Optimal. Leaf size=61 \[ -\frac {A \left (b x^2+c x^4\right )^{3/2}}{5 b x^8}-\frac {(5 b B-2 A c) \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6} \]
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Rubi [A]
time = 0.10, 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 = {2059, 806, 664}
\begin {gather*} -\frac {\left (b x^2+c x^4\right )^{3/2} (5 b B-2 A c)}{15 b^2 x^6}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{5 b x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 806
Rule 2059
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^7} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) \sqrt {b x+c x^2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{5 b x^8}+\frac {\left (-4 (-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^3} \, dx,x,x^2\right )}{5 b}\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{5 b x^8}-\frac {(5 b B-2 A c) \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 44, normalized size = 0.72 \begin {gather*} -\frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (3 A b+5 b B x^2-2 A c x^2\right )}{15 b^2 x^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 48, normalized size = 0.79
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (-2 A c \,x^{2}+5 b B \,x^{2}+3 A b \right ) \sqrt {x^{4} c +b \,x^{2}}}{15 b^{2} x^{6}}\) | \(48\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (-2 A c \,x^{2}+5 b B \,x^{2}+3 A b \right ) \sqrt {x^{4} c +b \,x^{2}}}{15 b^{2} x^{6}}\) | \(48\) |
trager | \(-\frac {\left (-2 A \,c^{2} x^{4}+5 x^{4} b B c +A b c \,x^{2}+5 b^{2} B \,x^{2}+3 b^{2} A \right ) \sqrt {x^{4} c +b \,x^{2}}}{15 b^{2} x^{6}}\) | \(62\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (-2 A \,c^{2} x^{4}+5 x^{4} b B c +A b c \,x^{2}+5 b^{2} B \,x^{2}+3 b^{2} A \right )}{15 x^{6} b^{2}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs.
\(2 (53) = 106\).
time = 0.30, size = 111, normalized size = 1.82 \begin {gather*} -\frac {1}{3} \, B {\left (\frac {\sqrt {c x^{4} + b x^{2}} c}{b x^{2}} + \frac {\sqrt {c x^{4} + b x^{2}}}{x^{4}}\right )} + \frac {1}{15} \, A {\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.78, size = 59, normalized size = 0.97 \begin {gather*} -\frac {{\left ({\left (5 \, B b c - 2 \, A c^{2}\right )} x^{4} + 3 \, A b^{2} + {\left (5 \, B b^{2} + A b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15 \, b^{2} x^{6}} \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^{7}}\, 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. 250 vs.
\(2 (53) = 106\).
time = 1.08, size = 250, normalized size = 4.10 \begin {gather*} \frac {2 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) - 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{2} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) - 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{3} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{2} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 5 \, B b^{4} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) - 2 \, A b^{3} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )\right )}}{15 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 113, normalized size = 1.85 \begin {gather*} \frac {\left (A\,c^2+B\,b\,c\right )\,\sqrt {c\,x^4+b\,x^2}}{5\,b^2\,x^2}-\frac {\left (5\,B\,b^2+A\,c\,b\right )\,\sqrt {c\,x^4+b\,x^2}}{15\,b^2\,x^4}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{5\,x^6}-\frac {c\,\left (A\,c+8\,B\,b\right )\,\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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