Optimal. Leaf size=170 \[ -\frac {A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac {(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac {2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac {8 c^2 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}+\frac {16 c^3 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6} \]
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Rubi [A]
time = 0.20, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {16 c^3 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^6}-\frac {8 c^2 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^8}+\frac {2 c \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^{10}}-\frac {\left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^{12}}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \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^{13}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) \sqrt {b x+c x^2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac {\left (-7 (-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^6} \, dx,x,x^2\right )}{11 b}\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac {(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac {(c (11 b B-8 A c)) \text {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x^5} \, dx,x,x^2\right )}{33 b^2}\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac {(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac {2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac {\left (4 c^2 (11 b B-8 A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x^4} \, dx,x,x^2\right )}{231 b^3}\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac {(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac {2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac {8 c^2 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac {\left (8 c^3 (11 b B-8 A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x^3} \, dx,x,x^2\right )}{1155 b^4}\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac {(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac {2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac {8 c^2 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}+\frac {16 c^3 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 110, normalized size = 0.65 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (11 b B x^2 \left (-35 b^3+30 b^2 c x^2-24 b c^2 x^4+16 c^3 x^6\right )+A \left (-315 b^4+280 b^3 c x^2-240 b^2 c^2 x^4+192 b c^3 x^6-128 c^4 x^8\right )\right )}{3465 b^5 x^{14}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 118, normalized size = 0.69
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (128 A \,c^{4} x^{8}-176 B b \,c^{3} x^{8}-192 A b \,c^{3} x^{6}+264 B \,b^{2} c^{2} x^{6}+240 A \,b^{2} c^{2} x^{4}-330 B \,b^{3} c \,x^{4}-280 A \,b^{3} c \,x^{2}+385 B \,b^{4} x^{2}+315 A \,b^{4}\right ) \sqrt {x^{4} c +b \,x^{2}}}{3465 b^{5} x^{12}}\) | \(118\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (128 A \,c^{4} x^{8}-176 B b \,c^{3} x^{8}-192 A b \,c^{3} x^{6}+264 B \,b^{2} c^{2} x^{6}+240 A \,b^{2} c^{2} x^{4}-330 B \,b^{3} c \,x^{4}-280 A \,b^{3} c \,x^{2}+385 B \,b^{4} x^{2}+315 A \,b^{4}\right ) \sqrt {x^{4} c +b \,x^{2}}}{3465 b^{5} x^{12}}\) | \(118\) |
trager | \(-\frac {\left (128 A \,c^{5} x^{10}-176 B b \,c^{4} x^{10}-64 A b \,c^{4} x^{8}+88 B \,b^{2} c^{3} x^{8}+48 A \,b^{2} c^{3} x^{6}-66 B \,b^{3} c^{2} x^{6}-40 A \,b^{3} c^{2} x^{4}+55 B \,b^{4} c \,x^{4}+35 A \,b^{4} c \,x^{2}+385 b^{5} B \,x^{2}+315 b^{5} A \right ) \sqrt {x^{4} c +b \,x^{2}}}{3465 b^{5} x^{12}}\) | \(135\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (128 A \,c^{5} x^{10}-176 B b \,c^{4} x^{10}-64 A b \,c^{4} x^{8}+88 B \,b^{2} c^{3} x^{8}+48 A \,b^{2} c^{3} x^{6}-66 B \,b^{3} c^{2} x^{6}-40 A \,b^{3} c^{2} x^{4}+55 B \,b^{4} c \,x^{4}+35 A \,b^{4} c \,x^{2}+385 b^{5} B \,x^{2}+315 b^{5} A \right )}{3465 x^{12} b^{5}}\) | \(135\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 257, normalized size = 1.51 \begin {gather*} \frac {1}{315} \, B {\left (\frac {16 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{4} x^{2}} - \frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{3} x^{4}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{2} x^{6}} - \frac {5 \, \sqrt {c x^{4} + b x^{2}} c}{b x^{8}} - \frac {35 \, \sqrt {c x^{4} + b x^{2}}}{x^{10}}\right )} - \frac {1}{3465} \, A {\left (\frac {128 \, \sqrt {c x^{4} + b x^{2}} c^{5}}{b^{5} x^{2}} - \frac {64 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{4} x^{4}} + \frac {48 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{3} x^{6}} - \frac {40 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{2} x^{8}} + \frac {35 \, \sqrt {c x^{4} + b x^{2}} c}{b x^{10}} + \frac {315 \, \sqrt {c x^{4} + b x^{2}}}{x^{12}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.74, size = 133, normalized size = 0.78 \begin {gather*} \frac {{\left (16 \, {\left (11 \, B b c^{4} - 8 \, A c^{5}\right )} x^{10} - 8 \, {\left (11 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} x^{8} + 6 \, {\left (11 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} x^{6} - 315 \, A b^{5} - 5 \, {\left (11 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} x^{4} - 35 \, {\left (11 \, B b^{5} + A b^{4} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{3465 \, b^{5} x^{12}} \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^{13}}\, 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. 430 vs.
\(2 (150) = 300\).
time = 2.20, size = 430, normalized size = 2.53 \begin {gather*} \frac {32 \, {\left (3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{14} B c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 4851 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} B b c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} A c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B b^{2} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 7392 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} A b c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 165 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b^{3} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 2640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A b^{2} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 1815 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{4} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 1320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b^{3} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 605 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{5} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{4} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 121 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{6} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{5} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 11 \, B b^{7} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 8 \, A b^{6} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right )\right )}}{3465 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.41, size = 260, normalized size = 1.53 \begin {gather*} \frac {8\,A\,c^2\,\sqrt {c\,x^4+b\,x^2}}{693\,b^2\,x^8}-\frac {B\,\sqrt {c\,x^4+b\,x^2}}{9\,x^{10}}-\frac {A\,c\,\sqrt {c\,x^4+b\,x^2}}{99\,b\,x^{10}}-\frac {B\,c\,\sqrt {c\,x^4+b\,x^2}}{63\,b\,x^8}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{11\,x^{12}}-\frac {16\,A\,c^3\,\sqrt {c\,x^4+b\,x^2}}{1155\,b^3\,x^6}+\frac {64\,A\,c^4\,\sqrt {c\,x^4+b\,x^2}}{3465\,b^4\,x^4}-\frac {128\,A\,c^5\,\sqrt {c\,x^4+b\,x^2}}{3465\,b^5\,x^2}+\frac {2\,B\,c^2\,\sqrt {c\,x^4+b\,x^2}}{105\,b^2\,x^6}-\frac {8\,B\,c^3\,\sqrt {c\,x^4+b\,x^2}}{315\,b^3\,x^4}+\frac {16\,B\,c^4\,\sqrt {c\,x^4+b\,x^2}}{315\,b^4\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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