Optimal. Leaf size=41 \[ -\frac {A b-a B}{b^2 \sqrt {a+b x^2}}+\frac {B \sqrt {a+b x^2}}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 45}
\begin {gather*} \frac {B \sqrt {a+b x^2}}{b^2}-\frac {A b-a B}{b^2 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 455
Rubi steps
\begin {align*} \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A b-a B}{b (a+b x)^{3/2}}+\frac {B}{b \sqrt {a+b x}}\right ) \, dx,x,x^2\right )\\ &=-\frac {A b-a B}{b^2 \sqrt {a+b x^2}}+\frac {B \sqrt {a+b x^2}}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 30, normalized size = 0.73 \begin {gather*} \frac {-A b+2 a B+b B x^2}{b^2 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 51, normalized size = 1.24
method | result | size |
gosper | \(-\frac {-b B \,x^{2}+A b -2 B a}{\sqrt {b \,x^{2}+a}\, b^{2}}\) | \(30\) |
trager | \(-\frac {-b B \,x^{2}+A b -2 B a}{\sqrt {b \,x^{2}+a}\, b^{2}}\) | \(30\) |
risch | \(\frac {B \sqrt {b \,x^{2}+a}}{b^{2}}-\frac {A b -B a}{\sqrt {b \,x^{2}+a}\, b^{2}}\) | \(38\) |
default | \(B \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )-\frac {A}{b \sqrt {b \,x^{2}+a}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 49, normalized size = 1.20 \begin {gather*} \frac {B x^{2}}{\sqrt {b x^{2} + a} b} + \frac {2 \, B a}{\sqrt {b x^{2} + a} b^{2}} - \frac {A}{\sqrt {b x^{2} + a} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.55, size = 40, normalized size = 0.98 \begin {gather*} \frac {{\left (B b x^{2} + 2 \, B a - A b\right )} \sqrt {b x^{2} + a}}{b^{3} x^{2} + a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.28, size = 66, normalized size = 1.61 \begin {gather*} \begin {cases} - \frac {A}{b \sqrt {a + b x^{2}}} + \frac {2 B a}{b^{2} \sqrt {a + b x^{2}}} + \frac {B x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{4}}{4}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.93, size = 36, normalized size = 0.88 \begin {gather*} \frac {\sqrt {b x^{2} + a} B}{b^{2}} + \frac {B a - A b}{\sqrt {b x^{2} + a} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.29, size = 30, normalized size = 0.73 \begin {gather*} \frac {B\,a-A\,b+B\,\left (b\,x^2+a\right )}{b^2\,\sqrt {b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________