Optimal. Leaf size=189 \[ \frac{b^3 \sqrt{a+b x^2} (3 A b-10 a B)}{256 a^2 x^2}-\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{5/2}}+\frac{b^2 \sqrt{a+b x^2} (3 A b-10 a B)}{128 a x^4}+\frac{b \left (a+b x^2\right )^{3/2} (3 A b-10 a B)}{96 a x^6}+\frac{\left (a+b x^2\right )^{5/2} (3 A b-10 a B)}{80 a x^8}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}} \]
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Rubi [A] time = 0.148023, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{b^3 \sqrt{a+b x^2} (3 A b-10 a B)}{256 a^2 x^2}-\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{5/2}}+\frac{b^2 \sqrt{a+b x^2} (3 A b-10 a B)}{128 a x^4}+\frac{b \left (a+b x^2\right )^{3/2} (3 A b-10 a B)}{96 a x^6}+\frac{\left (a+b x^2\right )^{5/2} (3 A b-10 a B)}{80 a x^8}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}} \]
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 )^{5/2} \left (A+B x^2\right )}{x^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2} (A+B x)}{x^6} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}+\frac{\left (-\frac{3 A b}{2}+5 a B\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^5} \, dx,x,x^2\right )}{10 a}\\ &=\frac{(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}-\frac{(b (3 A b-10 a B)) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^4} \, dx,x,x^2\right )}{32 a}\\ &=\frac{b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac{(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}-\frac{\left (b^2 (3 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,x^2\right )}{64 a}\\ &=\frac{b^2 (3 A b-10 a B) \sqrt{a+b x^2}}{128 a x^4}+\frac{b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac{(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}-\frac{\left (b^3 (3 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{256 a}\\ &=\frac{b^2 (3 A b-10 a B) \sqrt{a+b x^2}}{128 a x^4}+\frac{b^3 (3 A b-10 a B) \sqrt{a+b x^2}}{256 a^2 x^2}+\frac{b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac{(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}+\frac{\left (b^4 (3 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{512 a^2}\\ &=\frac{b^2 (3 A b-10 a B) \sqrt{a+b x^2}}{128 a x^4}+\frac{b^3 (3 A b-10 a B) \sqrt{a+b x^2}}{256 a^2 x^2}+\frac{b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac{(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}+\frac{\left (b^3 (3 A b-10 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{256 a^2}\\ &=\frac{b^2 (3 A b-10 a B) \sqrt{a+b x^2}}{128 a x^4}+\frac{b^3 (3 A b-10 a B) \sqrt{a+b x^2}}{256 a^2 x^2}+\frac{b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac{(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}-\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.026164, size = 62, normalized size = 0.33 \[ -\frac{\left (a+b x^2\right )^{7/2} \left (7 a^5 A+b^4 x^{10} (10 a B-3 A b) \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{b x^2}{a}+1\right )\right )}{70 a^6 x^{10}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 353, normalized size = 1.9 \begin{align*} -{\frac{B}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bb}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{b}^{2}}{192\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,B{b}^{4}}{384\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,B{b}^{4}}{128\,{a}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{A}{10\,a{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,Ab}{80\,{a}^{2}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{b}^{2}}{160\,{a}^{3}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{b}^{3}}{640\,{a}^{4}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,A{b}^{4}}{1280\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,A{b}^{5}}{1280\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{b}^{5}}{256\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{b}^{5}}{256}\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}^{5}}{256\,{a}^{3}}\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 = 2.09433, size = 761, normalized size = 4.03 \begin{align*} \left [-\frac{15 \,{\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} \sqrt{a} x^{10} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (15 \,{\left (10 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{8} + 10 \,{\left (118 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{6} + 384 \, A a^{5} + 8 \,{\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{7680 \, a^{3} x^{10}}, -\frac{15 \,{\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} \sqrt{-a} x^{10} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (15 \,{\left (10 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{8} + 10 \,{\left (118 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{6} + 384 \, A a^{5} + 8 \,{\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3840 \, a^{3} x^{10}}\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 = 1.17197, size = 311, normalized size = 1.65 \begin{align*} -\frac{\frac{15 \,{\left (10 \, B a b^{5} - 3 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{150 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} B a b^{5} + 580 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a^{2} b^{5} - 1280 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{3} b^{5} + 700 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 150 \, \sqrt{b x^{2} + a} B a^{5} b^{5} - 45 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} A b^{6} + 210 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A a b^{6} + 384 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a^{2} b^{6} - 210 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 45 \, \sqrt{b x^{2} + a} A a^{4} b^{6}}{a^{2} b^{5} x^{10}}}{3840 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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