)^3*F^a*F^(1/(d*x+c)*b)*c^2+1/120*b^4*ln(F)^4*F^a*F^(1/(d*x+c)*b)*x+1/120/d*b^4*ln(F)^4*F^a*F^(1/(d*x+c)*b
)*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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{120} \, {\left (24 \, F^{a} d^{4} x^{5} + 6 \, {\left (F^{a} b d^{3} \log \relax (F) + 20 \, F^{a} c d^{3}\right )} x^{4} + 2 \, {\left (F^{a} b^{2} d^{2} \log \relax (F)^{2} + 12 \, F^{a} b c d^{2} \log \relax (F) + 120 \, F^{a} c^{2} d^{2}\right )} x^{3} + {\left (F^{a} b^{3} d \log \relax (F)^{3} + 6 \, F^{a} b^{2} c d \log \relax (F)^{2} + 36 \, F^{a} b c^{2} d \log \relax (F) + 240 \, F^{a} c^{3} d\right )} x^{2} + {\left (F^{a} b^{4} \log \relax (F)^{4} + 2 \, F^{a} b^{3} c \log \relax (F)^{3} + 6 \, F^{a} b^{2} c^{2} \log \relax (F)^{2} + 24 \, F^{a} b c^{3} \log \relax (F) + 120 \, F^{a} c^{4}\right )} x\right )} F^{\frac {b}{d x + c}} + \int \frac {{\left (F^{a} b^{5} d x \log \relax (F)^{5} - F^{a} b^{4} c^{2} \log \relax (F)^{4} - 2 \, F^{a} b^{3} c^{3} \log \relax (F)^{3} - 6 \, F^{a} b^{2} c^{4} \log \relax (F)^{2} - 24 \, F^{a} b c^{5} \log \relax (F)\right )} F^{\frac {b}{d x + c}}}{120 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c))*(d*x+c)^4,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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mupad [B]  time = 3.67, size = 181, normalized size = 6.24 \[ \frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^5}{5\,d}+\frac {F^a\,b^5\,{\ln \relax (F)}^5\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{c+d\,x}\right )}{120\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^2\,{\ln \relax (F)}^2\,{\left (c+d\,x\right )}^3}{60\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^3\,{\ln \relax (F)}^3\,{\left (c+d\,x\right )}^2}{120\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \relax (F)\,{\left (c+d\,x\right )}^4}{20\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^4\,{\ln \relax (F)}^4\,\left (c+d\,x\right )}{120\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^a*F^(b/(c + d*x))*(c + d*x)^5)/(5*d) + (F^a*b^5*log(F)^5*expint(-(b*log(F))/(c + d*x)))/(120*d) + (F^a*F^(b
/(c + d*x))*b^2*log(F)^2*(c + d*x)^3)/(60*d) + (F^a*F^(b/(c + d*x))*b^3*log(F)^3*(c + d*x)^2)/(120*d) + (F^a*F
^(b/(c + d*x))*b*log(F)*(c + d*x)^4)/(20*d) + (F^a*F^(b/(c + d*x))*b^4*log(F)^4*(c + d*x))/(120*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{c + d x}} \left (c + d x\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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