Optimal. Leaf size=100 \[ \frac{3}{8} a^3 x \sqrt{a^2-b^2 x^2}+\frac{1}{4} a x \left (a^2-b^2 x^2\right )^{3/2}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+\frac{3 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b} \]
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Rubi [A] time = 0.0262414, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {641, 195, 217, 203} \[ \frac{3}{8} a^3 x \sqrt{a^2-b^2 x^2}+\frac{1}{4} a x \left (a^2-b^2 x^2\right )^{3/2}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+\frac{3 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b} \]
Antiderivative was successfully verified.
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Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int (a+b x) \left (a^2-b^2 x^2\right )^{3/2} \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+a \int \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=\frac{1}{4} a x \left (a^2-b^2 x^2\right )^{3/2}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+\frac{1}{4} \left (3 a^3\right ) \int \sqrt{a^2-b^2 x^2} \, dx\\ &=\frac{3}{8} a^3 x \sqrt{a^2-b^2 x^2}+\frac{1}{4} a x \left (a^2-b^2 x^2\right )^{3/2}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+\frac{1}{8} \left (3 a^5\right ) \int \frac{1}{\sqrt{a^2-b^2 x^2}} \, dx\\ &=\frac{3}{8} a^3 x \sqrt{a^2-b^2 x^2}+\frac{1}{4} a x \left (a^2-b^2 x^2\right )^{3/2}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+\frac{1}{8} \left (3 a^5\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{3}{8} a^3 x \sqrt{a^2-b^2 x^2}+\frac{1}{4} a x \left (a^2-b^2 x^2\right )^{3/2}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+\frac{3 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b}\\ \end{align*}
Mathematica [A] time = 0.145797, size = 112, normalized size = 1.12 \[ \frac{\sqrt{a^2-b^2 x^2} \left (\sqrt{1-\frac{b^2 x^2}{a^2}} \left (16 a^2 b^2 x^2+25 a^3 b x-8 a^4-10 a b^3 x^3-8 b^4 x^4\right )+15 a^4 \sin ^{-1}\left (\frac{b x}{a}\right )\right )}{40 b \sqrt{1-\frac{b^2 x^2}{a^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 91, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{ax}{4} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,x{a}^{3}}{8}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{3\,{a}^{5}}{8}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68208, size = 112, normalized size = 1.12 \begin{align*} \frac{3 \, a^{5} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{8 \, \sqrt{b^{2}}} + \frac{3}{8} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{3} x + \frac{1}{4} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a x - \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}}}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11073, size = 200, normalized size = 2. \begin{align*} -\frac{30 \, a^{5} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) +{\left (8 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} - 16 \, a^{2} b^{2} x^{2} - 25 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{40 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.83493, size = 439, normalized size = 4.39 \begin{align*} a^{3} \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 ) + a^{2} 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 ) - a 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 ) - b^{3} \left (\begin{cases} - \frac{2 a^{4} \sqrt{a^{2} - b^{2} x^{2}}}{15 b^{4}} - \frac{a^{2} x^{2} \sqrt{a^{2} - b^{2} x^{2}}}{15 b^{2}} + \frac{x^{4} \sqrt{a^{2} - b^{2} x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{x^{4} \sqrt{a^{2}}}{4} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2365, size = 109, normalized size = 1.09 \begin{align*} \frac{3 \, a^{5} \arcsin \left (\frac{b x}{a}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (b\right )}{8 \,{\left | b \right |}} - \frac{1}{40} \, \sqrt{-b^{2} x^{2} + a^{2}}{\left (\frac{8 \, a^{4}}{b} -{\left (25 \, a^{3} + 2 \,{\left (8 \, a^{2} b -{\left (4 \, b^{3} x + 5 \, a b^{2}\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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