Optimal. Leaf size=61 \[ -\frac {\left (b x^2+c x^4\right )^{5/2} (7 b B-2 A c)}{35 b^2 x^{10}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}} \]
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Rubi [A] time = 0.17, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2034, 792, 650} \[ -\frac {\left (b x^2+c x^4\right )^{5/2} (7 b B-2 A c)}{35 b^2 x^{10}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}} \]
Antiderivative was successfully verified.
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Rule 650
Rule 792
Rule 2034
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}+\frac {\left (-6 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{7 b}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}-\frac {(7 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{35 b^2 x^{10}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 0.72 \[ -\frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (5 A b-2 A c x^2+7 b B x^2\right )}{35 b^2 x^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 82, normalized size = 1.34 \[ -\frac {{\left ({\left (7 \, B b c^{2} - 2 \, A c^{3}\right )} x^{6} + {\left (14 \, B b^{2} c + A b c^{2}\right )} x^{4} + 5 \, A b^{3} + {\left (7 \, B b^{3} + 8 \, A b^{2} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{35 \, b^{2} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.89, size = 370, normalized size = 6.07 \[ \frac {2 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} B c^{\frac {5}{2}} \mathrm {sgn}\relax (x) - 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B b c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} A c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b^{2} c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A b c^{\frac {7}{2}} \mathrm {sgn}\relax (x) - 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{3} c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b^{2} c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 77 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{4} c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{3} c^{\frac {7}{2}} \mathrm {sgn}\relax (x) - 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{5} c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{4} c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 7 \, B b^{6} c^{\frac {5}{2}} \mathrm {sgn}\relax (x) - 2 \, A b^{5} c^{\frac {7}{2}} \mathrm {sgn}\relax (x)\right )}}{35 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 0.79 \[ -\frac {\left (c \,x^{2}+b \right ) \left (-2 A c \,x^{2}+7 B b \,x^{2}+5 A b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{35 b^{2} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.56, size = 193, normalized size = 3.16 \[ -\frac {1}{10} \, B {\left (\frac {2 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}} c}{x^{4}} - \frac {3 \, \sqrt {c x^{4} + b x^{2}} b}{x^{6}} + \frac {5 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{8}}\right )} + \frac {1}{140} \, A {\left (\frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{2} x^{2}} - \frac {4 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} c}{x^{6}} + \frac {15 \, \sqrt {c x^{4} + b x^{2}} b}{x^{8}} - \frac {35 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{10}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 156, normalized size = 2.56 \[ \frac {2\,A\,c^3\,\sqrt {c\,x^4+b\,x^2}}{35\,b^2\,x^2}-\frac {8\,A\,c\,\sqrt {c\,x^4+b\,x^2}}{35\,x^6}-\frac {B\,b\,\sqrt {c\,x^4+b\,x^2}}{5\,x^6}-\frac {2\,B\,c\,\sqrt {c\,x^4+b\,x^2}}{5\,x^4}-\frac {A\,c^2\,\sqrt {c\,x^4+b\,x^2}}{35\,b\,x^4}-\frac {A\,b\,\sqrt {c\,x^4+b\,x^2}}{7\,x^8}-\frac {B\,c^2\,\sqrt {c\,x^4+b\,x^2}}{5\,b\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{11}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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