Optimal. Leaf size=151 \[ -\frac {3 \left (b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 x^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}-\frac {3 \left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 \sqrt {a}}+\frac {3}{4} b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1128, 746, 826,
857, 635, 212, 738} \begin {gather*} -\frac {3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 \sqrt {a}}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}-\frac {3 \left (b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 x^2}+\frac {3}{4} b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 826
Rule 857
Rule 1128
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}+\frac {3}{8} \text {Subst}\left (\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 \left (b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 x^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}-\frac {3}{16} \text {Subst}\left (\int \frac {-b^2-4 a c-4 b c x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {3 \left (b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 x^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}+\frac {1}{4} (3 b c) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (3 \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 )\\ &=-\frac {3 \left (b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 x^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}+\frac {1}{2} (3 b c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )-\frac {1}{8} \left (3 \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 )\\ &=-\frac {3 \left (b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 x^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{4 x^4}-\frac {3 \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 )}{16 \sqrt {a}}+\frac {3}{4} b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 131, normalized size = 0.87 \begin {gather*} \frac {1}{8} \left (\frac {\sqrt {a+b x^2+c x^4} \left (-2 a-5 b x^2+4 c x^4\right )}{x^4}+\frac {3 \left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{\sqrt {a}}-6 b \sqrt {c} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 174, normalized size = 1.15
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (5 b \,x^{2}+2 a \right )}{8 x^{4}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{2}+\frac {3 b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4}-\frac {3 \sqrt {a}\, c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4}-\frac {3 b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 \sqrt {a}}\) | \(163\) |
default | \(\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{2}+\frac {3 b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 x^{4}}-\frac {5 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 x^{2}}-\frac {3 b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 \sqrt {a}}-\frac {3 \sqrt {a}\, c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4}\) | \(174\) |
elliptic | \(\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{2}+\frac {3 b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 x^{4}}-\frac {5 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 x^{2}}-\frac {3 b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 \sqrt {a}}-\frac {3 \sqrt {a}\, c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4}\) | \(174\) |
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 = 0.43, size = 713, normalized size = 4.72 \begin {gather*} \left [\frac {12 \, a b \sqrt {c} x^{4} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {a} x^{4} \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 (4 \, a c x^{4} - 5 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{32 \, a x^{4}}, -\frac {24 \, a b \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {a} x^{4} \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 (4 \, a c x^{4} - 5 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{32 \, a x^{4}}, \frac {6 \, a b \sqrt {c} x^{4} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-a} x^{4} \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 (4 \, a c x^{4} - 5 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{16 \, a x^{4}}, -\frac {12 \, a b \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-a} x^{4} \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 (4 \, a c x^{4} - 5 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{16 \, a x^{4}}\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 {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{5}}\, 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. 302 vs.
\(2 (125) = 250\).
time = 4.87, size = 302, normalized size = 2.00 \begin {gather*} -\frac {3}{4} \, b \sqrt {c} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right ) + \frac {1}{2} \, \sqrt {c x^{4} + b x^{2} + a} c + \frac {3 \, {\left (b^{2} + 4 \, a c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a}} + \frac {5 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} b^{2} + 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a c + 16 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a b \sqrt {c} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a b^{2} + 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} c - 8 \, a^{2} b \sqrt {c}}{8 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{2}} \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 {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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