Optimal. Leaf size=107 \[ \frac{5}{8} a^2 x \sqrt{a^2-b^2 x^2}-\frac{5 a \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{5 a^4 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b} \]
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Rubi [A] time = 0.0339475, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {671, 641, 195, 217, 203} \[ \frac{5}{8} a^2 x \sqrt{a^2-b^2 x^2}-\frac{5 a \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{5 a^4 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b} \]
Antiderivative was successfully verified.
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Rule 671
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int (a+b x)^2 \sqrt{a^2-b^2 x^2} \, dx &=-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{1}{4} (5 a) \int (a+b x) \sqrt{a^2-b^2 x^2} \, dx\\ &=-\frac{5 a \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{1}{4} \left (5 a^2\right ) \int \sqrt{a^2-b^2 x^2} \, dx\\ &=\frac{5}{8} a^2 x \sqrt{a^2-b^2 x^2}-\frac{5 a \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{1}{8} \left (5 a^4\right ) \int \frac{1}{\sqrt{a^2-b^2 x^2}} \, dx\\ &=\frac{5}{8} a^2 x \sqrt{a^2-b^2 x^2}-\frac{5 a \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{1}{8} \left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+b^2 x^2} \, dx,x,\frac{x}{\sqrt{a^2-b^2 x^2}}\right )\\ &=\frac{5}{8} a^2 x \sqrt{a^2-b^2 x^2}-\frac{5 a \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{5 a^4 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b}\\ \end{align*}
Mathematica [A] time = 0.179711, size = 101, normalized size = 0.94 \[ \frac{\sqrt{a^2-b^2 x^2} \left (\sqrt{1-\frac{b^2 x^2}{a^2}} \left (9 a^2 b x-16 a^3+16 a b^2 x^2+6 b^3 x^3\right )+15 a^3 \sin ^{-1}\left (\frac{b x}{a}\right )\right )}{24 b \sqrt{1-\frac{b^2 x^2}{a^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 91, normalized size = 0.9 \begin{align*} -{\frac{x}{4} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{8}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{5\,{a}^{4}}{8}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{2\,a}{3\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67389, size = 112, normalized size = 1.05 \begin{align*} \frac{5 \, a^{4} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{8 \, \sqrt{b^{2}}} + \frac{5}{8} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{2} x - \frac{1}{4} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} x - \frac{2 \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82214, size = 177, normalized size = 1.65 \begin{align*} -\frac{30 \, a^{4} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) -{\left (6 \, b^{3} x^{3} + 16 \, a b^{2} x^{2} + 9 \, a^{2} b x - 16 \, a^{3}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.28458, size = 354, normalized size = 3.31 \begin{align*} a^{2} \left (\begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b} - \frac{i a x}{2 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \frac{\left |{b^{2} x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\\frac{a^{2} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b} + \frac{a x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + 2 a b \left (\begin{cases} \frac{x^{2} \sqrt{a^{2}}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} & \text{otherwise} \end{cases}\right ) + b^{2} \left (\begin{cases} - \frac{i a^{4} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{8 b^{3}} + \frac{i a^{3} x}{8 b^{2} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{3 i a x^{3}}{8 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{5}}{4 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \frac{\left |{b^{2} x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\\frac{a^{4} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{8 b^{3}} - \frac{a^{3} x}{8 b^{2} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{3 a x^{3}}{8 \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{b^{2} x^{5}}{4 a \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24354, size = 93, normalized size = 0.87 \begin{align*} \frac{5 \, a^{4} \arcsin \left (\frac{b x}{a}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (b\right )}{8 \,{\left | b \right |}} - \frac{1}{24} \, \sqrt{-b^{2} x^{2} + a^{2}}{\left (\frac{16 \, a^{3}}{b} -{\left (9 \, a^{2} + 2 \,{\left (3 \, b^{2} x + 8 \, a b\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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