Optimal. Leaf size=115 \[ -\frac {8 b^2 x (6 A b-5 a B)}{15 a^4 \sqrt {a+b x^2}}-\frac {4 b (6 A b-5 a B)}{15 a^3 x \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}-\frac {A}{5 a x^5 \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 115, 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, 191} \[ -\frac {8 b^2 x (6 A b-5 a B)}{15 a^4 \sqrt {a+b x^2}}-\frac {4 b (6 A b-5 a B)}{15 a^3 x \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}-\frac {A}{5 a x^5 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 271
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac {A}{5 a x^5 \sqrt {a+b x^2}}-\frac {(6 A b-5 a B) \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2}} \, dx}{5 a}\\ &=-\frac {A}{5 a x^5 \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}+\frac {(4 b (6 A b-5 a B)) \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{15 a^2}\\ &=-\frac {A}{5 a x^5 \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}-\frac {4 b (6 A b-5 a B)}{15 a^3 x \sqrt {a+b x^2}}-\frac {\left (8 b^2 (6 A b-5 a B)\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^3}\\ &=-\frac {A}{5 a x^5 \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}-\frac {4 b (6 A b-5 a B)}{15 a^3 x \sqrt {a+b x^2}}-\frac {8 b^2 (6 A b-5 a B) x}{15 a^4 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.52 \[ \frac {x^2 \left (a^2-4 a b x^2-8 b^2 x^4\right ) (6 A b-5 a B)-3 a^3 A}{15 a^4 x^5 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 94, normalized size = 0.82 \[ \frac {{\left (8 \, {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{6} + 4 \, {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{4} - 3 \, A a^{3} - {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 294, normalized size = 2.56 \[ \frac {{\left (B a b^{2} - A b^{3}\right )} x}{\sqrt {b x^{2} + a} a^{4}} - \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} + 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {5}{2}} + 160 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} - 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 110 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} + 25 \, B a^{5} b^{\frac {3}{2}} - 33 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 83, normalized size = 0.72 \[ -\frac {48 A \,b^{3} x^{6}-40 B a \,b^{2} x^{6}+24 x^{4} A a \,b^{2}-20 B \,a^{2} b \,x^{4}-6 A \,a^{2} b \,x^{2}+5 B \,a^{3} x^{2}+3 A \,a^{3}}{15 \sqrt {b \,x^{2}+a}\, a^{4} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 134, normalized size = 1.17 \[ \frac {8 \, B b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {16 \, A b^{3} x}{5 \, \sqrt {b x^{2} + a} a^{4}} + \frac {4 \, B b}{3 \, \sqrt {b x^{2} + a} a^{2} x} - \frac {8 \, A b^{2}}{5 \, \sqrt {b x^{2} + a} a^{3} x} - \frac {B}{3 \, \sqrt {b x^{2} + a} a x^{3}} + \frac {2 \, A b}{5 \, \sqrt {b x^{2} + a} a^{2} x^{3}} - \frac {A}{5 \, \sqrt {b x^{2} + a} a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 82, normalized size = 0.71 \[ -\frac {5\,B\,a^3\,x^2+3\,A\,a^3-20\,B\,a^2\,b\,x^4-6\,A\,a^2\,b\,x^2-40\,B\,a\,b^2\,x^6+24\,A\,a\,b^2\,x^4+48\,A\,b^3\,x^6}{15\,a^4\,x^5\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.52, size = 593, normalized size = 5.16 \[ A \left (- \frac {a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {5 a^{3} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {30 a^{2} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {40 a b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {16 b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}}\right ) + B \left (- \frac {a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {3 a^{2} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {12 a b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {8 b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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