\(\int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx\) [296]

Optimal result

Integrand size = 21, antiderivative size = 49 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx=-\frac {F^a \Gamma \left (-\frac {1}{3},-b (c+d x)^3 \log (F)\right ) \sqrt [3]{-b (c+d x)^3 \log (F)}}{3 d (c+d x)} \]

[Out]

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

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} \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx=-\frac {F^a \sqrt [3]{-b \log (F) (c+d x)^3} \Gamma \left (-\frac {1}{3},-b (c+d x)^3 \log (F)\right )}{3 d (c+d x)} \]

[In]

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

[Out]

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

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 {1}{3},-b (c+d x)^3 \log (F)\right ) \sqrt [3]{-b (c+d x)^3 \log (F)}}{3 d (c+d x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx=-\frac {F^a \Gamma \left (-\frac {1}{3},-b (c+d x)^3 \log (F)\right ) \sqrt [3]{-b (c+d x)^3 \log (F)}}{3 d (c+d x)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{2}}d x\]

[In]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x)

[Out]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (43) = 86\).

Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.24 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx=\frac {\left (-b d^{3} \log \left (F\right )\right )^{\frac {1}{3}} {\left (d x + c\right )} F^{a} \Gamma \left (\frac {2}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) - F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} d}{d^{3} x + c d^{2}} \]

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx=\int \frac {F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**2,x)

[Out]

Integral(F**(a + b*(c + d*x)**3)/(c + d*x)**2, x)

Maxima [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.51 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx=-\frac {F^a\,\left (F^{b\,{\left (c+d\,x\right )}^3}-\Gamma \left (\frac {2}{3},-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{1/3}+\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{1/3}\right )}{d\,\left (c+d\,x\right )} \]

[In]

int(F^(a + b*(c + d*x)^3)/(c + d*x)^2,x)

[Out]

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