3.347 \(\int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^4} \, dx\)

Optimal. Leaf size=27 \[ -\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F)} \]

[Out]

-F^(a + b/(c + d*x)^3)/(3*b*d*Log[F])

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Rubi [A]  time = 0.0425225, antiderivative size = 27, 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, Rules used = {2209} \[ -\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)/(c + d*x)^4,x]

[Out]

-F^(a + b/(c + d*x)^3)/(3*b*d*Log[F])

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

Mathematica [A]  time = 0.0119075, size = 27, normalized size = 1. \[ -\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^4,x]

[Out]

-F^(a + b/(c + d*x)^3)/(3*b*d*Log[F])

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Maple [A]  time = 0.003, size = 26, normalized size = 1. \begin{align*} -{\frac{1}{3\,\ln \left ( F \right ) bd}{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^3)/(d*x+c)^4,x)

[Out]

-1/3*F^(a+b/(d*x+c)^3)/b/d/ln(F)

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Maxima [A]  time = 0.974407, size = 34, normalized size = 1.26 \begin{align*} -\frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{3 \, b d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^4,x, algorithm="maxima")

[Out]

-1/3*F^(a + b/(d*x + c)^3)/(b*d*log(F))

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Fricas [B]  time = 1.57842, size = 161, normalized size = 5.96 \begin{align*} -\frac{F^{\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, b d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^4,x, algorithm="fricas")

[Out]

-1/3*F^((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(b*d*
log(F))

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Sympy [A]  time = 0.521485, size = 66, normalized size = 2.44 \begin{align*} \begin{cases} - \frac{F^{a + \frac{b}{\left (c + d x\right )^{3}}}}{3 b d \log{\left (F \right )}} & \text{for}\: 3 b d \log{\left (F \right )} \neq 0 \\- \frac{1}{3 c^{3} d + 9 c^{2} d^{2} x + 9 c d^{3} x^{2} + 3 d^{4} x^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**3)/(d*x+c)**4,x)

[Out]

Piecewise((-F**(a + b/(c + d*x)**3)/(3*b*d*log(F)), Ne(3*b*d*log(F), 0)), (-1/(3*c**3*d + 9*c**2*d**2*x + 9*c*
d**3*x**2 + 3*d**4*x**3), True))

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Giac [A]  time = 1.38827, size = 34, normalized size = 1.26 \begin{align*} -\frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{3 \, b d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^4,x, algorithm="giac")

[Out]

-1/3*F^(a + b/(d*x + c)^3)/(b*d*log(F))