Optimal. Leaf size=76 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-\frac{3 \sqrt{a^2-b^2 x^2}}{b}-\frac{3 a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
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Rubi [A] time = 0.0277218, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {663, 665, 217, 203} \[ -\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-\frac{3 \sqrt{a^2-b^2 x^2}}{b}-\frac{3 a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 663
Rule 665
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx &=-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-3 \int \frac{\sqrt{a^2-b^2 x^2}}{a+b x} \, dx\\ &=-\frac{3 \sqrt{a^2-b^2 x^2}}{b}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-(3 a) \int \frac{1}{\sqrt{a^2-b^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{a^2-b^2 x^2}}{b}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-(3 a) \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 \sqrt{a^2-b^2 x^2}}{b}-\frac{2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-\frac{3 a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0853772, size = 60, normalized size = 0.79 \[ -\frac{\frac{\sqrt{a^2-b^2 x^2} (5 a+b x)}{a+b x}+3 a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 206, normalized size = 2.7 \begin{align*} -{\frac{1}{{b}^{4}a} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-3}}-2\,{\frac{1}{{b}^{3}{a}^{2}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab \right ) ^{5/2} \left ( x+{\frac{a}{b}} \right ) ^{-2}}-2\,{\frac{1}{b{a}^{2}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab \right ) ^{3/2}}-3\,{\frac{x}{a}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab}}-3\,{\frac{a}{\sqrt{{b}^{2}}}\arctan \left ({\sqrt{{b}^{2}}x{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15592, size = 174, normalized size = 2.29 \begin{align*} -\frac{5 \, a b x + 5 \, a^{2} - 6 \,{\left (a b x + a^{2}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) + \sqrt{-b^{2} x^{2} + a^{2}}{\left (b x + 5 \, a\right )}}{b^{2} x + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac{3}{2}}}{\left (a + b x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2186, size = 104, normalized size = 1.37 \begin{align*} -\frac{3 \, a \arcsin \left (\frac{b x}{a}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (b\right )}{{\left | b \right |}} - \frac{\sqrt{-b^{2} x^{2} + a^{2}}}{b} + \frac{8 \, a}{{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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