Optimal. Leaf size=56 \[ -\frac{8 b \sqrt{f^x} (a+b x)}{\log ^2(f)}+\frac{2 \sqrt{f^x} (a+b x)^2}{\log (f)}+\frac{16 b^2 \sqrt{f^x}}{\log ^3(f)} \]
[Out]
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Rubi [A] time = 0.0674831, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{8 b \sqrt{f^x} (a+b x)}{\log ^2(f)}+\frac{2 \sqrt{f^x} (a+b x)^2}{\log (f)}+\frac{16 b^2 \sqrt{f^x}}{\log ^3(f)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[f^x]*(a + b*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 7.92601, size = 54, normalized size = 0.96 \[ \frac{16 b^{2} \sqrt{f^{x}}}{\log{\left (f \right )}^{3}} - \frac{8 b \left (a + b x\right ) \sqrt{f^{x}}}{\log{\left (f \right )}^{2}} + \frac{2 \left (a + b x\right )^{2} \sqrt{f^{x}}}{\log{\left (f \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2*(f**x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0235811, size = 41, normalized size = 0.73 \[ \frac{2 \sqrt{f^x} \left (\log ^2(f) (a+b x)^2-4 b \log (f) (a+b x)+8 b^2\right )}{\log ^3(f)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[f^x]*(a + b*x)^2,x]
[Out]
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Maple [A] time = 0.012, size = 60, normalized size = 1.1 \[ 2\,{\frac{ \left ({b}^{2}{x}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}+2\, \left ( \ln \left ( f \right ) \right ) ^{2}abx+ \left ( \ln \left ( f \right ) \right ) ^{2}{a}^{2}-4\,\ln \left ( f \right ){b}^{2}x-4\,\ln \left ( f \right ) ba+8\,{b}^{2} \right ) \sqrt{{f}^{x}}}{ \left ( \ln \left ( f \right ) \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2*(f^x)^(1/2),x)
[Out]
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Maxima [A] time = 0.801437, size = 85, normalized size = 1.52 \[ \frac{4 \,{\left (x \log \left (f\right ) - 2\right )} a b \sqrt{f^{x}}}{\log \left (f\right )^{2}} + \frac{2 \, a^{2} \sqrt{f^{x}}}{\log \left (f\right )} + \frac{2 \,{\left (x^{2} \log \left (f\right )^{2} - 4 \, x \log \left (f\right ) + 8\right )} b^{2} \sqrt{f^{x}}}{\log \left (f\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2*sqrt(f^x),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2*sqrt(f^x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.164867, size = 94, normalized size = 1.68 \[ \begin{cases} \frac{\left (2 a^{2} \log{\left (f \right )}^{2} + 4 a b x \log{\left (f \right )}^{2} - 8 a b \log{\left (f \right )} + 2 b^{2} x^{2} \log{\left (f \right )}^{2} - 8 b^{2} x \log{\left (f \right )} + 16 b^{2}\right ) \sqrt{f^{x}}}{\log{\left (f \right )}^{3}} & \text{for}\: \log{\left (f \right )}^{3} \neq 0 \\a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2*(f**x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.24893, size = 1, normalized size = 0.02 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2*sqrt(f^x),x, algorithm="giac")
[Out]