3.599 \(\int \frac{a+c x^2}{(f+g x)^{3/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 \left (a g^2+c f^2\right )}{g^3 \sqrt{f+g x}}+\frac{2 c (f+g x)^{3/2}}{3 g^3}-\frac{4 c f \sqrt{f+g x}}{g^3} \]

[Out]

(-2*(c*f^2 + a*g^2))/(g^3*Sqrt[f + g*x]) - (4*c*f*Sqrt[f + g*x])/g^3 + (2*c*(f + g*x)^(3/2))/(3*g^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0257772, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ -\frac{2 \left (a g^2+c f^2\right )}{g^3 \sqrt{f+g x}}+\frac{2 c (f+g x)^{3/2}}{3 g^3}-\frac{4 c f \sqrt{f+g x}}{g^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/(f + g*x)^(3/2),x]

[Out]

(-2*(c*f^2 + a*g^2))/(g^3*Sqrt[f + g*x]) - (4*c*f*Sqrt[f + g*x])/g^3 + (2*c*(f + g*x)^(3/2))/(3*g^3)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{a+c x^2}{(f+g x)^{3/2}} \, dx &=\int \left (\frac{c f^2+a g^2}{g^2 (f+g x)^{3/2}}-\frac{2 c f}{g^2 \sqrt{f+g x}}+\frac{c \sqrt{f+g x}}{g^2}\right ) \, dx\\ &=-\frac{2 \left (c f^2+a g^2\right )}{g^3 \sqrt{f+g x}}-\frac{4 c f \sqrt{f+g x}}{g^3}+\frac{2 c (f+g x)^{3/2}}{3 g^3}\\ \end{align*}

Mathematica [A]  time = 0.0281284, size = 43, normalized size = 0.73 \[ \frac{2 \left (c \left (-8 f^2-4 f g x+g^2 x^2\right )-3 a g^2\right )}{3 g^3 \sqrt{f+g x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)/(f + g*x)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 41, normalized size = 0.7 \begin{align*} -{\frac{-2\,c{x}^{2}{g}^{2}+8\,cfxg+6\,a{g}^{2}+16\,c{f}^{2}}{3\,{g}^{3}}{\frac{1}{\sqrt{gx+f}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)/(g*x+f)^(3/2),x)

[Out]

-2/3/(g*x+f)^(1/2)*(-c*g^2*x^2+4*c*f*g*x+3*a*g^2+8*c*f^2)/g^3

________________________________________________________________________________________

Maxima [A]  time = 1.01421, size = 73, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (\frac{{\left (g x + f\right )}^{\frac{3}{2}} c - 6 \, \sqrt{g x + f} c f}{g^{2}} - \frac{3 \,{\left (c f^{2} + a g^{2}\right )}}{\sqrt{g x + f} g^{2}}\right )}}{3 \, g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(3/2),x, algorithm="maxima")

[Out]

2/3*(((g*x + f)^(3/2)*c - 6*sqrt(g*x + f)*c*f)/g^2 - 3*(c*f^2 + a*g^2)/(sqrt(g*x + f)*g^2))/g

________________________________________________________________________________________

Fricas [A]  time = 1.68355, size = 107, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (c g^{2} x^{2} - 4 \, c f g x - 8 \, c f^{2} - 3 \, a g^{2}\right )} \sqrt{g x + f}}{3 \,{\left (g^{4} x + f g^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

2/3*(c*g^2*x^2 - 4*c*f*g*x - 8*c*f^2 - 3*a*g^2)*sqrt(g*x + f)/(g^4*x + f*g^3)

________________________________________________________________________________________

Sympy [A]  time = 5.66873, size = 58, normalized size = 0.98 \begin{align*} - \frac{4 c f \sqrt{f + g x}}{g^{3}} + \frac{2 c \left (f + g x\right )^{\frac{3}{2}}}{3 g^{3}} - \frac{2 \left (a g^{2} + c f^{2}\right )}{g^{3} \sqrt{f + g x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(g*x+f)**(3/2),x)

[Out]

-4*c*f*sqrt(f + g*x)/g**3 + 2*c*(f + g*x)**(3/2)/(3*g**3) - 2*(a*g**2 + c*f**2)/(g**3*sqrt(f + g*x))

________________________________________________________________________________________

Giac [A]  time = 1.14379, size = 76, normalized size = 1.29 \begin{align*} -\frac{2 \,{\left (c f^{2} + a g^{2}\right )}}{\sqrt{g x + f} g^{3}} + \frac{2 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} c g^{6} - 6 \, \sqrt{g x + f} c f g^{6}\right )}}{3 \, g^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

-2*(c*f^2 + a*g^2)/(sqrt(g*x + f)*g^3) + 2/3*((g*x + f)^(3/2)*c*g^6 - 6*sqrt(g*x + f)*c*f*g^6)/g^9