Optimal. Leaf size=164 \[ \frac{9}{16} a^5 x \sqrt{a^2-b^2 x^2}+\frac{3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{9 a^7 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b} \]
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Rubi [A] time = 0.0629918, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{9}{16} a^5 x \sqrt{a^2-b^2 x^2}+\frac{3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{9 a^7 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b} \]
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)^3 \left (a^2-b^2 x^2\right )^{3/2} \, dx &=-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{1}{7} (9 a) \int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{1}{2} \left (3 a^2\right ) \int (a+b x) \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=-\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{1}{2} \left (3 a^3\right ) \int \left (a^2-b^2 x^2\right )^{3/2} \, dx\\ &=\frac{3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{1}{8} \left (9 a^5\right ) \int \sqrt{a^2-b^2 x^2} \, dx\\ &=\frac{9}{16} a^5 x \sqrt{a^2-b^2 x^2}+\frac{3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{1}{16} \left (9 a^7\right ) \int \frac{1}{\sqrt{a^2-b^2 x^2}} \, dx\\ &=\frac{9}{16} a^5 x \sqrt{a^2-b^2 x^2}+\frac{3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{1}{16} \left (9 a^7\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{9}{16} a^5 x \sqrt{a^2-b^2 x^2}+\frac{3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{9 a^7 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b}\\ \end{align*}
Mathematica [A] time = 0.224929, size = 134, normalized size = 0.82 \[ \frac{\sqrt{a^2-b^2 x^2} \left (\sqrt{1-\frac{b^2 x^2}{a^2}} \left (656 a^4 b^2 x^2+350 a^3 b^3 x^3-208 a^2 b^4 x^4+245 a^5 b x-368 a^6-280 a b^5 x^5-80 b^6 x^6\right )+315 a^6 \sin ^{-1}\left (\frac{b x}{a}\right )\right )}{560 b \sqrt{1-\frac{b^2 x^2}{a^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 134, normalized size = 0.8 \begin{align*} -{\frac{b{x}^{2}}{7} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{23\,{a}^{2}}{35\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{2} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{3\,x{a}^{3}}{8} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{5}x}{16}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{9\,{a}^{7}}{16}\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 = 1.56827, size = 170, normalized size = 1.04 \begin{align*} \frac{9 \, a^{7} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{16 \, \sqrt{b^{2}}} + \frac{9}{16} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{5} x + \frac{3}{8} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a^{3} x - \frac{1}{7} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} b x^{2} - \frac{1}{2} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} a x - \frac{23 \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} a^{2}}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05022, size = 259, normalized size = 1.58 \begin{align*} -\frac{630 \, a^{7} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) +{\left (80 \, b^{6} x^{6} + 280 \, a b^{5} x^{5} + 208 \, a^{2} b^{4} x^{4} - 350 \, a^{3} b^{3} x^{3} - 656 \, a^{4} b^{2} x^{2} - 245 \, a^{5} b x + 368 \, a^{6}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{560 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 13.6904, size = 821, normalized size = 5.01 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26588, size = 140, normalized size = 0.85 \begin{align*} \frac{9 \, a^{7} \arcsin \left (\frac{b x}{a}\right ) \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (b\right )}{16 \,{\left | b \right |}} - \frac{1}{560} \,{\left (\frac{368 \, a^{6}}{b} -{\left (245 \, a^{5} + 2 \,{\left (328 \, a^{4} b +{\left (175 \, a^{3} b^{2} - 4 \,{\left (26 \, a^{2} b^{3} + 5 \,{\left (2 \, b^{5} x + 7 \, a b^{4}\right )} x\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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