Optimal. Leaf size=189 \[ -\frac {b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{9/2}}+\frac {b^3 \sqrt {a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac {b^2 \sqrt {a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}+\frac {b \sqrt {a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac {\sqrt {a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}} \]
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Rubi [A] time = 0.15, antiderivative size = 189, normalized size of antiderivative = 1.00, 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} \[ \frac {b^3 \sqrt {a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac {b^2 \sqrt {a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}-\frac {b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{9/2}}+\frac {b \sqrt {a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac {\sqrt {a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}+\frac {\left (-\frac {7 A b}{2}+5 a B\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^5} \, dx,x,x^2\right )}{10 a}\\ &=\frac {(7 A b-10 a B) \sqrt {a+b x^2}}{80 a x^8}-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}-\frac {(b (7 A b-10 a B)) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,x^2\right )}{160 a}\\ &=\frac {(7 A b-10 a B) \sqrt {a+b x^2}}{80 a x^8}+\frac {b (7 A b-10 a B) \sqrt {a+b x^2}}{480 a^2 x^6}-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}+\frac {\left (b^2 (7 A b-10 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^2\right )}{192 a^2}\\ &=\frac {(7 A b-10 a B) \sqrt {a+b x^2}}{80 a x^8}+\frac {b (7 A b-10 a B) \sqrt {a+b x^2}}{480 a^2 x^6}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x^2}}{384 a^3 x^4}-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}-\frac {\left (b^3 (7 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^3}\\ &=\frac {(7 A b-10 a B) \sqrt {a+b x^2}}{80 a x^8}+\frac {b (7 A b-10 a B) \sqrt {a+b x^2}}{480 a^2 x^6}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x^2}}{384 a^3 x^4}+\frac {b^3 (7 A b-10 a B) \sqrt {a+b x^2}}{256 a^4 x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}+\frac {\left (b^4 (7 A b-10 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{512 a^4}\\ &=\frac {(7 A b-10 a B) \sqrt {a+b x^2}}{80 a x^8}+\frac {b (7 A b-10 a B) \sqrt {a+b x^2}}{480 a^2 x^6}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x^2}}{384 a^3 x^4}+\frac {b^3 (7 A b-10 a B) \sqrt {a+b x^2}}{256 a^4 x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}+\frac {\left (b^3 (7 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^4}\\ &=\frac {(7 A b-10 a B) \sqrt {a+b x^2}}{80 a x^8}+\frac {b (7 A b-10 a B) \sqrt {a+b x^2}}{480 a^2 x^6}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x^2}}{384 a^3 x^4}+\frac {b^3 (7 A b-10 a B) \sqrt {a+b x^2}}{256 a^4 x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{10 a x^{10}}-\frac {b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 62, normalized size = 0.33 \[ -\frac {\left (a+b x^2\right )^{3/2} \left (3 a^5 A+b^4 x^{10} (10 a B-7 A b) \, _2F_1\left (\frac {3}{2},5;\frac {5}{2};\frac {b x^2}{a}+1\right )\right )}{30 a^6 x^{10}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 317, normalized size = 1.68 \[ \left [-\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt {a} x^{10} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (15 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{8} - 10 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{6} + 384 \, A a^{5} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{4} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{7680 \, a^{5} x^{10}}, -\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt {-a} x^{10} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{8} - 10 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{6} + 384 \, A a^{5} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{4} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3840 \, a^{5} x^{10}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 230, normalized size = 1.22 \[ -\frac {\frac {15 \, {\left (10 \, B a b^{5} - 7 \, A b^{6}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {150 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B a b^{5} - 700 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} b^{5} + 1280 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{3} b^{5} - 580 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{4} b^{5} - 150 \, \sqrt {b x^{2} + a} B a^{5} b^{5} - 105 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} A b^{6} + 490 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a b^{6} - 896 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2} b^{6} + 790 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{3} b^{6} + 105 \, \sqrt {b x^{2} + a} A a^{4} b^{6}}{a^{4} b^{5} x^{10}}}{3840 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 281, normalized size = 1.49 \[ -\frac {7 A \,b^{5} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{256 a^{\frac {9}{2}}}+\frac {5 B \,b^{4} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {7}{2}}}+\frac {7 \sqrt {b \,x^{2}+a}\, A \,b^{5}}{256 a^{5}}-\frac {5 \sqrt {b \,x^{2}+a}\, B \,b^{4}}{128 a^{4}}-\frac {7 \left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{4}}{256 a^{5} x^{2}}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,b^{3}}{128 a^{4} x^{2}}+\frac {7 \left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{3}}{128 a^{4} x^{4}}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,b^{2}}{64 a^{3} x^{4}}-\frac {7 \left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{2}}{96 a^{3} x^{6}}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B b}{48 a^{2} x^{6}}+\frac {7 \left (b \,x^{2}+a \right )^{\frac {3}{2}} A b}{80 a^{2} x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B}{8 a \,x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A}{10 a \,x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 258, normalized size = 1.37 \[ \frac {5 \, B b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {7}{2}}} - \frac {7 \, A b^{5} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{256 \, a^{\frac {9}{2}}} - \frac {5 \, \sqrt {b x^{2} + a} B b^{4}}{128 \, a^{4}} + \frac {7 \, \sqrt {b x^{2} + a} A b^{5}}{256 \, a^{5}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3}}{128 \, a^{4} x^{2}} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{4}}{256 \, a^{5} x^{2}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{64 \, a^{3} x^{4}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{128 \, a^{4} x^{4}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{48 \, a^{2} x^{6}} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{96 \, a^{3} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{8 \, a x^{8}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{80 \, a^{2} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{10 \, a x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 209, normalized size = 1.11 \[ \frac {7\,A\,{\left (b\,x^2+a\right )}^{5/2}}{30\,a^2\,x^{10}}-\frac {5\,B\,\sqrt {b\,x^2+a}}{128\,x^8}-\frac {79\,A\,{\left (b\,x^2+a\right )}^{3/2}}{384\,a\,x^{10}}-\frac {7\,A\,\sqrt {b\,x^2+a}}{256\,x^{10}}-\frac {49\,A\,{\left (b\,x^2+a\right )}^{7/2}}{384\,a^3\,x^{10}}+\frac {7\,A\,{\left (b\,x^2+a\right )}^{9/2}}{256\,a^4\,x^{10}}-\frac {73\,B\,{\left (b\,x^2+a\right )}^{3/2}}{384\,a\,x^8}+\frac {55\,B\,{\left (b\,x^2+a\right )}^{5/2}}{384\,a^2\,x^8}-\frac {5\,B\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a^3\,x^8}+\frac {A\,b^5\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,7{}\mathrm {i}}{256\,a^{9/2}}-\frac {B\,b^4\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{128\,a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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