3.719 \(\int \sqrt{f^x} (a+b x)^2 \, dx\)

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]

(16*b^2*Sqrt[f^x])/Log[f]^3 - (8*b*Sqrt[f^x]*(a + b*x))/Log[f]^2 + (2*Sqrt[f^x]*
(a + b*x)^2)/Log[f]

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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]

(16*b^2*Sqrt[f^x])/Log[f]^3 - (8*b*Sqrt[f^x]*(a + b*x))/Log[f]^2 + (2*Sqrt[f^x]*
(a + b*x)^2)/Log[f]

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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]

16*b**2*sqrt(f**x)/log(f)**3 - 8*b*(a + b*x)*sqrt(f**x)/log(f)**2 + 2*(a + b*x)*
*2*sqrt(f**x)/log(f)

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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]

(2*Sqrt[f^x]*(8*b^2 - 4*b*(a + b*x)*Log[f] + (a + b*x)^2*Log[f]^2))/Log[f]^3

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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]

2*(b^2*x^2*ln(f)^2+2*ln(f)^2*a*b*x+ln(f)^2*a^2-4*ln(f)*b^2*x-4*ln(f)*b*a+8*b^2)*
(f^x)^(1/2)/ln(f)^3

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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]

4*(x*log(f) - 2)*a*b*sqrt(f^x)/log(f)^2 + 2*a^2*sqrt(f^x)/log(f) + 2*(x^2*log(f)
^2 - 4*x*log(f) + 8)*b^2*sqrt(f^x)/log(f)^3

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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]

Exception raised: TypeError

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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]

Piecewise(((2*a**2*log(f)**2 + 4*a*b*x*log(f)**2 - 8*a*b*log(f) + 2*b**2*x**2*lo
g(f)**2 - 8*b**2*x*log(f) + 16*b**2)*sqrt(f**x)/log(f)**3, Ne(log(f)**3, 0)), (a
**2*x + a*b*x**2 + b**2*x**3/3, True))

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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]

Done