3.259 \(\int F^{a+b (c+d x)^2} (c+d x)^3 \, dx\)

Optimal. Leaf size=62 \[ \frac{(c+d x)^2 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac{F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.163811, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{(c+d x)^2 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac{F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.04622, size = 49, normalized size = 0.79 \[ \frac{F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{2}}{2 b d \log{\left (F \right )}} - \frac{F^{a + b \left (c + d x\right )^{2}}}{2 b^{2} d \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0306243, size = 40, normalized size = 0.65 \[ \frac{F^{a+b (c+d x)^2} \left (b \log (F) (c+d x)^2-1\right )}{2 b^2 d \log ^2(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 63, normalized size = 1. \[{\frac{ \left ( \ln \left ( F \right ) b{d}^{2}{x}^{2}+2\,\ln \left ( F \right ) bcdx+\ln \left ( F \right ) b{c}^{2}-1 \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.03958, size = 1160, normalized size = 18.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b*c*d*(erf(sqrt(-(b*d^2*x*log(F)
+ b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)/((b*d^2*log(F))^(3/2)*sqrt(-(b*d^
2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - b*d^2*e^((b*d^2*x*log(F) + b*c*d
*log(F))^2/(b*d^2*log(F)))*log(F)/(b*d^2*log(F))^(3/2))*F^(b*c^2 + a)*c^2*d/(sqr
t(b*d^2*log(F))*F^(b*c^2)) + 3/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^2*c
^2*d^2*(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)
^2/((b*d^2*log(F))^(5/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))
) - 2*b^2*c*d^3*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2/(b
*d^2*log(F))^(5/2) - (b*d^2*x*log(F) + b*c*d*log(F))^3*gamma(3/2, -(b*d^2*x*log(
F) + b*c*d*log(F))^2/(b*d^2*log(F)))/((b*d^2*log(F))^(5/2)*(-(b*d^2*x*log(F) + b
*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*c*d^2/(sqrt(b*d^2*log(F))*F
^(b*c^2)) - 1/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^3*c^3*d^3*(erf(sqrt(
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^3/((b*d^2*log(F)
)^(7/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 3*b^3*c^2*d^4
*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/(b*d^2*log(F))^(7
/2) + b^2*d^4*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)
^2/(b*d^2*log(F))^(7/2) - 3*(b*d^2*x*log(F) + b*c*d*log(F))^3*b*c*d*gamma(3/2, -
(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)/((b*d^2*log(F))^(7/2)*(
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*d^3/(sq
rt(b*d^2*log(F))*F^(b*c^2)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^3*erf(sqrt(-b*log(F))
*d*x - b*c*log(F)/sqrt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)*d)

_______________________________________________________________________________________

Fricas [A]  time = 0.261042, size = 81, normalized size = 1.31 \[ \frac{{\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) - 1\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{2} d \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.455677, size = 100, normalized size = 1.61 \[ \begin{cases} \frac{F^{a + b \left (c + d x\right )^{2}} \left (b c^{2} \log{\left (F \right )} + 2 b c d x \log{\left (F \right )} + b d^{2} x^{2} \log{\left (F \right )} - 1\right )}{2 b^{2} d \log{\left (F \right )}^{2}} & \text{for}\: 2 b^{2} d \log{\left (F \right )}^{2} \neq 0 \\c^{3} x + \frac{3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac{d^{3} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**3,x)

[Out]

Piecewise((F**(a + b*(c + d*x)**2)*(b*c**2*log(F) + 2*b*c*d*x*log(F) + b*d**2*x*
*2*log(F) - 1)/(2*b**2*d*log(F)**2), Ne(2*b**2*d*log(F)**2, 0)), (c**3*x + 3*c**
2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.255136, size = 82, normalized size = 1.32 \[ \frac{{\left (b d^{2}{\left (x + \frac{c}{d}\right )}^{2}{\rm ln}\left (F\right ) - 1\right )} e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{2 \, b^{2} d{\rm ln}\left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*d^2*(x + c/d)^2*ln(F) - 1)*e^(b*d^2*x^2*ln(F) + 2*b*c*d*x*ln(F) + b*c^2*l
n(F) + a*ln(F))/(b^2*d*ln(F)^2)