Optimal. Leaf size=161 \[ -\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{128 a^3 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}+\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{256 a^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1128, 758, 820,
734, 738, 212} \begin {gather*} \frac {\left (b^2-4 a c\right ) \left (5 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 )}{256 a^{7/2}}-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{128 a^3 x^4}+\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 734
Rule 738
Rule 758
Rule 820
Rule 1128
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2+c x^4}}{x^9} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}-\frac {\text {Subst}\left (\int \frac {\left (\frac {5 b}{2}+c x\right ) \sqrt {a+b x+c x^2}}{x^4} \, dx,x,x^2\right )}{8 a}\\ &=-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}+\frac {\left (5 b^2-4 a c\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{32 a^2}\\ &=-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{128 a^3 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac {\left (\left (b^2-4 a c\right ) \left (5 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 )}{256 a^3}\\ &=-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{128 a^3 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}+\frac {\left (\left (b^2-4 a c\right ) \left (5 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 )}{128 a^3}\\ &=-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{128 a^3 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac {5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}+\frac {\left (b^2-4 a c\right ) \left (5 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 )}{256 a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 141, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+b x^2+c x^4} \left (-48 a^3-8 a^2 b x^2+10 a b^2 x^4-24 a^2 c x^4-15 b^3 x^6+52 a b c x^6\right )}{384 a^3 x^8}+\frac {\left (-5 b^4+24 a b^2 c-16 a^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{128 a^{7/2}} \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. \(386\) vs.
\(2(139)=278\).
time = 0.06, size = 387, normalized size = 2.40
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-52 a b c \,x^{6}+15 b^{3} x^{6}+24 a^{2} c \,x^{4}-10 a \,b^{2} x^{4}+8 a^{2} b \,x^{2}+48 a^{3}\right )}{384 x^{8} a^{3}}+\frac {c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {3 b^{2} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {5 b^{4} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{256 a^{\frac {7}{2}}}\) | \(197\) |
default | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {5 b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{48 a^{2} x^{6}}-\frac {5 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{64 a^{3} x^{4}}+\frac {5 b^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{128 a^{4} x^{2}}-\frac {5 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{4}}+\frac {5 b^{4} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{256 a^{\frac {7}{2}}}-\frac {5 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{128 a^{4}}+\frac {7 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 a^{3}}-\frac {3 b^{2} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{16 a^{2} x^{4}}-\frac {c b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 a^{3} x^{2}}+\frac {c^{2} b \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{32 a^{3}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{2}}+\frac {c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}\) | \(387\) |
elliptic | \(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {5 b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{48 a^{2} x^{6}}-\frac {5 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{64 a^{3} x^{4}}+\frac {5 b^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{128 a^{4} x^{2}}-\frac {5 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{128 a^{4}}+\frac {5 b^{4} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{256 a^{\frac {7}{2}}}-\frac {5 b^{3} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{128 a^{4}}+\frac {7 b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 a^{3}}-\frac {3 b^{2} c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {c \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{16 a^{2} x^{4}}-\frac {c b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 a^{3} x^{2}}+\frac {c^{2} b \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{32 a^{3}}-\frac {c^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{2}}+\frac {c^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}\) | \(387\) |
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 = 0.42, size = 325, normalized size = 2.02 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {a} x^{8} \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 (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{6} + 8 \, a^{3} b x^{2} - 2 \, {\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{4} + 48 \, a^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{1536 \, a^{4} x^{8}}, -\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-a} x^{8} \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 (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{6} + 8 \, a^{3} b x^{2} - 2 \, {\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{4} + 48 \, a^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{768 \, a^{4} x^{8}}\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^{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. 617 vs.
\(2 (139) = 278\).
time = 3.62, size = 617, normalized size = 3.83 \begin {gather*} -\frac {{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{128 \, \sqrt {-a} a^{3}} + \frac {15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} b^{4} - 72 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a b^{2} c + 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{2} c^{2} - 55 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a b^{4} + 264 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{2} b^{2} c + 336 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{3} c^{2} + 1152 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{3} b c^{\frac {3}{2}} + 73 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{2} b^{4} + 648 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{3} b^{2} c + 336 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{4} c^{2} + 384 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{3} b^{3} \sqrt {c} + 256 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{4} b c^{\frac {3}{2}} + 15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{3} b^{4} + 312 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{4} b^{2} c + 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{5} c^{2} + 128 \, a^{5} b c^{\frac {3}{2}}}{384 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{4} a^{3}} \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^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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