Optimal. Leaf size=177 \[ -\frac {A \sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {(5 A b-6 a B) \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}+\frac {\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{7/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1265, 848, 820,
738, 212} \begin {gather*} -\frac {\sqrt {a+b x^2+c x^4} \left (-16 a A c-18 a b B+15 A b^2\right )}{48 a^3 x^2}+\frac {(5 A b-6 a B) \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{7/2}}-\frac {A \sqrt {a+b x^2+c x^4}}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 820
Rule 848
Rule 1265
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^7 \sqrt {a+b x^2+c x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^4 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{6 a x^6}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (5 A b-6 a B)+2 A c x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{6 a}\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {(5 A b-6 a B) \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} \left (15 A b^2-18 a b B-16 a A c\right )+\frac {1}{2} (5 A b-6 a B) c x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{12 a^2}\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {(5 A b-6 a B) \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}-\frac {\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{32 a^3}\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {(5 A b-6 a B) \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}+\frac {\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\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 )}{16 a^3}\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{6 a x^6}+\frac {(5 A b-6 a B) \sqrt {a+b x^2+c x^4}}{24 a^2 x^4}-\frac {\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt {a+b x^2+c x^4}}{48 a^3 x^2}+\frac {\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 186, normalized size = 1.05 \begin {gather*} \frac {\sqrt {a+b x^2+c x^4} \left (-8 a^2 A+10 a A b x^2-12 a^2 B x^2-15 A b^2 x^4+18 a b B x^4+16 a A c x^4\right )}{48 a^3 x^6}+\frac {\left (-5 A b^3-8 a^2 B c\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{16 a^{7/2}}-\frac {3 b (b B+2 A c) \tanh ^{-1}\left (\frac {-\sqrt {c} x^2+\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{8 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 307, normalized size = 1.73
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-16 A a c \,x^{4}+15 A \,b^{2} x^{4}-18 B a b \,x^{4}-10 a A b \,x^{2}+12 a^{2} B \,x^{2}+8 a^{2} A \right )}{48 a^{3} x^{6}}-\frac {3 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) A b c}{8 a^{\frac {5}{2}}}+\frac {5 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) A \,b^{3}}{32 a^{\frac {7}{2}}}+\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) c B}{4 a^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) B \,b^{2}}{16 a^{\frac {5}{2}}}\) | \(238\) |
default | \(B \left (-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a \,x^{4}}+\frac {3 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2} x^{2}}-\frac {3 b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {5}{2}}}+\frac {c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\right )+A \left (-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{6 a \,x^{6}}+\frac {5 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{24 a^{2} x^{4}}-\frac {5 b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{3} x^{2}}+\frac {5 b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {7}{2}}}-\frac {3 b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 a^{\frac {5}{2}}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 a^{2} x^{2}}\right )\) | \(307\) |
elliptic | \(-\frac {B \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a \,x^{4}}+\frac {3 B b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2} x^{2}}-\frac {3 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) B \,b^{2}}{16 a^{\frac {5}{2}}}+\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) c B}{4 a^{\frac {3}{2}}}-\frac {A \sqrt {c \,x^{4}+b \,x^{2}+a}}{6 a \,x^{6}}+\frac {5 A b \sqrt {c \,x^{4}+b \,x^{2}+a}}{24 a^{2} x^{4}}-\frac {5 A \,b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{3} x^{2}}+\frac {5 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) A \,b^{3}}{32 a^{\frac {7}{2}}}-\frac {3 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) A b c}{8 a^{\frac {5}{2}}}+\frac {A c \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 a^{2} x^{2}}\) | \(311\) |
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 = 0.57, size = 339, normalized size = 1.92 \begin {gather*} \left [\frac {3 \, {\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} \sqrt {a} x^{6} \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 (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{4} - 8 \, A a^{3} - 2 \, {\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, a^{4} x^{6}}, \frac {3 \, {\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} \sqrt {-a} x^{6} \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 (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{4} - 8 \, A a^{3} - 2 \, {\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, a^{4} x^{6}}\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 {A + B x^{2}}{x^{7} \sqrt {a + b x^{2} + c x^{4}}}\, 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. 571 vs.
\(2 (155) = 310\).
time = 3.19, size = 571, normalized size = 3.23 \begin {gather*} \frac {{\left (6 \, B a b^{2} - 5 \, A b^{3} - 8 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a^{3}} - \frac {18 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} B a b^{2} - 15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} A b^{3} - 24 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} B a^{2} c + 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} A a b c - 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} B a^{2} b^{2} + 40 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} A a b^{3} - 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} A a^{2} b c - 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} B a^{3} b \sqrt {c} - 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} + 30 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} B a^{3} b^{2} - 33 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} A a^{2} b^{3} + 24 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} B a^{4} c - 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} A a^{3} b c + 48 \, B a^{4} b \sqrt {c} - 48 \, A a^{3} b^{2} \sqrt {c} + 32 \, A a^{4} c^{\frac {3}{2}}}{48 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{3} 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 {B\,x^2+A}{x^7\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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