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

Optimal. Leaf size=49 \[ \frac{F^a (c+d x)^4 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0479236, antiderivative size = 49, 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 = {2218} \[ \frac{F^a (c+d x)^4 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*(c + d*x)^4*Gamma[-4/3, -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(4/3))/(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)^3 \, dx &=\frac{F^a (c+d x)^4 \Gamma \left (-\frac{4}{3},-\frac{b \log (F)}{(c+d x)^3}\right ) \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{4/3}}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0280762, size = 49, normalized size = 1. \[ \frac{F^a (c+d x)^4 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) ^{3}\, 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)^3,x)

[Out]

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

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

[Out]

1/4*(F^a*d^3*x^4 + 4*F^a*c*d^2*x^3 + 6*F^a*c^2*d*x^2 + (4*F^a*c^3 + 3*F^a*b*log(F))*x)*F^(b/(d^3*x^3 + 3*c*d^2
*x^2 + 3*c^2*d*x + c^3)) + integrate(-3/4*(F^a*b*c^4*log(F) - 3*F^a*b^2*d*x*log(F)^2)*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 = 1.58905, size = 402, normalized size = 8.2 \begin{align*} -\frac{3 \, F^{a} b d \left (-\frac{b \log \left (F\right )}{d^{3}}\right )^{\frac{1}{3}} \Gamma \left (\frac{2}{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 ) \log \left (F\right ) -{\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} + 3 \,{\left (b d x + b c\right )} \log \left (F\right )\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}}}}{4 \, 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)^3,x, algorithm="fricas")

[Out]

-1/4*(3*F^a*b*d*(-b*log(F)/d^3)^(1/3)*gamma(2/3, -b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*log(F) -
(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4 + 3*(b*d*x + b*c)*log(F))*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)**3,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 )}^{3} 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)^3,x, algorithm="giac")

[Out]

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