Optimal. Leaf size=156 \[ \frac{b^2 \sqrt{a+b x^2} (3 A b-8 a B)}{128 a^2 x^2}-\frac{b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{b \sqrt{a+b x^2} (3 A b-8 a B)}{64 a x^4}+\frac{\left (a+b x^2\right )^{3/2} (3 A b-8 a B)}{48 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8} \]
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Rubi [A] time = 0.123087, 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} \[ \frac{b^2 \sqrt{a+b x^2} (3 A b-8 a B)}{128 a^2 x^2}-\frac{b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{b \sqrt{a+b x^2} (3 A b-8 a B)}{64 a x^4}+\frac{\left (a+b x^2\right )^{3/2} (3 A b-8 a B)}{48 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8} \]
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{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (A+B x)}{x^5} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac{\left (-\frac{3 A b}{2}+4 a B\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^4} \, dx,x,x^2\right )}{8 a}\\ &=\frac{(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac{(b (3 A b-8 a B)) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,x^2\right )}{32 a}\\ &=\frac{b (3 A b-8 a B) \sqrt{a+b x^2}}{64 a x^4}+\frac{(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac{\left (b^2 (3 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}\\ &=\frac{b (3 A b-8 a B) \sqrt{a+b x^2}}{64 a x^4}+\frac{b^2 (3 A b-8 a B) \sqrt{a+b x^2}}{128 a^2 x^2}+\frac{(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac{\left (b^3 (3 A b-8 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{256 a^2}\\ &=\frac{b (3 A b-8 a B) \sqrt{a+b x^2}}{64 a x^4}+\frac{b^2 (3 A b-8 a B) \sqrt{a+b x^2}}{128 a^2 x^2}+\frac{(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac{\left (b^2 (3 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^2}\\ &=\frac{b (3 A b-8 a B) \sqrt{a+b x^2}}{64 a x^4}+\frac{b^2 (3 A b-8 a B) \sqrt{a+b x^2}}{128 a^2 x^2}+\frac{(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac{b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0244022, size = 62, normalized size = 0.4 \[ -\frac{\left (a+b x^2\right )^{5/2} \left (5 a^4 A+b^3 x^8 (3 A b-8 a B) \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{b x^2}{a}+1\right )\right )}{40 a^5 x^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 275, normalized size = 1.8 \begin{align*} -{\frac{A}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ab}{16\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{A{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{A{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{3\,A{b}^{4}}{128\,{a}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{B}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{24\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{b}^{2}}{48\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B{b}^{3}}{48\,{a}^{3}} \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{3}{2}}}}-{\frac{B{b}^{3}}{16\,{a}^{2}}\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 = 1.7552, size = 621, normalized size = 3.98 \begin{align*} \left [-\frac{3 \,{\left (8 \, B a b^{3} - 3 \, 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} - 3 \, A a b^{3}\right )} x^{6} + 48 \, A a^{4} + 2 \,{\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + 8 \,{\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{768 \, a^{3} x^{8}}, -\frac{3 \,{\left (8 \, B a b^{3} - 3 \, 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} - 3 \, A a b^{3}\right )} x^{6} + 48 \, A a^{4} + 2 \,{\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + 8 \,{\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{384 \, a^{3} x^{8}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 136.73, size = 287, normalized size = 1.84 \begin{align*} - \frac{A a^{2}}{8 \sqrt{b} x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A a \sqrt{b}}{16 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{13 A b^{\frac{3}{2}}}{64 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{5}{2}}}{128 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 A b^{\frac{7}{2}}}{128 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{128 a^{\frac{5}{2}}} - \frac{B a^{2}}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{11 B a \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{17 B b^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B b^{\frac{5}{2}}}{16 a 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{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14682, size = 262, normalized size = 1.68 \begin{align*} -\frac{\frac{3 \,{\left (8 \, B a b^{4} - 3 \, A b^{5}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{24 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a b^{4} + 40 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{2} b^{4} - 88 \,{\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} - 9 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A b^{5} + 33 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a b^{5} + 33 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{2} b^{5} - 9 \, \sqrt{b x^{2} + a} A a^{3} b^{5}}{a^{2} b^{4} x^{8}}}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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