\(\int \frac {A+B x}{(a+b x^2)^{3/2}} \, dx\) [32]

Optimal result

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

[Out]

(A*b*x-B*a)/a/b/(b*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.00 (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} \[ \int \frac {A+B x}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {a B-A b x}{a b \sqrt {a+b x^2}} \]

[In]

Int[(A + B*x)/(a + b*x^2)^(3/2),x]

[Out]

-((a*B - A*b*x)/(a*b*Sqrt[a + b*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 B-A b x}{a b \sqrt {a+b x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-a B+A b x}{a b \sqrt {a+b x^2}} \]

[In]

Integrate[(A + B*x)/(a + b*x^2)^(3/2),x]

[Out]

(-(a*B) + A*b*x)/(a*b*Sqrt[a + b*x^2])

Maple [A] (verified)

Time = 3.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

method	result	size
gosper	\(\frac {A b x -B a}{a b \sqrt {b \,x^{2}+a}}\)	\(26\)
trager	\(\frac {A b x -B a}{a b \sqrt {b \,x^{2}+a}}\)	\(26\)
default	\(\frac {A x}{a \sqrt {b \,x^{2}+a}}-\frac {B}{b \sqrt {b \,x^{2}+a}}\)	\(32\)

[In]

int((B*x+A)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

(A*b*x-B*a)/a/b/(b*x^2+a)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (A b x - B a\right )} \sqrt {b x^{2} + a}}{a b^{2} x^{2} + a^{2} b} \]

[In]

integrate((B*x+A)/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

(A*b*x - B*a)*sqrt(b*x^2 + a)/(a*b^2*x^2 + a^2*b)

Sympy [A] (verification not implemented)

Time = 1.87 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {A+B x}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + B \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((B*x+A)/(b*x**2+a)**(3/2),x)

[Out]

A*x/(a**(3/2)*sqrt(1 + b*x**2/a)) + B*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True)
)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A x}{\sqrt {b x^{2} + a} a} - \frac {B}{\sqrt {b x^{2} + a} b} \]

[In]

integrate((B*x+A)/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

A*x/(sqrt(b*x^2 + a)*a) - B/(sqrt(b*x^2 + a)*b)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {A x}{a} - \frac {B}{b}}{\sqrt {b x^{2} + a}} \]

[In]

integrate((B*x+A)/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

(A*x/a - B/b)/sqrt(b*x^2 + a)

Mupad [B] (verification not implemented)

Time = 6.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {B}{b}-\frac {A\,x}{a}}{\sqrt {b\,x^2+a}} \]

[In]

int((A + B*x)/(a + b*x^2)^(3/2),x)

[Out]

-(B/b - (A*x)/a)/(a + b*x^2)^(1/2)