3.68 \(\int F^{a+b (c+d x)} (e+f x)^2 \, dx\)

Optimal. Leaf size=85 \[ \frac{2 f^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{2 f (e+f x) F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{(e+f x)^2 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.192087, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 f^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{2 f (e+f x) F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{(e+f x)^2 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.3629, size = 85, normalized size = 1. \[ \frac{F^{a + b c + b d x} \left (e + f x\right )^{2}}{b d \log{\left (F \right )}} - \frac{2 F^{a + b c + b d x} f \left (e + f x\right )}{b^{2} d^{2} \log{\left (F \right )}^{2}} + \frac{2 F^{a + b c + b d x} f^{2}}{b^{3} d^{3} \log{\left (F \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

F**(a + b*c + b*d*x)*(e + f*x)**2/(b*d*log(F)) - 2*F**(a + b*c + b*d*x)*f*(e + f
*x)/(b**2*d**2*log(F)**2) + 2*F**(a + b*c + b*d*x)*f**2/(b**3*d**3*log(F)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0568514, size = 58, normalized size = 0.68 \[ \frac{F^{a+b (c+d x)} \left (b^2 d^2 \log ^2(F) (e+f x)^2-2 b d f \log (F) (e+f x)+2 f^2\right )}{b^3 d^3 \log ^3(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 93, normalized size = 1.1 \[{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{f}^{2}{x}^{2}+2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}efx+ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}-2\,\ln \left ( F \right ) bd{f}^{2}x-2\,ef\ln \left ( F \right ) bd+2\,{f}^{2} \right ){F}^{bdx+cb+a}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(ln(F)^2*b^2*d^2*f^2*x^2+2*ln(F)^2*b^2*d^2*e*f*x+ln(F)^2*b^2*d^2*e^2-2*ln(F)*b*d
*f^2*x-2*e*f*ln(F)*b*d+2*f^2)*F^(b*d*x+b*c+a)/ln(F)^3/b^3/d^3

_______________________________________________________________________________________

Maxima [A]  time = 0.783351, size = 181, normalized size = 2.13 \[ \frac{F^{b d x + b c + a} e^{2}}{b d \log \left (F\right )} + \frac{2 \,{\left (F^{b c + a} b d x \log \left (F\right ) - F^{b c + a}\right )} F^{b d x} e f}{b^{2} d^{2} \log \left (F\right )^{2}} + \frac{{\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{3} d^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.253537, size = 115, normalized size = 1.35 \[ \frac{{\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} + 2 \, f^{2} - 2 \,{\left (b d f^{2} x + b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{3} d^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.408104, size = 134, normalized size = 1.58 \[ \begin{cases} \frac{F^{a + b \left (c + d x\right )} \left (b^{2} d^{2} e^{2} \log{\left (F \right )}^{2} + 2 b^{2} d^{2} e f x \log{\left (F \right )}^{2} + b^{2} d^{2} f^{2} x^{2} \log{\left (F \right )}^{2} - 2 b d e f \log{\left (F \right )} - 2 b d f^{2} x \log{\left (F \right )} + 2 f^{2}\right )}{b^{3} d^{3} \log{\left (F \right )}^{3}} & \text{for}\: b^{3} d^{3} \log{\left (F \right )}^{3} \neq 0 \\e^{2} x + e f x^{2} + \frac{f^{2} x^{3}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((F**(a + b*(c + d*x))*(b**2*d**2*e**2*log(F)**2 + 2*b**2*d**2*e*f*x*lo
g(F)**2 + b**2*d**2*f**2*x**2*log(F)**2 - 2*b*d*e*f*log(F) - 2*b*d*f**2*x*log(F)
+ 2*f**2)/(b**3*d**3*log(F)**3), Ne(b**3*d**3*log(F)**3, 0)), (e**2*x + e*f*x**
2 + f**2*x**3/3, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.292454, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done