Optimal. Leaf size=150 \[ -\frac{x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]
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Rubi [A] time = 0.165252, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1804, 819, 778, 217, 206} \[ -\frac{x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 819
Rule 778
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^5 (-6 a B-7 a C x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{\int \frac{x^3 \left (-24 a^2 B-35 a^2 C x\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{\int \frac{x \left (-48 a^3 B-105 a^3 C x\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b^3}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \int \frac{1}{\sqrt{a+b x^2}} \, dx}{b^4}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{b^4}\\ &=-\frac{x^6 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}+\frac{C \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.268011, size = 147, normalized size = 0.98 \[ \frac{\sqrt{a} C \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a+b x^2}}-\frac{14 a^2 b^2 x^4 (15 B+29 C x)+14 a^3 b x^2 (12 B+25 C x)+3 a^4 (16 B+35 C x)+a b^3 x^6 (105 B+176 C x)-15 A b^4 x^7}{105 a b^4 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 277, normalized size = 1.9 \begin{align*} -{\frac{C{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{C{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{C{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Cx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{C\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}-{\frac{B{x}^{6}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-2\,{\frac{aB{x}^{4}}{{b}^{2} \left ( b{x}^{2}+a \right ) ^{7/2}}}-{\frac{8\,B{x}^{2}{a}^{2}}{5\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{16\,B{a}^{3}}{35\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{A{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,aA{x}^{3}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,{a}^{2}Ax}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,aAx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Ax}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Ax}{7\,{b}^{3}a}{\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 [A] time = 1.74462, size = 1044, normalized size = 6.96 \begin{align*} \left [\frac{105 \,{\left (C a b^{4} x^{8} + 4 \, C a^{2} b^{3} x^{6} + 6 \, C a^{3} b^{2} x^{4} + 4 \, C a^{4} b x^{2} + C a^{5}\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (105 \, B a b^{4} x^{6} + 406 \, C a^{2} b^{3} x^{5} + 210 \, B a^{2} b^{3} x^{4} + 350 \, C a^{3} b^{2} x^{3} + 168 \, B a^{3} b^{2} x^{2} +{\left (176 \, C a b^{4} - 15 \, A b^{5}\right )} x^{7} + 105 \, C a^{4} b x + 48 \, B a^{4} b\right )} \sqrt{b x^{2} + a}}{210 \,{\left (a b^{9} x^{8} + 4 \, a^{2} b^{8} x^{6} + 6 \, a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}, -\frac{105 \,{\left (C a b^{4} x^{8} + 4 \, C a^{2} b^{3} x^{6} + 6 \, C a^{3} b^{2} x^{4} + 4 \, C a^{4} b x^{2} + C a^{5}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (105 \, B a b^{4} x^{6} + 406 \, C a^{2} b^{3} x^{5} + 210 \, B a^{2} b^{3} x^{4} + 350 \, C a^{3} b^{2} x^{3} + 168 \, B a^{3} b^{2} x^{2} +{\left (176 \, C a b^{4} - 15 \, A b^{5}\right )} x^{7} + 105 \, C a^{4} b x + 48 \, B a^{4} b\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a b^{9} x^{8} + 4 \, a^{2} b^{8} x^{6} + 6 \, a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}\right ] \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 = 1.22477, size = 186, normalized size = 1.24 \begin{align*} -\frac{{\left ({\left ({\left ({\left ({\left (x{\left (\frac{105 \, B}{b} + \frac{{\left (176 \, C a^{3} b^{7} - 15 \, A a^{2} b^{8}\right )} x}{a^{3} b^{8}}\right )} + \frac{406 \, C a}{b^{2}}\right )} x + \frac{210 \, B a}{b^{2}}\right )} x + \frac{350 \, C a^{2}}{b^{3}}\right )} x + \frac{168 \, B a^{2}}{b^{3}}\right )} x + \frac{105 \, C a^{3}}{b^{4}}\right )} x + \frac{48 \, B a^{3}}{b^{4}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{C \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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