Optimal. Leaf size=178 \[ -\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{9 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2048 b^{5/2}}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2} \]
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Rubi [A] time = 0.0855914, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ -\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{9 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2048 b^{5/2}}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^4 \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{14} (9 a) \int x^4 \left (a+b x^2\right )^{7/2} \, dx\\ &=\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{8} \left (3 a^2\right ) \int x^4 \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{16} \left (3 a^3\right ) \int x^4 \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{128} \left (9 a^4\right ) \int x^4 \sqrt{a+b x^2} \, dx\\ &=\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{256} \left (3 a^5\right ) \int \frac{x^4}{\sqrt{a+b x^2}} \, dx\\ &=\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}-\frac{\left (9 a^6\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{1024 b}\\ &=-\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{\left (9 a^7\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{2048 b^2}\\ &=-\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{\left (9 a^7\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2048 b^2}\\ &=-\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{9 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2048 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.184786, size = 127, normalized size = 0.71 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (44928 a^2 b^4 x^8+39056 a^3 b^3 x^6+14168 a^4 b^2 x^4+210 a^5 b x^2-315 a^6+24320 a b^5 x^{10}+5120 b^6 x^{12}\right )+\frac{315 a^{13/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{71680 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 149, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{14\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{ax}{56\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{a}^{2}x}{560\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,{a}^{3}x}{4480\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{a}^{4}x}{1280\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{a}^{5}x}{1024\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{6}x}{2048\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{9\,{a}^{7}}{2048}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \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 = 2.20406, size = 599, normalized size = 3.37 \begin{align*} \left [\frac{315 \, a^{7} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (5120 \, b^{7} x^{13} + 24320 \, a b^{6} x^{11} + 44928 \, a^{2} b^{5} x^{9} + 39056 \, a^{3} b^{4} x^{7} + 14168 \, a^{4} b^{3} x^{5} + 210 \, a^{5} b^{2} x^{3} - 315 \, a^{6} b x\right )} \sqrt{b x^{2} + a}}{143360 \, b^{3}}, -\frac{315 \, a^{7} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (5120 \, b^{7} x^{13} + 24320 \, a b^{6} x^{11} + 44928 \, a^{2} b^{5} x^{9} + 39056 \, a^{3} b^{4} x^{7} + 14168 \, a^{4} b^{3} x^{5} + 210 \, a^{5} b^{2} x^{3} - 315 \, a^{6} b x\right )} \sqrt{b x^{2} + a}}{71680 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.973, size = 231, normalized size = 1.3 \begin{align*} - \frac{9 a^{\frac{13}{2}} x}{2048 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{11}{2}} x^{3}}{2048 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{1027 a^{\frac{9}{2}} x^{5}}{5120 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{6653 a^{\frac{7}{2}} b x^{7}}{8960 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5249 a^{\frac{5}{2}} b^{2} x^{9}}{4480 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{541 a^{\frac{3}{2}} b^{3} x^{11}}{560 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 \sqrt{a} b^{4} x^{13}}{56 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{9 a^{7} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2048 b^{\frac{5}{2}}} + \frac{b^{5} x^{15}}{14 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.98651, size = 161, normalized size = 0.9 \begin{align*} -\frac{9 \, a^{7} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2048 \, b^{\frac{5}{2}}} - \frac{1}{71680} \,{\left (\frac{315 \, a^{6}}{b^{2}} - 2 \,{\left (\frac{105 \, a^{5}}{b} + 4 \,{\left (1771 \, a^{4} + 2 \,{\left (2441 \, a^{3} b + 8 \,{\left (351 \, a^{2} b^{2} + 10 \,{\left (4 \, b^{4} x^{2} + 19 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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