3.4.54 \(\int F^{a+\frac {b}{(c+d x)^3}} (c+d x) \, dx\) [354]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {2250} \begin {gather*} \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} \end {gather*}

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 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{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 = 0.02, size = 49, normalized size = 1.00 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{a +\frac {b}{\left (d x +c \right )^{3}}} \left (d x +c \right )\, dx\]

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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (43) = 86\).
time = 0.09, size = 142, normalized size = 2.90 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 4.99, size = 107, normalized size = 2.18 \begin {gather*} \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,{\left (c+d\,x\right )}^2}{2\,d}-\frac {F^a\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )\,{\left (c+d\,x\right )}^2\,{\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}^{2/3}}{2\,d}+\frac {\pi \,\sqrt {3}\,F^a\,{\left (c+d\,x\right )}^2\,{\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}^{2/3}}{3\,d\,\Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________