Optimal. Leaf size=100 \[ \frac {\left (-4 a B c-4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{5/2}}-\frac {\sqrt {a+b x^2+c x^4} \left (-4 A c+3 b B-2 B c x^2\right )}{8 c^2} \]
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Rubi [A] time = 0.09, 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 = {1251, 779, 621, 206} \[ \frac {\left (-4 a B c-4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{5/2}}-\frac {\sqrt {a+b x^2+c x^4} \left (-4 A c+3 b B-2 B c x^2\right )}{8 c^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 1251
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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.05, size = 101, normalized size = 1.01 \[ \frac {\left (-4 a B c-4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )+2 \sqrt {c} \sqrt {a+b x^2+c x^4} \left (4 A c-3 b B+2 B c x^2\right )}{16 c^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 233, normalized size = 2.33 \[ \left [-\frac {{\left (3 \, B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (2 \, B c^{2} x^{2} - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{32 \, c^{3}}, -\frac {{\left (3 \, B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 2 \, {\left (2 \, B c^{2} x^{2} - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{16 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 98, normalized size = 0.98 \[ \frac {1}{8} \, \sqrt {c x^{4} + b x^{2} + a} {\left (\frac {2 \, B x^{2}}{c} - \frac {3 \, B b - 4 \, A c}{c^{2}}\right )} - \frac {{\left (3 \, B b^{2} - 4 \, B a c - 4 \, A b c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 176, normalized size = 1.76 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, B \,x^{2}}{4 c}-\frac {A b \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}-\frac {B a \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}+\frac {3 B \,b^{2} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, A}{2 c}-\frac {3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, B b}{8 c^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^3\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
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 \[ \int \frac {x^{3} \left (A + B x^{2}\right )}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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