Optimal. Leaf size=131 \[ \frac{7}{16} a^4 x \sqrt{a^2-b^2 x^2}+\frac{7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{7 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b} \]
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Rubi [A] time = 0.0437812, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{7}{16} a^4 x \sqrt{a^2-b^2 x^2}+\frac{7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{7 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b} \]
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 \left (a^2-b^2 x^2\right )^{3/2} \, dx &=-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{1}{6} (7 a) \int (a+b x) \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{1}{6} \left (7 a^2\right ) \int \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=\frac{7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{1}{8} \left (7 a^4\right ) \int \sqrt{a^2-b^2 x^2} \, dx\\ &=\frac{7}{16} a^4 x \sqrt{a^2-b^2 x^2}+\frac{7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{1}{16} \left (7 a^6\right ) \int \frac{1}{\sqrt{a^2-b^2 x^2}} \, dx\\ &=\frac{7}{16} a^4 x \sqrt{a^2-b^2 x^2}+\frac{7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{1}{16} \left (7 a^6\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{7}{16} a^4 x \sqrt{a^2-b^2 x^2}+\frac{7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{7 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b}\\ \end{align*}
Mathematica [A] time = 0.183362, size = 123, normalized size = 0.94 \[ \frac{\sqrt{a^2-b^2 x^2} \left (\sqrt{1-\frac{b^2 x^2}{a^2}} \left (192 a^3 b^2 x^2+10 a^2 b^3 x^3+135 a^4 b x-96 a^5-96 a b^4 x^4-40 b^5 x^5\right )+105 a^5 \sin ^{-1}\left (\frac{b x}{a}\right )\right )}{240 b \sqrt{1-\frac{b^2 x^2}{a^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 111, normalized size = 0.9 \begin{align*} -{\frac{x}{6} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{a}^{2}x}{24} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{a}^{4}x}{16}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{7\,{a}^{6}}{16}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{2\,a}{5\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6062, size = 139, normalized size = 1.06 \begin{align*} \frac{7 \, a^{6} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{16 \, \sqrt{b^{2}}} + \frac{7}{16} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{4} x + \frac{7}{24} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a^{2} x - \frac{1}{6} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} x - \frac{2 \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} a}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08246, size = 231, normalized size = 1.76 \begin{align*} -\frac{210 \, a^{6} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) +{\left (40 \, b^{5} x^{5} + 96 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} - 192 \, a^{3} b^{2} x^{2} - 135 \, a^{4} b x + 96 \, a^{5}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{240 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.46096, size = 498, normalized size = 3.8 \begin{align*} a^{4} \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^{3} 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 ) - 2 a 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 ) - b^{4} \left (\begin{cases} - \frac{i a^{6} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{16 b^{5}} + \frac{i a^{5} x}{16 b^{4} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{i a^{3} x^{3}}{48 b^{2} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{5 i a x^{5}}{24 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{7}}{6 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^{6} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{16 b^{5}} - \frac{a^{5} x}{16 b^{4} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{a^{3} x^{3}}{48 b^{2} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{5 a x^{5}}{24 \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{b^{2} x^{7}}{6 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 = 1.24505, size = 124, normalized size = 0.95 \begin{align*} \frac{7 \, a^{6} \arcsin \left (\frac{b x}{a}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (b\right )}{16 \,{\left | b \right |}} - \frac{1}{240} \,{\left (\frac{96 \, a^{5}}{b} -{\left (135 \, a^{4} + 2 \,{\left (96 \, a^{3} b +{\left (5 \, a^{2} b^{2} - 4 \,{\left (5 \, b^{4} x + 12 \, a b^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-b^{2} x^{2} + a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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