Optimal. Leaf size=76 \[ \frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {a^3 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {641, 195, 217, 203} \begin {gather*} \frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {a^3 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 641
Rubi steps
\begin {align*} \int (a+b x) \sqrt {a^2-b^2 x^2} \, dx &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+a \int \sqrt {a^2-b^2 x^2} \, dx\\ &=\frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {1}{2} a^3 \int \frac {1}{\sqrt {a^2-b^2 x^2}} \, dx\\ &=\frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{1+b^2 x^2} \, dx,x,\frac {x}{\sqrt {a^2-b^2 x^2}}\right )\\ &=\frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {a^3 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 90, normalized size = 1.18 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (\left (-2 a^2+3 a b x+2 b^2 x^2\right ) \sqrt {1-\frac {b^2 x^2}{a^2}}+3 a^2 \sin ^{-1}\left (\frac {b x}{a}\right )\right )}{6 b \sqrt {1-\frac {b^2 x^2}{a^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 92, normalized size = 1.21 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (-2 a^2+3 a b x+2 b^2 x^2\right )}{6 b}+\frac {a^3 \sqrt {-b^2} \log \left (\sqrt {a^2-b^2 x^2}-\sqrt {-b^2} x\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 73, normalized size = 0.96 \begin {gather*} -\frac {6 \, a^{3} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) - {\left (2 \, b^{2} x^{2} + 3 \, a b x - 2 \, a^{2}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 56, normalized size = 0.74 \begin {gather*} \frac {a^{3} \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (b)}{2 \, {\left | b \right |}} + \frac {1}{6} \, \sqrt {-b^{2} x^{2} + a^{2}} {\left ({\left (2 \, b x + 3 \, a\right )} x - \frac {2 \, a^{2}}{b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 71, normalized size = 0.93 \begin {gather*} \frac {a^{3} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}+\frac {\sqrt {-b^{2} x^{2}+a^{2}}\, a x}{2}-\frac {\left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{3 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 53, normalized size = 0.70 \begin {gather*} \frac {a^{3} \arcsin \left (\frac {b x}{a}\right )}{2 \, b} + \frac {1}{2} \, \sqrt {-b^{2} x^{2} + a^{2}} a x - \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 75, normalized size = 0.99 \begin {gather*} \frac {a^3\,\ln \left (x\,\sqrt {-b^2}+\sqrt {a^2-b^2\,x^2}\right )}{2\,\sqrt {-b^2}}-\frac {{\left (a^2-b^2\,x^2\right )}^{3/2}}{3\,b}+\frac {a\,x\,\sqrt {a^2-b^2\,x^2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.37, size = 144, normalized size = 1.89 \begin {gather*} a \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}\: \left |{\frac {b^{2} x^{2}}{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 ) + 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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