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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0999681, 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 \[ -\frac{F^a \left (-b \log (F) (c+d x)^3\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.29846, size = 49, normalized size = 1. \[ - \frac{F^{a} \left (- b \left (c + d x\right )^{3} \log{\left (F \right )}\right )^{\frac{2}{3}} \Gamma{\left (- \frac{2}{3},- b \left (c + d x\right )^{3} \log{\left (F \right )} \right )}}{3 d \left (c + d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.113851, size = 63, normalized size = 1.29 \[ -\frac{F^a \left (F^{b (c+d x)^3}-\left (-b \log (F) (c+d x)^3\right )^{2/3} \text{Gamma}\left (\frac{1}{3},-b \log (F) (c+d x)^3\right )\right )}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{ \left ( dx+c \right ) ^{3}}}\, dx \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.250853, size = 200, normalized size = 4.08 \[ -\frac{{\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} F^{a} \Gamma \left (\frac{1}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) \log \left (F\right ) + \left (-b d^{3} \log \left (F\right )\right )^{\frac{1}{3}} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )} \left (-b d^{3} \log \left (F\right )\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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