Optimal. Leaf size=117 \[ \frac {8 b^2 \sqrt {a+b x^2} (6 A b-7 a B)}{105 a^4 x}-\frac {4 b \sqrt {a+b x^2} (6 A b-7 a B)}{105 a^3 x^3}+\frac {\sqrt {a+b x^2} (6 A b-7 a B)}{35 a^2 x^5}-\frac {A \sqrt {a+b x^2}}{7 a x^7} \]
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Rubi [A] time = 0.05, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ \frac {8 b^2 \sqrt {a+b x^2} (6 A b-7 a B)}{105 a^4 x}-\frac {4 b \sqrt {a+b x^2} (6 A b-7 a B)}{105 a^3 x^3}+\frac {\sqrt {a+b x^2} (6 A b-7 a B)}{35 a^2 x^5}-\frac {A \sqrt {a+b x^2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^8 \sqrt {a+b x^2}} \, dx &=-\frac {A \sqrt {a+b x^2}}{7 a x^7}-\frac {(6 A b-7 a B) \int \frac {1}{x^6 \sqrt {a+b x^2}} \, dx}{7 a}\\ &=-\frac {A \sqrt {a+b x^2}}{7 a x^7}+\frac {(6 A b-7 a B) \sqrt {a+b x^2}}{35 a^2 x^5}+\frac {(4 b (6 A b-7 a B)) \int \frac {1}{x^4 \sqrt {a+b x^2}} \, dx}{35 a^2}\\ &=-\frac {A \sqrt {a+b x^2}}{7 a x^7}+\frac {(6 A b-7 a B) \sqrt {a+b x^2}}{35 a^2 x^5}-\frac {4 b (6 A b-7 a B) \sqrt {a+b x^2}}{105 a^3 x^3}-\frac {\left (8 b^2 (6 A b-7 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{105 a^3}\\ &=-\frac {A \sqrt {a+b x^2}}{7 a x^7}+\frac {(6 A b-7 a B) \sqrt {a+b x^2}}{35 a^2 x^5}-\frac {4 b (6 A b-7 a B) \sqrt {a+b x^2}}{105 a^3 x^3}+\frac {8 b^2 (6 A b-7 a B) \sqrt {a+b x^2}}{105 a^4 x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 84, normalized size = 0.72 \[ \frac {\left (\frac {b x^2}{a}+1\right ) \left (3 a^2-4 a b x^2+8 b^2 x^4\right ) (6 A b-7 a B)}{105 a^3 x^5 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{7 a x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 82, normalized size = 0.70 \[ -\frac {{\left (8 \, {\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{6} - 4 \, {\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 3 \, {\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 232, normalized size = 1.98 \[ \frac {16 \, {\left (70 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B b^{\frac {5}{2}} - 175 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a b^{\frac {5}{2}} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A b^{\frac {7}{2}} + 147 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{2} b^{\frac {5}{2}} - 126 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a b^{\frac {7}{2}} - 49 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{3} b^{\frac {5}{2}} + 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{2} b^{\frac {7}{2}} + 7 \, B a^{4} b^{\frac {5}{2}} - 6 \, A a^{3} b^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 83, normalized size = 0.71 \[ -\frac {\sqrt {b \,x^{2}+a}\, \left (-48 A \,b^{3} x^{6}+56 B a \,b^{2} x^{6}+24 x^{4} A a \,b^{2}-28 B \,a^{2} b \,x^{4}-18 A \,a^{2} b \,x^{2}+21 B \,a^{3} x^{2}+15 A \,a^{3}\right )}{105 a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 138, normalized size = 1.18 \[ -\frac {8 \, \sqrt {b x^{2} + a} B b^{2}}{15 \, a^{3} x} + \frac {16 \, \sqrt {b x^{2} + a} A b^{3}}{35 \, a^{4} x} + \frac {4 \, \sqrt {b x^{2} + a} B b}{15 \, a^{2} x^{3}} - \frac {8 \, \sqrt {b x^{2} + a} A b^{2}}{35 \, a^{3} x^{3}} - \frac {\sqrt {b x^{2} + a} B}{5 \, a x^{5}} + \frac {6 \, \sqrt {b x^{2} + a} A b}{35 \, a^{2} x^{5}} - \frac {\sqrt {b x^{2} + a} A}{7 \, a x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 105, normalized size = 0.90 \[ \frac {\sqrt {b\,x^2+a}\,\left (6\,A\,b-7\,B\,a\right )}{35\,a^2\,x^5}+\frac {\sqrt {b\,x^2+a}\,\left (48\,A\,b^3-56\,B\,a\,b^2\right )}{105\,a^4\,x}-\frac {\left (24\,A\,b^2-28\,B\,a\,b\right )\,\sqrt {b\,x^2+a}}{105\,a^3\,x^3}-\frac {A\,\sqrt {b\,x^2+a}}{7\,a\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.68, size = 819, normalized size = 7.00 \[ - \frac {5 A a^{6} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {9 A a^{5} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {5 A a^{4} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac {5 A a^{3} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac {30 A a^{2} b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac {40 A a b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac {16 A b^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {3 B a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {2 B a^{3} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {3 B a^{2} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {12 B a b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {8 B b^{\frac {17}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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