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

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 = 0.0407702, 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, Rules used = {2176, 2194} \[ -\frac{8 b \sqrt{f^x} (a+b x)}{\log ^2(f)}+\frac{2 \sqrt{f^x} (a+b x)^2}{\log (f)}+\frac{16 b^2 \sqrt{f^x}}{\log ^3(f)} \]

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]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \sqrt{f^x} (a+b x)^2 \, dx &=\frac{2 \sqrt{f^x} (a+b x)^2}{\log (f)}-\frac{(4 b) \int \sqrt{f^x} (a+b x) \, dx}{\log (f)}\\ &=-\frac{8 b \sqrt{f^x} (a+b x)}{\log ^2(f)}+\frac{2 \sqrt{f^x} (a+b x)^2}{\log (f)}+\frac{\left (8 b^2\right ) \int \sqrt{f^x} \, dx}{\log ^2(f)}\\ &=\frac{16 b^2 \sqrt{f^x}}{\log ^3(f)}-\frac{8 b \sqrt{f^x} (a+b x)}{\log ^2(f)}+\frac{2 \sqrt{f^x} (a+b x)^2}{\log (f)}\\ \end{align*}

Mathematica [A]  time = 0.038849, 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 = 0.043, size = 60, normalized size = 1.1 \begin{align*} 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}}} \end{align*}

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 = 1.00998, size = 85, normalized size = 1.52 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(f^x)^(1/2),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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(f^x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.147631, size = 94, normalized size = 1.68 \begin{align*} \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} \end{align*}

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*log(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 [B]  time = 1.30601, size = 1910, normalized size = 34.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(f^x)^(1/2),x, algorithm="giac")

[Out]

-2*((2*(pi*b^2*x^2*log(abs(f))*sgn(f) - pi*b^2*x^2*log(abs(f)) + 2*pi*a*b*x*log(abs(f))*sgn(f) - 2*pi*a*b*x*lo
g(abs(f)) - 2*pi*b^2*x*sgn(f) + pi*a^2*log(abs(f))*sgn(f) + 2*pi*b^2*x - pi*a^2*log(abs(f)) - 2*pi*a*b*sgn(f)
+ 2*pi*a*b)*(pi^3*sgn(f) - 3*pi*log(abs(f))^2*sgn(f) - pi^3 + 3*pi*log(abs(f))^2)/((pi^3*sgn(f) - 3*pi*log(abs
(f))^2*sgn(f) - pi^3 + 3*pi*log(abs(f))^2)^2 + (3*pi^2*log(abs(f))*sgn(f) - 3*pi^2*log(abs(f)) + 2*log(abs(f))
^3)^2) - (pi^2*b^2*x^2*sgn(f) - pi^2*b^2*x^2 + 2*b^2*x^2*log(abs(f))^2 + 2*pi^2*a*b*x*sgn(f) - 2*pi^2*a*b*x +
4*a*b*x*log(abs(f))^2 + pi^2*a^2*sgn(f) - pi^2*a^2 - 8*b^2*x*log(abs(f)) + 2*a^2*log(abs(f))^2 - 8*a*b*log(abs
(f)) + 16*b^2)*(3*pi^2*log(abs(f))*sgn(f) - 3*pi^2*log(abs(f)) + 2*log(abs(f))^3)/((pi^3*sgn(f) - 3*pi*log(abs
(f))^2*sgn(f) - pi^3 + 3*pi*log(abs(f))^2)^2 + (3*pi^2*log(abs(f))*sgn(f) - 3*pi^2*log(abs(f)) + 2*log(abs(f))
^3)^2))*cos(-1/4*pi*x*sgn(f) + 1/4*pi*x) - ((pi^2*b^2*x^2*sgn(f) - pi^2*b^2*x^2 + 2*b^2*x^2*log(abs(f))^2 + 2*
pi^2*a*b*x*sgn(f) - 2*pi^2*a*b*x + 4*a*b*x*log(abs(f))^2 + pi^2*a^2*sgn(f) - pi^2*a^2 - 8*b^2*x*log(abs(f)) +
2*a^2*log(abs(f))^2 - 8*a*b*log(abs(f)) + 16*b^2)*(pi^3*sgn(f) - 3*pi*log(abs(f))^2*sgn(f) - pi^3 + 3*pi*log(a
bs(f))^2)/((pi^3*sgn(f) - 3*pi*log(abs(f))^2*sgn(f) - pi^3 + 3*pi*log(abs(f))^2)^2 + (3*pi^2*log(abs(f))*sgn(f
) - 3*pi^2*log(abs(f)) + 2*log(abs(f))^3)^2) + 2*(pi*b^2*x^2*log(abs(f))*sgn(f) - pi*b^2*x^2*log(abs(f)) + 2*p
i*a*b*x*log(abs(f))*sgn(f) - 2*pi*a*b*x*log(abs(f)) - 2*pi*b^2*x*sgn(f) + pi*a^2*log(abs(f))*sgn(f) + 2*pi*b^2
*x - pi*a^2*log(abs(f)) - 2*pi*a*b*sgn(f) + 2*pi*a*b)*(3*pi^2*log(abs(f))*sgn(f) - 3*pi^2*log(abs(f)) + 2*log(
abs(f))^3)/((pi^3*sgn(f) - 3*pi*log(abs(f))^2*sgn(f) - pi^3 + 3*pi*log(abs(f))^2)^2 + (3*pi^2*log(abs(f))*sgn(
f) - 3*pi^2*log(abs(f)) + 2*log(abs(f))^3)^2))*sin(-1/4*pi*x*sgn(f) + 1/4*pi*x))*abs(f)^(1/2*x) + abs(f)^(1/2*
x)*((pi^2*b^2*i*x^2*sgn(f) - pi^2*b^2*i*x^2 + 2*b^2*i*x^2*log(abs(f))^2 + 2*pi^2*a*b*i*x*sgn(f) - 2*pi*b^2*x^2
*log(abs(f))*sgn(f) - 2*pi^2*a*b*i*x + 2*pi*b^2*x^2*log(abs(f)) + 4*a*b*i*x*log(abs(f))^2 + pi^2*a^2*i*sgn(f)
- 4*pi*a*b*x*log(abs(f))*sgn(f) - pi^2*a^2*i + 4*pi*a*b*x*log(abs(f)) - 8*b^2*i*x*log(abs(f)) + 2*a^2*i*log(ab
s(f))^2 + 4*pi*b^2*x*sgn(f) - 2*pi*a^2*log(abs(f))*sgn(f) - 4*pi*b^2*x + 2*pi*a^2*log(abs(f)) - 8*a*b*i*log(ab
s(f)) + 4*pi*a*b*sgn(f) - 4*pi*a*b + 16*b^2*i)*e^(1/4*pi*i*x*(sgn(f) - 1))/(pi^3*i*sgn(f) - 3*pi*i*log(abs(f))
^2*sgn(f) - pi^3*i + 3*pi*i*log(abs(f))^2 - 3*pi^2*log(abs(f))*sgn(f) + 3*pi^2*log(abs(f)) - 2*log(abs(f))^3)
+ (pi^2*b^2*i*x^2*sgn(f) - pi^2*b^2*i*x^2 + 2*b^2*i*x^2*log(abs(f))^2 + 2*pi^2*a*b*i*x*sgn(f) + 2*pi*b^2*x^2*l
og(abs(f))*sgn(f) - 2*pi^2*a*b*i*x - 2*pi*b^2*x^2*log(abs(f)) + 4*a*b*i*x*log(abs(f))^2 + pi^2*a^2*i*sgn(f) +
4*pi*a*b*x*log(abs(f))*sgn(f) - pi^2*a^2*i - 4*pi*a*b*x*log(abs(f)) - 8*b^2*i*x*log(abs(f)) + 2*a^2*i*log(abs(
f))^2 - 4*pi*b^2*x*sgn(f) + 2*pi*a^2*log(abs(f))*sgn(f) + 4*pi*b^2*x - 2*pi*a^2*log(abs(f)) - 8*a*b*i*log(abs(
f)) - 4*pi*a*b*sgn(f) + 4*pi*a*b + 16*b^2*i)*e^(-1/4*pi*i*x*(sgn(f) - 1))/(pi^3*i*sgn(f) - 3*pi*i*log(abs(f))^
2*sgn(f) - pi^3*i + 3*pi*i*log(abs(f))^2 + 3*pi^2*log(abs(f))*sgn(f) - 3*pi^2*log(abs(f)) + 2*log(abs(f))^3))/
i