\(\int \frac {d+e x}{(a+c x^2)^{3/2}} \, dx\) [572]

Optimal result

Integrand size = 17, antiderivative size = 28 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {a e-c d x}{a c \sqrt {a+c x^2}} \]

[Out]

(c*d*x-a*e)/a/c/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {651} \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {a e-c d x}{a c \sqrt {a+c x^2}} \]

[In]

Int[(d + e*x)/(a + c*x^2)^(3/2),x]

[Out]

-((a*e - c*d*x)/(a*c*Sqrt[a + c*x^2]))

Rule 651

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((-a)*e + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {a e-c d x}{a c \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {-a e+c d x}{a c \sqrt {a+c x^2}} \]

[In]

Integrate[(d + e*x)/(a + c*x^2)^(3/2),x]

[Out]

(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2])

Maple [A] (verified)

Time = 1.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96

method	result	size
gosper	\(-\frac {-c d x +a e}{\sqrt {c \,x^{2}+a}\, a c}\)	\(27\)
trager	\(-\frac {-c d x +a e}{\sqrt {c \,x^{2}+a}\, a c}\)	\(27\)
default	\(\frac {d x}{a \sqrt {c \,x^{2}+a}}-\frac {e}{c \sqrt {c \,x^{2}+a}}\)	\(32\)

[In]

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

[Out]

-(-c*d*x+a*e)/(c*x^2+a)^(1/2)/a/c

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c^{2} x^{2} + a^{2} c} \]

[In]

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

[Out]

(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c^2*x^2 + a^2*c)

Sympy [A] (verification not implemented)

Time = 1.91 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=e \left (\begin {cases} - \frac {1}{c \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {d x}{a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{2}}{a}}} \]

[In]

integrate((e*x+d)/(c*x**2+a)**(3/2),x)

[Out]

e*Piecewise((-1/(c*sqrt(a + c*x**2)), Ne(c, 0)), (x**2/(2*a**(3/2)), True)) + d*x/(a**(3/2)*sqrt(1 + c*x**2/a)
)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {d x}{\sqrt {c x^{2} + a} a} - \frac {e}{\sqrt {c x^{2} + a} c} \]

[In]

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

[Out]

d*x/(sqrt(c*x^2 + a)*a) - e/(sqrt(c*x^2 + a)*c)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\frac {d x}{a} - \frac {e}{c}}{\sqrt {c x^{2} + a}} \]

[In]

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

[Out]

(d*x/a - e/c)/sqrt(c*x^2 + a)

Mupad [B] (verification not implemented)

Time = 9.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {\frac {e}{c}-\frac {d\,x}{a}}{\sqrt {c\,x^2+a}} \]

[In]

int((d + e*x)/(a + c*x^2)^(3/2),x)

[Out]

-(e/c - (d*x)/a)/(a + c*x^2)^(1/2)