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

Optimal. Leaf size=90 \[ \frac{2 F^{a+\frac{b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)}-\frac{2 F^{a+\frac{b}{c+d x}}}{b^3 d \log ^3(F)}-\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F) (c+d x)^2} \]

[Out]

(-2*F^(a + b/(c + d*x)))/(b^3*d*Log[F]^3) + (2*F^(a + b/(c + d*x)))/(b^2*d*(c + d*x)*Log[F]^2) - F^(a + b/(c +
d*x))/(b*d*(c + d*x)^2*Log[F])

________________________________________________________________________________________

Rubi [A]  time = 0.133095, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac{2 F^{a+\frac{b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)}-\frac{2 F^{a+\frac{b}{c+d x}}}{b^3 d \log ^3(F)}-\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F) (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*F^(a + b/(c + d*x)))/(b^3*d*Log[F]^3) + (2*F^(a + b/(c + d*x)))/(b^2*d*(c + d*x)*Log[F]^2) - F^(a + b/(c +
d*x))/(b*d*(c + d*x)^2*Log[F])

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2209

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

Rubi steps

\begin{align*} \int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x)^4} \, dx &=-\frac{F^{a+\frac{b}{c+d x}}}{b d (c+d x)^2 \log (F)}-\frac{2 \int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x)^3} \, dx}{b \log (F)}\\ &=\frac{2 F^{a+\frac{b}{c+d x}}}{b^2 d (c+d x) \log ^2(F)}-\frac{F^{a+\frac{b}{c+d x}}}{b d (c+d x)^2 \log (F)}+\frac{2 \int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x)^2} \, dx}{b^2 \log ^2(F)}\\ &=-\frac{2 F^{a+\frac{b}{c+d x}}}{b^3 d \log ^3(F)}+\frac{2 F^{a+\frac{b}{c+d x}}}{b^2 d (c+d x) \log ^2(F)}-\frac{F^{a+\frac{b}{c+d x}}}{b d (c+d x)^2 \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0247779, size = 60, normalized size = 0.67 \[ -\frac{F^{a+\frac{b}{c+d x}} \left (b^2 \log ^2(F)-2 b \log (F) (c+d x)+2 (c+d x)^2\right )}{b^3 d \log ^3(F) (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((F^(a + b/(c + d*x))*(2*(c + d*x)^2 - 2*b*(c + d*x)*Log[F] + b^2*Log[F]^2))/(b^3*d*(c + d*x)^2*Log[F]^3))

________________________________________________________________________________________

Maple [A]  time = 0.028, size = 169, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( dx+c \right ) ^{3}} \left ( -2\,{\frac{{d}^{2}{x}^{3}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-4\,bc\ln \left ( F \right ) +6\,{c}^{2} \right ) x}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}+2\,{\frac{d \left ( b\ln \left ( F \right ) -3\,c \right ){x}^{2}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-2\,bc\ln \left ( F \right ) +2\,{c}^{2} \right ) c}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}d}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*d^2/ln(F)^3/b^3*x^3*exp((a+b/(d*x+c))*ln(F))-(ln(F)^2*b^2-4*b*c*ln(F)+6*c^2)/ln(F)^3/b^3*x*exp((a+b/(d*x+c
))*ln(F))+2*d*(b*ln(F)-3*c)/ln(F)^3/b^3*x^2*exp((a+b/(d*x+c))*ln(F))-(ln(F)^2*b^2-2*b*c*ln(F)+2*c^2)*c/b^3/ln(
F)^3/d*exp((a+b/(d*x+c))*ln(F)))/(d*x+c)^3

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (d x + c\right )}^{4}}\,{d x} \end{align*}

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]

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

________________________________________________________________________________________

Fricas [A]  time = 1.62484, size = 212, normalized size = 2.36 \begin{align*} -\frac{{\left (2 \, d^{2} x^{2} + b^{2} \log \left (F\right )^{2} + 4 \, c d x + 2 \, c^{2} - 2 \,{\left (b d x + b c\right )} \log \left (F\right )\right )} F^{\frac{a d x + a c + b}{d x + c}}}{{\left (b^{3} d^{3} x^{2} + 2 \, b^{3} c d^{2} x + b^{3} c^{2} d\right )} \log \left (F\right )^{3}} \end{align*}

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]

-(2*d^2*x^2 + b^2*log(F)^2 + 4*c*d*x + 2*c^2 - 2*(b*d*x + b*c)*log(F))*F^((a*d*x + a*c + b)/(d*x + c))/((b^3*d
^3*x^2 + 2*b^3*c*d^2*x + b^3*c^2*d)*log(F)^3)

________________________________________________________________________________________

Sympy [A]  time = 0.236293, size = 102, normalized size = 1.13 \begin{align*} \frac{F^{a + \frac{b}{c + d x}} \left (- b^{2} \log{\left (F \right )}^{2} + 2 b c \log{\left (F \right )} + 2 b d x \log{\left (F \right )} - 2 c^{2} - 4 c d x - 2 d^{2} x^{2}\right )}{b^{3} c^{2} d \log{\left (F \right )}^{3} + 2 b^{3} c d^{2} x \log{\left (F \right )}^{3} + b^{3} d^{3} x^{2} \log{\left (F \right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F**(a + b/(c + d*x))*(-b**2*log(F)**2 + 2*b*c*log(F) + 2*b*d*x*log(F) - 2*c**2 - 4*c*d*x - 2*d**2*x**2)/(b**3*
c**2*d*log(F)**3 + 2*b**3*c*d**2*x*log(F)**3 + b**3*d**3*x**2*log(F)**3)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (d x + c\right )}^{4}}\,{d x} \end{align*}

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(F^(a + b/(d*x + c))/(d*x + c)^4, x)