Optimal. Leaf size=146 \[ -\frac {16 b^2 x (8 A b-5 a B)}{15 a^5 \sqrt {a+b x^2}}-\frac {8 b^2 x (8 A b-5 a B)}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {453, 271, 192, 191} \[ -\frac {16 b^2 x (8 A b-5 a B)}{15 a^5 \sqrt {a+b x^2}}-\frac {8 b^2 x (8 A b-5 a B)}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 271
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}-\frac {(8 A b-5 a B) \int \frac {1}{x^4 \left (a+b x^2\right )^{5/2}} \, dx}{5 a}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}+\frac {(2 b (8 A b-5 a B)) \int \frac {1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx}{5 a^2}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac {\left (8 b^2 (8 A b-5 a B)\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{5 a^3}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac {8 b^2 (8 A b-5 a B) x}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 b^2 (8 A b-5 a B)\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^4}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac {8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac {2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac {8 b^2 (8 A b-5 a B) x}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac {16 b^2 (8 A b-5 a B) x}{15 a^5 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 0.49 \[ \frac {a x^2 \left (a^3-6 a^2 b x^2-24 a b^2 x^4-16 b^3 x^6\right ) (8 A b-5 a B)-3 a^5 A}{15 a^6 x^5 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 129, normalized size = 0.88 \[ \frac {{\left (16 \, {\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 24 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} - 3 \, A a^{4} + 6 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 336, normalized size = 2.30 \[ \frac {x {\left (\frac {{\left (8 \, B a^{5} b^{4} - 11 \, A a^{4} b^{5}\right )} x^{2}}{a^{9} b} + \frac {3 \, {\left (3 \, B a^{6} b^{3} - 4 \, A a^{5} b^{4}\right )}}{a^{9} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} - 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} + 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {5}{2}} + 250 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} - 490 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 170 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} + 320 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} + 40 \, B a^{5} b^{\frac {3}{2}} - 73 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 107, normalized size = 0.73 \[ -\frac {128 A \,b^{4} x^{8}-80 B a \,b^{3} x^{8}+192 A a \,b^{3} x^{6}-120 B \,a^{2} b^{2} x^{6}+48 A \,a^{2} b^{2} x^{4}-30 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+5 B \,a^{4} x^{2}+3 A \,a^{4}}{15 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{5} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 172, normalized size = 1.18 \[ \frac {16 \, B b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, B b^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {128 \, A b^{3} x}{15 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, A b^{3} x}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} + \frac {2 \, B b}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x} - \frac {16 \, A b^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} x} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{3}} + \frac {8 \, A b}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x^{3}} - \frac {A}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 231, normalized size = 1.58 \[ \frac {\frac {a\,\left (\frac {b^2\,\left (73\,A\,b-40\,B\,a\right )}{18\,a^4}+\frac {b^2\,\left (86\,A\,b-35\,B\,a\right )}{30\,a^4}+\frac {a\,\left (\frac {28\,A\,b^4-10\,B\,a\,b^3}{45\,a^5}-\frac {b^3\,\left (86\,A\,b-35\,B\,a\right )}{18\,a^5}\right )}{b}\right )}{b}-\frac {b\,\left (73\,A\,b-40\,B\,a\right )}{30\,a^3}}{x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {x^2\,\left (\frac {28\,A\,b^3-10\,B\,a\,b^2}{15\,a^5}-\frac {2\,b^2\,\left (26\,A\,b-15\,B\,a\right )}{5\,a^5}\right )-\frac {b\,\left (26\,A\,b-15\,B\,a\right )}{5\,a^4}}{x\,\sqrt {b\,x^2+a}}-\frac {\sqrt {b\,x^2+a}\,\left (5\,B\,a^3-14\,A\,a^2\,b\right )}{15\,a^6\,x^3}-\frac {A\,\sqrt {b\,x^2+a}}{5\,a^3\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 60.74, size = 944, normalized size = 6.47 \[ A \left (- \frac {3 a^{6} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} + \frac {2 a^{5} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {35 a^{4} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {280 a^{3} b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {560 a^{2} b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {448 a b^{\frac {43}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}} - \frac {128 b^{\frac {45}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{6} + 90 a^{7} b^{18} x^{8} + 60 a^{6} b^{19} x^{10} + 15 a^{5} b^{20} x^{12}}\right ) + B \left (- \frac {a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {5 a^{3} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {30 a^{2} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {40 a b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {16 b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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