Optimal. Leaf size=54 \[ \frac{1}{2} \left (2 a^2+b^2\right ) \sin ^{-1}(x)-\frac{3}{2} a b \sqrt{1-x^2}-\frac{1}{2} b \sqrt{1-x^2} (a+b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.023162, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {743, 641, 216} \[ \frac{1}{2} \left (2 a^2+b^2\right ) \sin ^{-1}(x)-\frac{3}{2} a b \sqrt{1-x^2}-\frac{1}{2} b \sqrt{1-x^2} (a+b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 743
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{\sqrt{1-x^2}} \, dx &=-\frac{1}{2} b (a+b x) \sqrt{1-x^2}-\frac{1}{2} \int \frac{-2 a^2-b^2-3 a b x}{\sqrt{1-x^2}} \, dx\\ &=-\frac{3}{2} a b \sqrt{1-x^2}-\frac{1}{2} b (a+b x) \sqrt{1-x^2}-\frac{1}{2} \left (-2 a^2-b^2\right ) \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{3}{2} a b \sqrt{1-x^2}-\frac{1}{2} b (a+b x) \sqrt{1-x^2}+\frac{1}{2} \left (2 a^2+b^2\right ) \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0322044, size = 38, normalized size = 0.7 \[ \frac{1}{2} \left (\left (2 a^2+b^2\right ) \sin ^{-1}(x)-b \sqrt{1-x^2} (4 a+b x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 42, normalized size = 0.8 \begin{align*}{b}^{2} \left ( -{\frac{x}{2}\sqrt{-{x}^{2}+1}}+{\frac{\arcsin \left ( x \right ) }{2}} \right ) -2\,ab\sqrt{-{x}^{2}+1}+{a}^{2}\arcsin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.75586, size = 57, normalized size = 1.06 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1} b^{2} x + a^{2} \arcsin \left (x\right ) + \frac{1}{2} \, b^{2} \arcsin \left (x\right ) - 2 \, \sqrt{-x^{2} + 1} a b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.85792, size = 113, normalized size = 2.09 \begin{align*} -{\left (2 \, a^{2} + b^{2}\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) - \frac{1}{2} \,{\left (b^{2} x + 4 \, a b\right )} \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.244047, size = 42, normalized size = 0.78 \begin{align*} a^{2} \operatorname{asin}{\left (x \right )} - 2 a b \sqrt{1 - x^{2}} - \frac{b^{2} x \sqrt{1 - x^{2}}}{2} + \frac{b^{2} \operatorname{asin}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.76367, size = 47, normalized size = 0.87 \begin{align*} \frac{1}{2} \,{\left (2 \, a^{2} + b^{2}\right )} \arcsin \left (x\right ) - \frac{1}{2} \,{\left (b^{2} x + 4 \, a b\right )} \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]