Optimal. Leaf size=199 \[ \frac {b \left (7 b^2-12 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^4 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{80 a^2 x^8}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 a^3 x^6}-\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{9/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1128, 758, 848,
820, 734, 738, 212} \begin {gather*} -\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{9/2}}+\frac {b \left (7 b^2-12 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^4 x^4}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 a^3 x^6}+\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{80 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 734
Rule 738
Rule 758
Rule 820
Rule 848
Rule 1128
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2+c x^4}}{x^{11}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}}-\frac {\text {Subst}\left (\int \frac {\left (\frac {7 b}{2}+2 c x\right ) \sqrt {a+b x+c x^2}}{x^5} \, dx,x,x^2\right )}{10 a}\\ &=-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{80 a^2 x^8}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{4} \left (35 b^2-32 a c\right )+\frac {7 b c x}{2}\right ) \sqrt {a+b x+c x^2}}{x^4} \, dx,x,x^2\right )}{40 a^2}\\ &=-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{80 a^2 x^8}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 a^3 x^6}-\frac {\left (b \left (7 b^2-12 a c\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{64 a^3}\\ &=\frac {b \left (7 b^2-12 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^4 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{80 a^2 x^8}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 a^3 x^6}+\frac {\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{512 a^4}\\ &=\frac {b \left (7 b^2-12 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^4 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{80 a^2 x^8}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 a^3 x^6}-\frac {\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}}\right )}{256 a^4}\\ &=\frac {b \left (7 b^2-12 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^4 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{10 a x^{10}}+\frac {7 b \left (a+b x^2+c x^4\right )^{3/2}}{80 a^2 x^8}-\frac {\left (35 b^2-32 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 a^3 x^6}-\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 176, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+b x^2+c x^4} \left (-384 a^4-48 a^3 b x^2+56 a^2 b^2 x^4-128 a^3 c x^4-70 a b^3 x^6+232 a^2 b c x^6+105 b^4 x^8-460 a b^2 c x^8+256 a^2 c^2 x^8\right )}{3840 a^4 x^{10}}+\frac {\left (7 b^5-40 a b^3 c+48 a^2 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{256 a^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs.
\(2(173)=346\).
time = 0.08, size = 442, normalized size = 2.22
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-256 a^{2} c^{2} x^{8}+460 a \,b^{2} c \,x^{8}-105 b^{4} x^{8}-232 a^{2} b c \,x^{6}+70 a \,b^{3} x^{6}+128 a^{3} c \,x^{4}-56 a^{2} b^{2} x^{4}+48 a^{3} b \,x^{2}+384 a^{4}\right )}{3840 x^{10} a^{4}}-\frac {3 b \,c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {5 b^{3} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{64 a^{\frac {7}{2}}}-\frac {7 b^{5} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {9}{2}}}\) | \(232\) |
default | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{10 a \,x^{10}}+\frac {7 b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{80 a^{2} x^{8}}-\frac {7 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{96 a^{3} x^{6}}+\frac {7 b^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{128 a^{4} x^{4}}-\frac {7 b^{4} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{256 a^{5} x^{2}}+\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 a^{5}}-\frac {7 b^{5} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {9}{2}}}+\frac {7 b^{4} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{256 a^{5}}-\frac {13 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{4}}+\frac {5 b^{3} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{64 a^{\frac {7}{2}}}-\frac {3 b c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 a^{3} x^{4}}+\frac {3 b^{2} c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{64 a^{4} x^{2}}-\frac {3 b^{2} c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{64 a^{4}}+\frac {3 b \,c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 a^{3}}-\frac {3 b \,c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{6}}\) | \(442\) |
elliptic | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{10 a \,x^{10}}+\frac {7 b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{80 a^{2} x^{8}}-\frac {7 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{96 a^{3} x^{6}}+\frac {7 b^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{128 a^{4} x^{4}}-\frac {7 b^{4} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{256 a^{5} x^{2}}+\frac {7 b^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 a^{5}}-\frac {7 b^{5} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{512 a^{\frac {9}{2}}}+\frac {7 b^{4} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{256 a^{5}}-\frac {13 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{4}}+\frac {5 b^{3} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{64 a^{\frac {7}{2}}}-\frac {3 b c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 a^{3} x^{4}}+\frac {3 b^{2} c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{64 a^{4} x^{2}}-\frac {3 b^{2} c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{64 a^{4}}+\frac {3 b \,c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{32 a^{3}}-\frac {3 b \,c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{6}}\) | \(442\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 389, normalized size = 1.95 \begin {gather*} \left [\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {a} x^{10} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, {\left ({\left (105 \, a b^{4} - 460 \, a^{2} b^{2} c + 256 \, a^{3} c^{2}\right )} x^{8} - 48 \, a^{4} b x^{2} - 2 \, {\left (35 \, a^{2} b^{3} - 116 \, a^{3} b c\right )} x^{6} - 384 \, a^{5} + 8 \, {\left (7 \, a^{3} b^{2} - 16 \, a^{4} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{15360 \, a^{5} x^{10}}, \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {-a} x^{10} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (105 \, a b^{4} - 460 \, a^{2} b^{2} c + 256 \, a^{3} c^{2}\right )} x^{8} - 48 \, a^{4} b x^{2} - 2 \, {\left (35 \, a^{2} b^{3} - 116 \, a^{3} b c\right )} x^{6} - 384 \, a^{5} + 8 \, {\left (7 \, a^{3} b^{2} - 16 \, a^{4} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{7680 \, a^{5} x^{10}}\right ] \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 {a + b x^{2} + c x^{4}}}{x^{11}}\, 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. 842 vs.
\(2 (173) = 346\).
time = 5.50, size = 842, normalized size = 4.23 \begin {gather*} \frac {{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{256 \, \sqrt {-a} a^{4}} - \frac {105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} b^{5} - 600 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a b^{3} c + 720 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a^{2} b c^{2} - 490 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a b^{5} + 2800 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{2} b^{3} c - 3360 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{3} b c^{2} - 7680 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{6} a^{4} c^{\frac {5}{2}} + 896 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{2} b^{5} - 5120 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{3} b^{3} c - 15360 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{4} b c^{2} - 24320 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{4} b^{2} c^{\frac {3}{2}} - 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{5} c^{\frac {5}{2}} - 790 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{3} b^{5} - 9200 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{4} b^{3} c - 12000 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{5} b c^{2} - 3840 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{4} b^{4} \sqrt {c} - 5120 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{5} b^{2} c^{\frac {3}{2}} - 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{6} c^{\frac {5}{2}} - 105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{4} b^{5} - 3240 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{5} b^{3} c - 720 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{6} b c^{2} - 1280 \, a^{6} b^{2} c^{\frac {3}{2}} + 512 \, a^{7} c^{\frac {5}{2}}}{3840 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^{11}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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