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

Optimal. Leaf size=328 \[ \frac{24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac{12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac{24 f^2 x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac{12 e f x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac{12 f^2 x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{2 e^2 x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{6 e f x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{4 f^2 x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{e^2 x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac{2 e f x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x^4 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

(24*f^2*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) - (12*e*f*F^(a + b*c + b*d*x))/(
b^4*d^4*Log[F]^4) - (24*f^2*F^(a + b*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) + (2*e^2*F
^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) + (12*e*f*F^(a + b*c + b*d*x)*x)/(b^3*d^3
*Log[F]^3) + (12*f^2*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) - (2*e^2*F^(a +
b*c + b*d*x)*x)/(b^2*d^2*Log[F]^2) - (6*e*f*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*L
og[F]^2) - (4*f^2*F^(a + b*c + b*d*x)*x^3)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c
+ b*d*x)*x^2)/(b*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^3)/(b*d*Log[F]) + (f^2
*F^(a + b*c + b*d*x)*x^4)/(b*d*Log[F])

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Rubi [A]  time = 0.86392, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{24 f^2 F^{a+b c+b d x}}{b^5 d^5 \log ^5(F)}-\frac{12 e f F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}-\frac{24 f^2 x F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{2 e^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac{12 e f x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac{12 f^2 x^2 F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{2 e^2 x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{6 e f x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{4 f^2 x^3 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{e^2 x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac{2 e f x^3 F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x^4 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

(24*f^2*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) - (12*e*f*F^(a + b*c + b*d*x))/(
b^4*d^4*Log[F]^4) - (24*f^2*F^(a + b*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) + (2*e^2*F
^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) + (12*e*f*F^(a + b*c + b*d*x)*x)/(b^3*d^3
*Log[F]^3) + (12*f^2*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) - (2*e^2*F^(a +
b*c + b*d*x)*x)/(b^2*d^2*Log[F]^2) - (6*e*f*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*L
og[F]^2) - (4*f^2*F^(a + b*c + b*d*x)*x^3)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c
+ b*d*x)*x^2)/(b*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^3)/(b*d*Log[F]) + (f^2
*F^(a + b*c + b*d*x)*x^4)/(b*d*Log[F])

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Rubi in Sympy [A]  time = 68.9908, size = 352, normalized size = 1.07 \[ \frac{F^{a + b c + b d x} e^{2} x^{2}}{b d \log{\left (F \right )}} + \frac{2 F^{a + b c + b d x} e f x^{3}}{b d \log{\left (F \right )}} + \frac{F^{a + b c + b d x} f^{2} x^{4}}{b d \log{\left (F \right )}} - \frac{2 F^{a + b c + b d x} e^{2} x}{b^{2} d^{2} \log{\left (F \right )}^{2}} - \frac{6 F^{a + b c + b d x} e f x^{2}}{b^{2} d^{2} \log{\left (F \right )}^{2}} - \frac{4 F^{a + b c + b d x} f^{2} x^{3}}{b^{2} d^{2} \log{\left (F \right )}^{2}} + \frac{2 F^{a + b c + b d x} e^{2}}{b^{3} d^{3} \log{\left (F \right )}^{3}} + \frac{12 F^{a + b c + b d x} e f x}{b^{3} d^{3} \log{\left (F \right )}^{3}} + \frac{12 F^{a + b c + b d x} f^{2} x^{2}}{b^{3} d^{3} \log{\left (F \right )}^{3}} - \frac{12 F^{a + b c + b d x} e f}{b^{4} d^{4} \log{\left (F \right )}^{4}} - \frac{24 F^{a + b c + b d x} f^{2} x}{b^{4} d^{4} \log{\left (F \right )}^{4}} + \frac{24 F^{a + b c + b d x} f^{2}}{b^{5} d^{5} \log{\left (F \right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

F**(a + b*c + b*d*x)*e**2*x**2/(b*d*log(F)) + 2*F**(a + b*c + b*d*x)*e*f*x**3/(b
*d*log(F)) + F**(a + b*c + b*d*x)*f**2*x**4/(b*d*log(F)) - 2*F**(a + b*c + b*d*x
)*e**2*x/(b**2*d**2*log(F)**2) - 6*F**(a + b*c + b*d*x)*e*f*x**2/(b**2*d**2*log(
F)**2) - 4*F**(a + b*c + b*d*x)*f**2*x**3/(b**2*d**2*log(F)**2) + 2*F**(a + b*c
+ b*d*x)*e**2/(b**3*d**3*log(F)**3) + 12*F**(a + b*c + b*d*x)*e*f*x/(b**3*d**3*l
og(F)**3) + 12*F**(a + b*c + b*d*x)*f**2*x**2/(b**3*d**3*log(F)**3) - 12*F**(a +
b*c + b*d*x)*e*f/(b**4*d**4*log(F)**4) - 24*F**(a + b*c + b*d*x)*f**2*x/(b**4*d
**4*log(F)**4) + 24*F**(a + b*c + b*d*x)*f**2/(b**5*d**5*log(F)**5)

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Mathematica [A]  time = 0.0891082, size = 121, normalized size = 0.37 \[ \frac{F^{a+b (c+d x)} \left (b^4 d^4 x^2 \log ^4(F) (e+f x)^2-2 b^3 d^3 x \log ^3(F) \left (e^2+3 e f x+2 f^2 x^2\right )+2 b^2 d^2 \log ^2(F) \left (e^2+6 e f x+6 f^2 x^2\right )-12 b d f \log (F) (e+2 f x)+24 f^2\right )}{b^5 d^5 \log ^5(F)} \]

Antiderivative was successfully verified.

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

[Out]

(F^(a + b*(c + d*x))*(24*f^2 - 12*b*d*f*(e + 2*f*x)*Log[F] + 2*b^2*d^2*(e^2 + 6*
e*f*x + 6*f^2*x^2)*Log[F]^2 - 2*b^3*d^3*x*(e^2 + 3*e*f*x + 2*f^2*x^2)*Log[F]^3 +
b^4*d^4*x^2*(e + f*x)^2*Log[F]^4))/(b^5*d^5*Log[F]^5)

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Maple [A]  time = 0.014, size = 197, normalized size = 0.6 \[{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}{f}^{2}{x}^{4}+2\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}ef{x}^{3}+ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}{e}^{2}{x}^{2}-4\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{f}^{2}{x}^{3}-6\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}ef{x}^{2}-2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{e}^{2}x+12\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{f}^{2}{x}^{2}+12\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}efx+2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}-24\,\ln \left ( F \right ) bd{f}^{2}x-12\,ef\ln \left ( F \right ) bd+24\,{f}^{2} \right ){F}^{bdx+cb+a}}{ \left ( \ln \left ( F \right ) \right ) ^{5}{b}^{5}{d}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(ln(F)^4*b^4*d^4*f^2*x^4+2*ln(F)^4*b^4*d^4*e*f*x^3+ln(F)^4*b^4*d^4*e^2*x^2-4*ln(
F)^3*b^3*d^3*f^2*x^3-6*ln(F)^3*b^3*d^3*e*f*x^2-2*ln(F)^3*b^3*d^3*e^2*x+12*ln(F)^
2*b^2*d^2*f^2*x^2+12*ln(F)^2*b^2*d^2*e*f*x+2*ln(F)^2*b^2*d^2*e^2-24*ln(F)*b*d*f^
2*x-12*e*f*ln(F)*b*d+24*f^2)*F^(b*d*x+b*c+a)/ln(F)^5/b^5/d^5

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Maxima [A]  time = 0.821868, size = 354, normalized size = 1.08 \[ \frac{{\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} F^{b d x} e^{2}}{b^{3} d^{3} \log \left (F\right )^{3}} + \frac{2 \,{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} e f}{b^{4} d^{4} \log \left (F\right )^{4}} + \frac{{\left (F^{b c + a} b^{4} d^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{b c + a} b d x \log \left (F\right ) + 24 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{5} d^{5} \log \left (F\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(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)*e^2/(b^3*d^3*log(F)^3) + 2*(F^(b*c + a)*b^3*d^3*x^3*log(F)^3 - 3*F^(b*
c + a)*b^2*d^2*x^2*log(F)^2 + 6*F^(b*c + a)*b*d*x*log(F) - 6*F^(b*c + a))*F^(b*d
*x)*e*f/(b^4*d^4*log(F)^4) + (F^(b*c + a)*b^4*d^4*x^4*log(F)^4 - 4*F^(b*c + a)*b
^3*d^3*x^3*log(F)^3 + 12*F^(b*c + a)*b^2*d^2*x^2*log(F)^2 - 24*F^(b*c + a)*b*d*x
*log(F) + 24*F^(b*c + a))*F^(b*d*x)*f^2/(b^5*d^5*log(F)^5)

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Fricas [A]  time = 0.263431, size = 240, normalized size = 0.73 \[ \frac{{\left ({\left (b^{4} d^{4} f^{2} x^{4} + 2 \, b^{4} d^{4} e f x^{3} + b^{4} d^{4} e^{2} x^{2}\right )} \log \left (F\right )^{4} - 2 \,{\left (2 \, b^{3} d^{3} f^{2} x^{3} + 3 \, b^{3} d^{3} e f x^{2} + b^{3} d^{3} e^{2} x\right )} \log \left (F\right )^{3} + 2 \,{\left (6 \, b^{2} d^{2} f^{2} x^{2} + 6 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} + 24 \, f^{2} - 12 \,{\left (2 \, b d f^{2} x + b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{5} d^{5} \log \left (F\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b^4*d^4*f^2*x^4 + 2*b^4*d^4*e*f*x^3 + b^4*d^4*e^2*x^2)*log(F)^4 - 2*(2*b^3*d^3
*f^2*x^3 + 3*b^3*d^3*e*f*x^2 + b^3*d^3*e^2*x)*log(F)^3 + 2*(6*b^2*d^2*f^2*x^2 +
6*b^2*d^2*e*f*x + b^2*d^2*e^2)*log(F)^2 + 24*f^2 - 12*(2*b*d*f^2*x + b*d*e*f)*lo
g(F))*F^(b*d*x + b*c + a)/(b^5*d^5*log(F)^5)

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Sympy [A]  time = 0.546149, size = 260, normalized size = 0.79 \[ \begin{cases} \frac{F^{a + b \left (c + d x\right )} \left (b^{4} d^{4} e^{2} x^{2} \log{\left (F \right )}^{4} + 2 b^{4} d^{4} e f x^{3} \log{\left (F \right )}^{4} + b^{4} d^{4} f^{2} x^{4} \log{\left (F \right )}^{4} - 2 b^{3} d^{3} e^{2} x \log{\left (F \right )}^{3} - 6 b^{3} d^{3} e f x^{2} \log{\left (F \right )}^{3} - 4 b^{3} d^{3} f^{2} x^{3} \log{\left (F \right )}^{3} + 2 b^{2} d^{2} e^{2} \log{\left (F \right )}^{2} + 12 b^{2} d^{2} e f x \log{\left (F \right )}^{2} + 12 b^{2} d^{2} f^{2} x^{2} \log{\left (F \right )}^{2} - 12 b d e f \log{\left (F \right )} - 24 b d f^{2} x \log{\left (F \right )} + 24 f^{2}\right )}{b^{5} d^{5} \log{\left (F \right )}^{5}} & \text{for}\: b^{5} d^{5} \log{\left (F \right )}^{5} \neq 0 \\\frac{e^{2} x^{3}}{3} + \frac{e f x^{4}}{2} + \frac{f^{2} x^{5}}{5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((F**(a + b*(c + d*x))*(b**4*d**4*e**2*x**2*log(F)**4 + 2*b**4*d**4*e*f
*x**3*log(F)**4 + b**4*d**4*f**2*x**4*log(F)**4 - 2*b**3*d**3*e**2*x*log(F)**3 -
6*b**3*d**3*e*f*x**2*log(F)**3 - 4*b**3*d**3*f**2*x**3*log(F)**3 + 2*b**2*d**2*
e**2*log(F)**2 + 12*b**2*d**2*e*f*x*log(F)**2 + 12*b**2*d**2*f**2*x**2*log(F)**2
- 12*b*d*e*f*log(F) - 24*b*d*f**2*x*log(F) + 24*f**2)/(b**5*d**5*log(F)**5), Ne
(b**5*d**5*log(F)**5, 0)), (e**2*x**3/3 + e*f*x**4/2 + f**2*x**5/5, True))

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GIAC/XCAS [A]  time = 0.341357, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done