Optimal. Leaf size=156 \[ -\frac{b^2 \sqrt{a+b x^2} (5 A b-8 a B)}{128 a^3 x^2}+\frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{7/2}}+\frac{b \sqrt{a+b x^2} (5 A b-8 a B)}{192 a^2 x^4}+\frac{\sqrt{a+b x^2} (5 A b-8 a B)}{48 a x^6}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8} \]
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Rubi [A] time = 0.119913, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {446, 78, 47, 51, 63, 208} \[ -\frac{b^2 \sqrt{a+b x^2} (5 A b-8 a B)}{128 a^3 x^2}+\frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{7/2}}+\frac{b \sqrt{a+b x^2} (5 A b-8 a B)}{192 a^2 x^4}+\frac{\sqrt{a+b x^2} (5 A b-8 a B)}{48 a x^6}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x} (A+B x)}{x^5} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8}+\frac{\left (-\frac{5 A b}{2}+4 a B\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^4} \, dx,x,x^2\right )}{8 a}\\ &=\frac{(5 A b-8 a B) \sqrt{a+b x^2}}{48 a x^6}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8}-\frac{(b (5 A b-8 a B)) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^2\right )}{96 a}\\ &=\frac{(5 A b-8 a B) \sqrt{a+b x^2}}{48 a x^6}+\frac{b (5 A b-8 a B) \sqrt{a+b x^2}}{192 a^2 x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8}+\frac{\left (b^2 (5 A b-8 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{128 a^2}\\ &=\frac{(5 A b-8 a B) \sqrt{a+b x^2}}{48 a x^6}+\frac{b (5 A b-8 a B) \sqrt{a+b x^2}}{192 a^2 x^4}-\frac{b^2 (5 A b-8 a B) \sqrt{a+b x^2}}{128 a^3 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8}-\frac{\left (b^3 (5 A b-8 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{256 a^3}\\ &=\frac{(5 A b-8 a B) \sqrt{a+b x^2}}{48 a x^6}+\frac{b (5 A b-8 a B) \sqrt{a+b x^2}}{192 a^2 x^4}-\frac{b^2 (5 A b-8 a B) \sqrt{a+b x^2}}{128 a^3 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8}-\frac{\left (b^2 (5 A b-8 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{128 a^3}\\ &=\frac{(5 A b-8 a B) \sqrt{a+b x^2}}{48 a x^6}+\frac{b (5 A b-8 a B) \sqrt{a+b x^2}}{192 a^2 x^4}-\frac{b^2 (5 A b-8 a B) \sqrt{a+b x^2}}{128 a^3 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8}+\frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.02195, size = 62, normalized size = 0.4 \[ -\frac{\left (a+b x^2\right )^{3/2} \left (3 a^4 A+b^3 x^8 (5 A b-8 a B) \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};\frac{b x^2}{a}+1\right )\right )}{24 a^5 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 239, normalized size = 1.5 \begin{align*} -{\frac{B}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{8\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{B{b}^{2}}{16\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{B{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{B{b}^{3}}{16\,{a}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{A}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ab}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,A{b}^{4}}{128\,{a}^{4}}\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 = 2.13965, size = 612, normalized size = 3.92 \begin{align*} \left [-\frac{3 \,{\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} \sqrt{a} x^{8} \log \left (-\frac{b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{6} - 48 \, A a^{4} - 2 \,{\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{4} - 8 \,{\left (8 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{768 \, a^{4} x^{8}}, \frac{3 \,{\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} \sqrt{-a} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (3 \,{\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{6} - 48 \, A a^{4} - 2 \,{\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{4} - 8 \,{\left (8 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{384 \, a^{4} x^{8}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 98.7911, size = 286, normalized size = 1.83 \begin{align*} - \frac{A a}{8 \sqrt{b} x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{7 A \sqrt{b}}{48 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{3}{2}}}{192 a x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A b^{\frac{5}{2}}}{384 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A b^{\frac{7}{2}}}{128 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 A b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{128 a^{\frac{7}{2}}} - \frac{B a}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 B \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{3}{2}}}{48 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{5}{2}}}{16 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.61277, size = 262, normalized size = 1.68 \begin{align*} \frac{\frac{3 \,{\left (8 \, B a b^{4} - 5 \, A b^{5}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{24 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a b^{4} - 88 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{2} b^{4} + 40 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{3} b^{4} + 24 \, \sqrt{b x^{2} + a} B a^{4} b^{4} - 15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A b^{5} + 55 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a b^{5} - 73 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{2} b^{5} - 15 \, \sqrt{b x^{2} + a} A a^{3} b^{5}}{a^{3} b^{4} x^{8}}}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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