Optimal. Leaf size=84 \[ -\frac {2 b \sqrt {a+b x^2} (4 A b-5 a B)}{15 a^3 x}+\frac {\sqrt {a+b x^2} (4 A b-5 a B)}{15 a^2 x^3}-\frac {A \sqrt {a+b x^2}}{5 a x^5} \]
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Rubi [A] time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 264} \begin {gather*} -\frac {2 b \sqrt {a+b x^2} (4 A b-5 a B)}{15 a^3 x}+\frac {\sqrt {a+b x^2} (4 A b-5 a B)}{15 a^2 x^3}-\frac {A \sqrt {a+b x^2}}{5 a x^5} \end {gather*}
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^6 \sqrt {a+b x^2}} \, dx &=-\frac {A \sqrt {a+b x^2}}{5 a x^5}-\frac {(4 A b-5 a B) \int \frac {1}{x^4 \sqrt {a+b x^2}} \, dx}{5 a}\\ &=-\frac {A \sqrt {a+b x^2}}{5 a x^5}+\frac {(4 A b-5 a B) \sqrt {a+b x^2}}{15 a^2 x^3}+\frac {(2 b (4 A b-5 a B)) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{15 a^2}\\ &=-\frac {A \sqrt {a+b x^2}}{5 a x^5}+\frac {(4 A b-5 a B) \sqrt {a+b x^2}}{15 a^2 x^3}-\frac {2 b (4 A b-5 a B) \sqrt {a+b x^2}}{15 a^3 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {a+b x^2} \left (a^2 \left (3 A+5 B x^2\right )-2 a b x^2 \left (2 A+5 B x^2\right )+8 A b^2 x^4\right )}{15 a^3 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 62, normalized size = 0.74 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-3 a^2 A-5 a^2 B x^2+4 a A b x^2+10 a b B x^4-8 A b^2 x^4\right )}{15 a^3 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 58, normalized size = 0.69 \begin {gather*} \frac {{\left (2 \, {\left (5 \, B a b - 4 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} - {\left (5 \, B a^{2} - 4 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, a^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 176, normalized size = 2.10 \begin {gather*} \frac {4 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B b^{\frac {3}{2}} - 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a b^{\frac {3}{2}} + 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A b^{\frac {5}{2}} + 25 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} b^{\frac {3}{2}} - 20 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {5}{2}} - 5 \, B a^{3} b^{\frac {3}{2}} + 4 \, A a^{2} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {b \,x^{2}+a}\, \left (8 A \,b^{2} x^{4}-10 B a b \,x^{4}-4 A a b \,x^{2}+5 B \,a^{2} x^{2}+3 a^{2} A \right )}{15 a^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 96, normalized size = 1.14 \begin {gather*} \frac {2 \, \sqrt {b x^{2} + a} B b}{3 \, a^{2} x} - \frac {8 \, \sqrt {b x^{2} + a} A b^{2}}{15 \, a^{3} x} - \frac {\sqrt {b x^{2} + a} B}{3 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a} A b}{15 \, a^{2} x^{3}} - \frac {\sqrt {b x^{2} + a} A}{5 \, a x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 58, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {b\,x^2+a}\,\left (5\,B\,a^2\,x^2+3\,A\,a^2-10\,B\,a\,b\,x^4-4\,A\,a\,b\,x^2+8\,A\,b^2\,x^4\right )}{15\,a^3\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.17, size = 355, normalized size = 4.23 \begin {gather*} - \frac {3 A 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 A 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 A 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 A 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 A 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}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {2 B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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