Optimal. Leaf size=53 \[ \frac {\left (a+b x^2\right )^{7/2} (2 A b-9 a B)}{63 a^2 x^7}-\frac {A \left (a+b x^2\right )^{7/2}}{9 a x^9} \]
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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 )^{7/2} (2 A b-9 a B)}{63 a^2 x^7}-\frac {A \left (a+b x^2\right )^{7/2}}{9 a x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx &=-\frac {A \left (a+b x^2\right )^{7/2}}{9 a x^9}-\frac {(2 A b-9 a B) \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{9 a}\\ &=-\frac {A \left (a+b x^2\right )^{7/2}}{9 a x^9}+\frac {(2 A b-9 a B) \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.75 \begin {gather*} -\frac {\left (a+b x^2\right )^{7/2} \left (7 a A+9 a B x^2-2 A b x^2\right )}{63 a^2 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.32, size = 110, normalized size = 2.08 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-7 a^4 A-9 a^4 B x^2-19 a^3 A b x^2-27 a^3 b B x^4-15 a^2 A b^2 x^4-27 a^2 b^2 B x^6-a A b^3 x^6-9 a b^3 B x^8+2 A b^4 x^8\right )}{63 a^2 x^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.11, size = 102, normalized size = 1.92 \begin {gather*} -\frac {{\left ({\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} x^{8} + {\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} x^{6} + 7 \, A a^{4} + 3 \, {\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{4} + {\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{63 \, a^{2} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 456, normalized size = 8.60 \begin {gather*} \frac {2 \, {\left (63 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} B b^{\frac {7}{2}} - 126 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} B a b^{\frac {7}{2}} + 126 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} A b^{\frac {9}{2}} + 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a^{2} b^{\frac {7}{2}} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A a b^{\frac {9}{2}} - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{3} b^{\frac {7}{2}} + 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a^{2} b^{\frac {9}{2}} + 504 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{4} b^{\frac {7}{2}} + 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{3} b^{\frac {9}{2}} - 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{5} b^{\frac {7}{2}} + 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{4} b^{\frac {9}{2}} + 198 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{6} b^{\frac {7}{2}} + 54 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{5} b^{\frac {9}{2}} - 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{7} b^{\frac {7}{2}} + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{6} b^{\frac {9}{2}} + 9 \, B a^{8} b^{\frac {7}{2}} - 2 \, A a^{7} b^{\frac {9}{2}}\right )}}{63 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \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 {7}{2}} \left (-2 A b \,x^{2}+9 B a \,x^{2}+7 A a \right )}{63 a^{2} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 56, normalized size = 1.06 \begin {gather*} -\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{7 \, a x^{7}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{63 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{9 \, a x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.72, size = 170, normalized size = 3.21 \begin {gather*} \frac {2\,A\,b^4\,\sqrt {b\,x^2+a}}{63\,a^2\,x}-\frac {5\,A\,b^2\,\sqrt {b\,x^2+a}}{21\,x^5}-\frac {B\,a^2\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {3\,B\,b^2\,\sqrt {b\,x^2+a}}{7\,x^3}-\frac {A\,b^3\,\sqrt {b\,x^2+a}}{63\,a\,x^3}-\frac {A\,a^2\,\sqrt {b\,x^2+a}}{9\,x^9}-\frac {B\,b^3\,\sqrt {b\,x^2+a}}{7\,a\,x}-\frac {19\,A\,a\,b\,\sqrt {b\,x^2+a}}{63\,x^7}-\frac {3\,B\,a\,b\,\sqrt {b\,x^2+a}}{7\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.95, size = 1489, normalized size = 28.09
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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