Optimal. Leaf size=33 \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0094301, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {651} \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 651
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^3} \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0325069, size = 39, normalized size = 1.18 \[ -\frac{(a-b x) \sqrt{a^2-b^2 x^2}}{3 a b (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 36, normalized size = 1.1 \begin{align*} -{\frac{-bx+a}{3\, \left ( bx+a \right ) ^{2}ba}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.7758, size = 134, normalized size = 4.06 \begin{align*} -\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - \sqrt{-b^{2} x^{2} + a^{2}}{\left (b x - a\right )}}{3 \,{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.28214, size = 100, normalized size = 3.03 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + 1\right )}}{3 \, a{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{3}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]