Optimal. Leaf size=214 \[ \frac{x^7 \left (a \left (15 a^2 b D-176 a^3 F+6 a b^2 C+8 b^3 B\right )+48 A b^4\right )}{105 a^4 b \left (a+b x^2\right )^{7/2}}+\frac{x^5 \left (a \left (-58 a^3 F+3 a b^2 C+4 b^3 B\right )+24 A b^4\right )}{15 a^3 b^2 \left (a+b x^2\right )^{7/2}}+\frac{x^3 \left (-10 a^4 F+a b^3 B+6 A b^4\right )}{3 a^2 b^3 \left (a+b x^2\right )^{7/2}}+\frac{x \left (A b^4-a^4 F\right )}{a b^4 \left (a+b x^2\right )^{7/2}}+\frac{F \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]
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Rubi [A] time = 0.409623, antiderivative size = 250, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {1814, 1157, 385, 217, 206} \[ \frac{x \left (\frac{15 a^2 b D-176 a^3 F+6 a b^2 C+8 b^3 B}{b^4}+\frac{48 A}{a}\right )}{105 a^3 \sqrt{a+b x^2}}+\frac{x \left (a \left (-45 a^2 b D+122 a^3 F+3 a b^2 C+4 b^3 B\right )+24 A b^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{x \left (\frac{15 a^2 b D-22 a^3 F-8 a b^2 C+b^3 B}{b^4}+\frac{6 A}{a}\right )}{35 a \left (a+b x^2\right )^{5/2}}+\frac{x \left (\frac{A}{a}-\frac{a^2 b D+a^3 (-F)-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}+\frac{F \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 1157
Rule 385
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac{\left (\frac{A}{a}-\frac{b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x}{7 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{-6 A-\frac{a \left (b^3 B-a b^2 C+a^2 b D-a^3 F\right )}{b^4}-\frac{7 a \left (b^2 C-a b D+a^2 F\right ) x^2}{b^3}-\frac{7 a (b D-a F) x^4}{b^2}-\frac{7 a F x^6}{b}}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac{\left (\frac{A}{a}-\frac{b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x}{7 \left (a+b x^2\right )^{7/2}}+\frac{\left (\frac{6 A}{a}+\frac{b^3 B-8 a b^2 C+15 a^2 b D-22 a^3 F}{b^4}\right ) x}{35 a \left (a+b x^2\right )^{5/2}}+\frac{\int \frac{\frac{24 A b^4+4 a b^3 B+3 a^2 b^2 C-10 a^3 b D+17 a^4 F}{b^4}+\frac{35 a^2 (b D-2 a F) x^2}{b^3}+\frac{35 a^2 F x^4}{b^2}}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac{\left (\frac{A}{a}-\frac{b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x}{7 \left (a+b x^2\right )^{7/2}}+\frac{\left (\frac{6 A}{a}+\frac{b^3 B-8 a b^2 C+15 a^2 b D-22 a^3 F}{b^4}\right ) x}{35 a \left (a+b x^2\right )^{5/2}}+\frac{\left (24 A b^4+a \left (4 b^3 B+3 a b^2 C-45 a^2 b D+122 a^3 F\right )\right ) x}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac{\int \frac{-\frac{48 A b^4+8 a b^3 B+6 a^2 b^2 C+15 a^3 b D-71 a^4 F}{b^4}-\frac{105 a^3 F x^2}{b^3}}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3}\\ &=\frac{\left (\frac{A}{a}-\frac{b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x}{7 \left (a+b x^2\right )^{7/2}}+\frac{\left (\frac{6 A}{a}+\frac{b^3 B-8 a b^2 C+15 a^2 b D-22 a^3 F}{b^4}\right ) x}{35 a \left (a+b x^2\right )^{5/2}}+\frac{\left (24 A b^4+a \left (4 b^3 B+3 a b^2 C-45 a^2 b D+122 a^3 F\right )\right ) x}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{\left (\frac{48 A}{a}+\frac{8 b^3 B+6 a b^2 C+15 a^2 b D-176 a^3 F}{b^4}\right ) x}{105 a^3 \sqrt{a+b x^2}}+\frac{F \int \frac{1}{\sqrt{a+b x^2}} \, dx}{b^4}\\ &=\frac{\left (\frac{A}{a}-\frac{b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x}{7 \left (a+b x^2\right )^{7/2}}+\frac{\left (\frac{6 A}{a}+\frac{b^3 B-8 a b^2 C+15 a^2 b D-22 a^3 F}{b^4}\right ) x}{35 a \left (a+b x^2\right )^{5/2}}+\frac{\left (24 A b^4+a \left (4 b^3 B+3 a b^2 C-45 a^2 b D+122 a^3 F\right )\right ) x}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{\left (\frac{48 A}{a}+\frac{8 b^3 B+6 a b^2 C+15 a^2 b D-176 a^3 F}{b^4}\right ) x}{105 a^3 \sqrt{a+b x^2}}+\frac{F \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{b^4}\\ &=\frac{\left (\frac{A}{a}-\frac{b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x}{7 \left (a+b x^2\right )^{7/2}}+\frac{\left (\frac{6 A}{a}+\frac{b^3 B-8 a b^2 C+15 a^2 b D-22 a^3 F}{b^4}\right ) x}{35 a \left (a+b x^2\right )^{5/2}}+\frac{\left (24 A b^4+a \left (4 b^3 B+3 a b^2 C-45 a^2 b D+122 a^3 F\right )\right ) x}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{\left (\frac{48 A}{a}+\frac{8 b^3 B+6 a b^2 C+15 a^2 b D-176 a^3 F}{b^4}\right ) x}{105 a^3 \sqrt{a+b x^2}}+\frac{F \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.445792, size = 197, normalized size = 0.92 \[ \frac{x \left (a^3 b^4 \left (105 A+35 B x^2+21 C x^4+15 D x^6\right )+2 a^2 b^5 x^2 \left (105 A+14 B x^2+3 C x^4\right )-406 a^5 b^2 F x^4-176 a^4 b^3 F x^6-350 a^6 b F x^2-105 a^7 F+8 a b^6 x^4 \left (21 A+B x^2\right )+48 A b^7 x^6\right )}{105 a^4 b^4 \left (a+b x^2\right )^{7/2}}+\frac{\sqrt{a} F \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 427, normalized size = 2. \begin{align*} -{\frac{F{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{F{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{F{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Fx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{F\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}-{\frac{D{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,D{x}^{3}a}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,{a}^{2}Dx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,aDx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Dx}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Dx}{7\,{b}^{3}a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{C{x}^{3}}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{3\,aCx}{28\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,Cx}{140\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Cx}{35\,a{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Cx}{35\,{b}^{2}{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{Bx}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{Bx}{35\,ab} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{4\,Bx}{105\,{a}^{2}b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,Bx}{105\,{a}^{3}b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{Ax}{7\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{6\,Ax}{35\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,Ax}{35\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,Ax}{35\,{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24285, size = 275, normalized size = 1.29 \begin{align*} -\frac{{\left ({\left (x^{2}{\left (\frac{{\left (176 \, F a^{4} b^{6} - 15 \, D a^{3} b^{7} - 6 \, C a^{2} b^{8} - 8 \, B a b^{9} - 48 \, A b^{10}\right )} x^{2}}{a^{4} b^{7}} + \frac{7 \,{\left (58 \, F a^{5} b^{5} - 3 \, C a^{3} b^{7} - 4 \, B a^{2} b^{8} - 24 \, A a b^{9}\right )}}{a^{4} b^{7}}\right )} + \frac{35 \,{\left (10 \, F a^{6} b^{4} - B a^{3} b^{7} - 6 \, A a^{2} b^{8}\right )}}{a^{4} b^{7}}\right )} x^{2} + \frac{105 \,{\left (F a^{7} b^{3} - A a^{3} b^{7}\right )}}{a^{4} b^{7}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{F \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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