Optimal. Leaf size=100 \[ -\frac {\left (3 b B-4 A c-2 B c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {\left (3 b^2 B-4 A b c-4 a B c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1265, 793, 635,
212} \begin {gather*} \frac {\left (-4 a B c-4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{5/2}}-\frac {\sqrt {a+b x^2+c x^4} \left (-4 A c+3 b B-2 B c x^2\right )}{8 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 793
Rule 1265
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x (A+B x)}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (3 b B-4 A c-2 B c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {\left (3 b^2 B-4 A b c-4 a B c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 c^2}\\ &=-\frac {\left (3 b B-4 A c-2 B c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {\left (3 b^2 B-4 A b c-4 a B c\right ) \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 )}{8 c^2}\\ &=-\frac {\left (3 b B-4 A c-2 B c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {\left (3 b^2 B-4 A b c-4 a B c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 104, normalized size = 1.04 \begin {gather*} \frac {\left (-3 b B+4 A c+2 B c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {\left (-3 b^2 B+4 A b c+4 a B c\right ) \log \left (b c^2+2 c^3 x^2-2 c^{5/2} \sqrt {a+b x^2+c x^4}\right )}{16 c^{5/2}} \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. \(175\) vs.
\(2(86)=172\).
time = 0.07, size = 176, normalized size = 1.76
method | result | size |
risch | \(\frac {\left (2 B c \,x^{2}+4 A c -3 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c^{2}}-\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) b A}{4 c^{\frac {3}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) a B}{4 c^{\frac {3}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) b^{2} B}{16 c^{\frac {5}{2}}}\) | \(147\) |
default | \(B \left (\frac {x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}-\frac {3 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c^{2}}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}-\frac {a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}\right )+A \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}-\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}\right )\) | \(176\) |
elliptic | \(\frac {B \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}-\frac {3 B b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c^{2}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) b^{2} B}{16 c^{\frac {5}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) a B}{4 c^{\frac {3}{2}}}+\frac {A \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}-\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) b A}{4 c^{\frac {3}{2}}}\) | \(176\) |
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 = 0.40, size = 233, normalized size = 2.33 \begin {gather*} \left [-\frac {{\left (3 \, B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} \sqrt {c} \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 ) - 4 \, {\left (2 \, B c^{2} x^{2} - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{32 \, c^{3}}, -\frac {{\left (3 \, B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} \sqrt {-c} \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 ) - 2 \, {\left (2 \, B c^{2} x^{2} - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{16 \, c^{3}}\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 {x^{3} \left (A + B x^{2}\right )}{\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 [A]
time = 5.27, size = 98, normalized size = 0.98 \begin {gather*} \frac {1}{8} \, \sqrt {c x^{4} + b x^{2} + a} {\left (\frac {2 \, B x^{2}}{c} - \frac {3 \, B b - 4 \, A c}{c^{2}}\right )} - \frac {{\left (3 \, B b^{2} - 4 \, B a c - 4 \, A b c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {5}{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 {x^3\,\left (B\,x^2+A\right )}{\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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