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

Optimal. Leaf size=57 \[ \frac{F^{a+\frac{b}{c+d x}}}{b^2 d \log ^2(F)}-\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F) (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.0849565, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2212, 2209} \[ \frac{F^{a+\frac{b}{c+d x}}}{b^2 d \log ^2(F)}-\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F) (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0174742, size = 41, normalized size = 0.72 \[ \frac{F^{a+\frac{b}{c+d x}} (-b \log (F)+c+d x)}{b^2 d \log ^2(F) (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.021, size = 106, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( dx+c \right ) ^{2}} \left ({\frac{d{x}^{2}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-{\frac{ \left ( b\ln \left ( F \right ) -2\,c \right ) x}{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-{\frac{c \left ( b\ln \left ( F \right ) -c \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}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)^3,x)

[Out]

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

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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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.59182, size = 117, normalized size = 2.05 \begin{align*} \frac{{\left (d x - b \log \left (F\right ) + c\right )} F^{\frac{a d x + a c + b}{d x + c}}}{{\left (b^{2} d^{2} x + b^{2} c d\right )} \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.198009, size = 44, normalized size = 0.77 \begin{align*} \frac{F^{a + \frac{b}{c + d x}} \left (- b \log{\left (F \right )} + c + d x\right )}{b^{2} c d \log{\left (F \right )}^{2} + b^{2} d^{2} x \log{\left (F \right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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