Optimal. Leaf size=96 \[ -\frac {A \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}-\frac {(7 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{35 b^2 x^8}+\frac {2 c (7 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^6} \]
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Rubi [A]
time = 0.14, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2059, 806, 672,
664} \begin {gather*} \frac {2 c \left (b x^2+c x^4\right )^{3/2} (7 b B-4 A c)}{105 b^3 x^6}-\frac {\left (b x^2+c x^4\right )^{3/2} (7 b B-4 A c)}{35 b^2 x^8}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
Rule 806
Rule 2059
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^9} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) \sqrt {b x+c x^2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}+\frac {\left (-5 (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x^4} \, dx,x,x^2\right )}{7 b}\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}-\frac {(7 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{35 b^2 x^8}-\frac {(c (7 b B-4 A c)) \text {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x^3} \, dx,x,x^2\right )}{35 b^2}\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}-\frac {(7 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{35 b^2 x^8}+\frac {2 c (7 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 66, normalized size = 0.69 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (7 b B x^2 \left (-3 b+2 c x^2\right )+A \left (-15 b^2+12 b c x^2-8 c^2 x^4\right )\right )}{105 b^3 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 70, normalized size = 0.73
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (8 A \,c^{2} x^{4}-14 x^{4} b B c -12 A b c \,x^{2}+21 b^{2} B \,x^{2}+15 b^{2} A \right ) \sqrt {x^{4} c +b \,x^{2}}}{105 b^{3} x^{8}}\) | \(70\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (8 A \,c^{2} x^{4}-14 x^{4} b B c -12 A b c \,x^{2}+21 b^{2} B \,x^{2}+15 b^{2} A \right ) \sqrt {x^{4} c +b \,x^{2}}}{105 b^{3} x^{8}}\) | \(70\) |
trager | \(-\frac {\left (8 A \,c^{3} x^{6}-14 x^{6} B b \,c^{2}-4 A b \,c^{2} x^{4}+7 x^{4} B \,b^{2} c +3 A \,b^{2} c \,x^{2}+21 x^{2} B \,b^{3}+15 A \,b^{3}\right ) \sqrt {x^{4} c +b \,x^{2}}}{105 b^{3} x^{8}}\) | \(87\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (8 A \,c^{3} x^{6}-14 x^{6} B b \,c^{2}-4 A b \,c^{2} x^{4}+7 x^{4} B \,b^{2} c +3 A \,b^{2} c \,x^{2}+21 x^{2} B \,b^{3}+15 A \,b^{3}\right )}{105 x^{8} b^{3}}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 161, normalized size = 1.68 \begin {gather*} \frac {1}{15} \, B {\left (\frac {2 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{2} x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}} c}{b x^{4}} - \frac {3 \, \sqrt {c x^{4} + b x^{2}}}{x^{6}}\right )} - \frac {1}{105} \, A {\left (\frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{3} x^{2}} - \frac {4 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{2} x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} c}{b x^{6}} + \frac {15 \, \sqrt {c x^{4} + b x^{2}}}{x^{8}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 85, normalized size = 0.89 \begin {gather*} \frac {{\left (2 \, {\left (7 \, B b c^{2} - 4 \, A c^{3}\right )} x^{6} - {\left (7 \, B b^{2} c - 4 \, A b c^{2}\right )} x^{4} - 15 \, A b^{3} - 3 \, {\left (7 \, B b^{3} + A b^{2} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{105 \, b^{3} x^{8}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 310 vs.
\(2 (84) = 168\).
time = 1.41, size = 310, normalized size = 3.23 \begin {gather*} \frac {4 \, {\left (105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) - 175 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{2} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 42 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{3} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{2} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 49 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{4} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) - 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{3} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 7 \, B b^{5} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 4 \, A b^{4} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right )\right )}}{105 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.68, size = 160, normalized size = 1.67 \begin {gather*} \frac {4\,A\,c^2\,\sqrt {c\,x^4+b\,x^2}}{105\,b^2\,x^4}-\frac {B\,\sqrt {c\,x^4+b\,x^2}}{5\,x^6}-\frac {A\,c\,\sqrt {c\,x^4+b\,x^2}}{35\,b\,x^6}-\frac {B\,c\,\sqrt {c\,x^4+b\,x^2}}{15\,b\,x^4}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{7\,x^8}-\frac {8\,A\,c^3\,\sqrt {c\,x^4+b\,x^2}}{105\,b^3\,x^2}+\frac {2\,B\,c^2\,\sqrt {c\,x^4+b\,x^2}}{15\,b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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