\(\int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx\) [599]

Optimal result

Integrand size = 17, antiderivative size = 59 \[ \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx=-\frac {2 \left (c f^2+a g^2\right )}{g^3 \sqrt {f+g x}}-\frac {4 c f \sqrt {f+g x}}{g^3}+\frac {2 c (f+g x)^{3/2}}{3 g^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx=-\frac {2 \left (a g^2+c f^2\right )}{g^3 \sqrt {f+g x}}+\frac {2 c (f+g x)^{3/2}}{3 g^3}-\frac {4 c f \sqrt {f+g x}}{g^3} \]

[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 711

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*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx=\frac {2 \left (-3 a g^2+c \left (-8 f^2-4 f g x+g^2 x^2\right )\right )}{3 g^3 \sqrt {f+g x}} \]

[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] (verified)

Time = 0.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (c \,x^{2}-3 a \right ) g^{2}}{3}-\frac {8 c f x g}{3}-\frac {16 c \,f^{2}}{3}}{\sqrt {g x +f}\, g^{3}}\)	\(39\)
gosper	\(-\frac {2 \left (-c \,x^{2} g^{2}+4 c f x g +3 a \,g^{2}+8 c \,f^{2}\right )}{3 \sqrt {g x +f}\, g^{3}}\)	\(41\)
trager	\(-\frac {2 \left (-c \,x^{2} g^{2}+4 c f x g +3 a \,g^{2}+8 c \,f^{2}\right )}{3 \sqrt {g x +f}\, g^{3}}\)	\(41\)
risch	\(-\frac {2 c \left (-g x +5 f \right ) \sqrt {g x +f}}{3 g^{3}}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{g^{3} \sqrt {g x +f}}\)	\(46\)
derivativedivides	\(\frac {\frac {2 c \left (g x +f \right )^{\frac {3}{2}}}{3}-4 c f \sqrt {g x +f}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\sqrt {g x +f}}}{g^{3}}\)	\(48\)
default	\(\frac {\frac {2 c \left (g x +f \right )^{\frac {3}{2}}}{3}-4 c f \sqrt {g x +f}-\frac {2 \left (a \,g^{2}+c \,f^{2}\right )}{\sqrt {g x +f}}}{g^{3}}\)	\(48\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (c g^{2} x^{2} - 4 \, c f g x - 8 \, c f^{2} - 3 \, a g^{2}\right )} \sqrt {g x + f}}{3 \, {\left (g^{4} x + f g^{3}\right )}} \]

[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] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {2 c f \sqrt {f + g x}}{g^{2}} + \frac {c \left (f + g x\right )^{\frac {3}{2}}}{3 g^{2}} - \frac {a g^{2} + c f^{2}}{g^{2} \sqrt {f + g x}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a x + \frac {c x^{3}}{3}}{f^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(-2*c*f*sqrt(f + g*x)/g**2 + c*(f + g*x)**(3/2)/(3*g**2) - (a*g**2 + c*f**2)/(g**2*sqrt(f + g*x))
)/g, Ne(g, 0)), ((a*x + c*x**3/3)/f**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {{\left (g x + f\right )}^{\frac {3}{2}} c - 6 \, \sqrt {g x + f} c f}{g^{2}} - \frac {3 \, {\left (c f^{2} + a g^{2}\right )}}{\sqrt {g x + f} g^{2}}\right )}}{3 \, g} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx=-\frac {2 \, {\left (c f^{2} + a g^{2}\right )}}{\sqrt {g x + f} g^{3}} + \frac {2 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} c g^{6} - 6 \, \sqrt {g x + f} c f g^{6}\right )}}{3 \, g^{9}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.75 \[ \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx=-\frac {6\,a\,g^2-2\,c\,{\left (f+g\,x\right )}^2+6\,c\,f^2+12\,c\,f\,\left (f+g\,x\right )}{3\,g^3\,\sqrt {f+g\,x}} \]

[In]

int((a + c*x^2)/(f + g*x)^(3/2),x)

[Out]

-(6*a*g^2 - 2*c*(f + g*x)^2 + 6*c*f^2 + 12*c*f*(f + g*x))/(3*g^3*(f + g*x)^(1/2))