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

Optimal. Leaf size=242 \[ -\frac{6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac{6 f^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{4 e f x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{3 f^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{e^2 x F^{a+b c+b d x}}{b d \log (F)}+\frac{2 e f x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x^3 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.583605, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac{4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac{6 f^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac{e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{4 e f x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac{3 f^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac{e^2 x F^{a+b c+b d x}}{b d \log (F)}+\frac{2 e f x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac{f^2 x^3 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.535, size = 253, normalized size = 1.05 \[ \frac{F^{a + b c + b d x} e^{2} x}{b d \log{\left (F \right )}} + \frac{2 F^{a + b c + b d x} e f x^{2}}{b d \log{\left (F \right )}} + \frac{F^{a + b c + b d x} f^{2} x^{3}}{b d \log{\left (F \right )}} - \frac{F^{a + b c + b d x} e^{2}}{b^{2} d^{2} \log{\left (F \right )}^{2}} - \frac{4 F^{a + b c + b d x} e f x}{b^{2} d^{2} \log{\left (F \right )}^{2}} - \frac{3 F^{a + b c + b d x} f^{2} x^{2}}{b^{2} d^{2} \log{\left (F \right )}^{2}} + \frac{4 F^{a + b c + b d x} e f}{b^{3} d^{3} \log{\left (F \right )}^{3}} + \frac{6 F^{a + b c + b d x} f^{2} x}{b^{3} d^{3} \log{\left (F \right )}^{3}} - \frac{6 F^{a + b c + b d x} f^{2}}{b^{4} d^{4} \log{\left (F \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0777754, size = 91, normalized size = 0.38 \[ \frac{F^{a+b (c+d x)} \left (b^3 d^3 x \log ^3(F) (e+f x)^2-b^2 d^2 \log ^2(F) \left (e^2+4 e f x+3 f^2 x^2\right )+2 b d f \log (F) (2 e+3 f x)-6 f^2\right )}{b^4 d^4 \log ^4(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 144, normalized size = 0.6 \[{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{f}^{2}{x}^{3}+2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}ef{x}^{2}+ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{e}^{2}x-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{f}^{2}{x}^{2}-4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}efx- \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}+6\,\ln \left ( F \right ) bd{f}^{2}x+4\,ef\ln \left ( F \right ) bd-6\,{f}^{2} \right ){F}^{bdx+cb+a}}{ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.828295, size = 265, normalized size = 1.1 \[ \frac{{\left (F^{b c + a} b d x \log \left (F\right ) - F^{b c + a}\right )} F^{b d x} e^{2}}{b^{2} d^{2} \log \left (F\right )^{2}} + \frac{2 \,{\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} F^{b d x} e f}{b^{3} d^{3} \log \left (F\right )^{3}} + \frac{{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{4} d^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273116, size = 178, normalized size = 0.74 \[ \frac{{\left ({\left (b^{3} d^{3} f^{2} x^{3} + 2 \, b^{3} d^{3} e f x^{2} + b^{3} d^{3} e^{2} x\right )} \log \left (F\right )^{3} -{\left (3 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} d^{2} e f x + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 6 \, f^{2} + 2 \,{\left (3 \, b d f^{2} x + 2 \, b d e f\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{4} d^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.46996, size = 199, normalized size = 0.82 \[ \begin{cases} \frac{F^{a + b \left (c + d x\right )} \left (b^{3} d^{3} e^{2} x \log{\left (F \right )}^{3} + 2 b^{3} d^{3} e f x^{2} \log{\left (F \right )}^{3} + b^{3} d^{3} f^{2} x^{3} \log{\left (F \right )}^{3} - b^{2} d^{2} e^{2} \log{\left (F \right )}^{2} - 4 b^{2} d^{2} e f x \log{\left (F \right )}^{2} - 3 b^{2} d^{2} f^{2} x^{2} \log{\left (F \right )}^{2} + 4 b d e f \log{\left (F \right )} + 6 b d f^{2} x \log{\left (F \right )} - 6 f^{2}\right )}{b^{4} d^{4} \log{\left (F \right )}^{4}} & \text{for}\: b^{4} d^{4} \log{\left (F \right )}^{4} \neq 0 \\\frac{e^{2} x^{2}}{2} + \frac{2 e f x^{3}}{3} + \frac{f^{2} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.346138, 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,x, algorithm="giac")

[Out]

Done