\(\int \frac {F^{a+b (c+d x)^3}}{(c+d x)^3} \, dx\) [297]

Optimal result

Integrand size = 21, antiderivative size = 49 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^3} \, dx=-\frac {F^a \Gamma \left (-\frac {2}{3},-b (c+d x)^3 \log (F)\right ) \left (-b (c+d x)^3 \log (F)\right )^{2/3}}{3 d (c+d x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2250} \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^3} \, dx=-\frac {F^a \left (-b \log (F) (c+d x)^3\right )^{2/3} \Gamma \left (-\frac {2}{3},-b (c+d x)^3 \log (F)\right )}{3 d (c+d x)^2} \]

[In]

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

[Out]

-1/3*(F^a*Gamma[-2/3, -(b*(c + d*x)^3*Log[F])]*(-(b*(c + d*x)^3*Log[F]))^(2/3))/(d*(c + d*x)^2)

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^3} \, dx=-\frac {F^a \Gamma \left (-\frac {2}{3},-b (c+d x)^3 \log (F)\right ) \left (-b (c+d x)^3 \log (F)\right )^{2/3}}{3 d (c+d x)^2} \]

[In]

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

[Out]

-1/3*(F^a*Gamma[-2/3, -(b*(c + d*x)^3*Log[F])]*(-(b*(c + d*x)^3*Log[F]))^(2/3))/(d*(c + d*x)^2)

Maple [F]

\[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{3}}d x\]

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (43) = 86\).

Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.76 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^3} \, dx=\frac {\left (-b d^{3} \log \left (F\right )\right )^{\frac {2}{3}} {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} F^{a} \Gamma \left (\frac {1}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) - F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} d^{2}}{2 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

[In]

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

[Out]

1/2*((-b*d^3*log(F))^(2/3)*(d^2*x^2 + 2*c*d*x + c^2)*F^a*gamma(1/3, -(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x
+ b*c^3)*log(F)) - F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*d^2)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3
)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.78 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^3} \, dx=-\frac {F^a\,\left (3\,F^{b\,{\left (c+d\,x\right )}^3}\,\Gamma \left (\frac {2}{3}\right )-3\,\Gamma \left (\frac {2}{3}\right )\,\Gamma \left (\frac {1}{3},-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{2/3}+2\,\pi \,\sqrt {3}\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{2/3}\right )}{6\,d\,\Gamma \left (\frac {2}{3}\right )\,{\left (c+d\,x\right )}^2} \]

[In]

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

[Out]

-(F^a*(3*F^(b*(c + d*x)^3)*gamma(2/3) - 3*gamma(2/3)*igamma(1/3, -b*log(F)*(c + d*x)^3)*(-b*log(F)*(c + d*x)^3
)^(2/3) + 2*3^(1/2)*pi*(-b*log(F)*(c + d*x)^3)^(2/3)))/(6*d*gamma(2/3)*(c + d*x)^2)