Optimal. Leaf size=53 \[ \frac {\left (a+b x^2\right )^{3/2} (2 A b-5 a B)}{15 a^2 x^3}-\frac {A \left (a+b x^2\right )^{3/2}}{5 a x^5} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {453, 264} \begin {gather*} \frac {\left (a+b x^2\right )^{3/2} (2 A b-5 a B)}{15 a^2 x^3}-\frac {A \left (a+b x^2\right )^{3/2}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^6} \, dx &=-\frac {A \left (a+b x^2\right )^{3/2}}{5 a x^5}-\frac {(2 A b-5 a B) \int \frac {\sqrt {a+b x^2}}{x^4} \, dx}{5 a}\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{5 a x^5}+\frac {(2 A b-5 a B) \left (a+b x^2\right )^{3/2}}{15 a^2 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.75 \begin {gather*} -\frac {\left (a+b x^2\right )^{3/2} \left (3 a A+5 a B x^2-2 A b x^2\right )}{15 a^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 62, normalized size = 1.17 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-3 a^2 A-5 a^2 B x^2-a A b x^2-5 a b B x^4+2 A b^2 x^4\right )}{15 a^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 55, normalized size = 1.04 \begin {gather*} -\frac {{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} + {\left (5 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 232, normalized size = 4.38 \begin {gather*} \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B b^{\frac {3}{2}} - 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a b^{\frac {3}{2}} + 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A b^{\frac {5}{2}} + 20 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{2} b^{\frac {3}{2}} + 10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a b^{\frac {5}{2}} - 10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{3} b^{\frac {3}{2}} + 10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{2} b^{\frac {5}{2}} + 5 \, B a^{4} b^{\frac {3}{2}} - 2 \, A a^{3} 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 = 37, normalized size = 0.70 \begin {gather*} -\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (-2 A b \,x^{2}+5 B a \,x^{2}+3 A a \right )}{15 a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 56, normalized size = 1.06 \begin {gather*} -\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{3 \, a x^{3}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{15 \, a^{2} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{5 \, a x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 97, normalized size = 1.83 \begin {gather*} \frac {\left (A\,b^2+B\,a\,b\right )\,\sqrt {b\,x^2+a}}{5\,a^2\,x}-\frac {\left (5\,B\,a^2+A\,b\,a\right )\,\sqrt {b\,x^2+a}}{15\,a^2\,x^3}-\frac {A\,\sqrt {b\,x^2+a}}{5\,x^5}-\frac {b\,\sqrt {b\,x^2+a}\,\left (A\,b+8\,B\,a\right )}{15\,a^2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.23, size = 119, normalized size = 2.25 \begin {gather*} - \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a x^{2}} + \frac {2 A b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{2}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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