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