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

Optimal. Leaf size=49 \[ \frac{F^a (c+d x)^2 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0273341, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2218} \[ \frac{F^a (c+d x)^2 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0211379, size = 49, normalized size = 1. \[ \frac{F^a (c+d x)^2 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (F^{a} d x^{2} + 2 \, F^{a} c x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int \frac{3 \,{\left (F^{a} b d^{2} x^{2} \log \left (F\right ) + 2 \, F^{a} b c d x \log \left (F\right )\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{2 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(F^a*d*x^2 + 2*F^a*c*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + integrate(3/2*(F^a*b*d^2*x^2*log
(F) + 2*F^a*b*c*d*x*log(F))*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2
*x^2 + 4*c^3*d*x + c^4), x)

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Fricas [B]  time = 1.63266, size = 313, normalized size = 6.39 \begin{align*} -\frac{F^{a} d^{2} \left (-\frac{b \log \left (F\right )}{d^{3}}\right )^{\frac{2}{3}} \Gamma \left (\frac{1}{3}, -\frac{b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) -{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} 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}}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} F^{a + \frac{b}{{\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)^3)*(d*x+c),x, algorithm="giac")

[Out]

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