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