Optimal. Leaf size=153 \[ \frac {\sqrt {a+b x^2+c x^4} \left (-16 a B c-2 c x^2 (5 b B-6 A c)-18 A b c+15 b^2 B\right )}{48 c^3}-\frac {\left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{7/2}}+\frac {B x^4 \sqrt {a+b x^2+c x^4}}{6 c} \]
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Rubi [A] time = 0.20, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1251, 832, 779, 621, 206} \[ \frac {\sqrt {a+b x^2+c x^4} \left (-16 a B c-2 c x^2 (5 b B-6 A c)-18 A b c+15 b^2 B\right )}{48 c^3}-\frac {\left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{7/2}}+\frac {B x^4 \sqrt {a+b x^2+c x^4}}{6 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rule 1251
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {B x^4 \sqrt {a+b x^2+c x^4}}{6 c}+\frac {\operatorname {Subst}\left (\int \frac {x \left (-2 a B-\frac {1}{2} (5 b B-6 A c) x\right )}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{6 c}\\ &=\frac {B x^4 \sqrt {a+b x^2+c x^4}}{6 c}+\frac {\left (15 b^2 B-18 A b c-16 a B c-2 c (5 b B-6 A c) x^2\right ) \sqrt {a+b x^2+c x^4}}{48 c^3}-\frac {\left (5 b^3 B-6 A b^2 c-12 a b B c+8 a A c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{32 c^3}\\ &=\frac {B x^4 \sqrt {a+b x^2+c x^4}}{6 c}+\frac {\left (15 b^2 B-18 A b c-16 a B c-2 c (5 b B-6 A c) x^2\right ) \sqrt {a+b x^2+c x^4}}{48 c^3}-\frac {\left (5 b^3 B-6 A b^2 c-12 a b B c+8 a A c^2\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 )}{16 c^3}\\ &=\frac {B x^4 \sqrt {a+b x^2+c x^4}}{6 c}+\frac {\left (15 b^2 B-18 A b c-16 a B c-2 c (5 b B-6 A c) x^2\right ) \sqrt {a+b x^2+c x^4}}{48 c^3}-\frac {\left (5 b^3 B-6 A b^2 c-12 a b B c+8 a A c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 139, normalized size = 0.91 \[ \frac {2 \sqrt {c} \sqrt {a+b x^2+c x^4} \left (4 c \left (-4 a B+3 A c x^2+2 B c x^4\right )-2 b c \left (9 A+5 B x^2\right )+15 b^2 B\right )-3 \left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{96 c^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 315, normalized size = 2.06 \[ \left [\frac {3 \, {\left (5 \, B b^{3} + 8 \, A a c^{2} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (8 \, B c^{3} x^{4} + 15 \, B b^{2} c - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{2} - 2 \, {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, c^{4}}, \frac {3 \, {\left (5 \, B b^{3} + 8 \, A a c^{2} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (8 \, B c^{3} x^{4} + 15 \, B b^{2} c - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{2} - 2 \, {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 138, normalized size = 0.90 \[ \frac {1}{48} \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (\frac {4 \, B x^{2}}{c} - \frac {5 \, B b c - 6 \, A c^{2}}{c^{3}}\right )} x^{2} + \frac {15 \, B b^{2} - 16 \, B a c - 18 \, A b c}{c^{3}}\right )} + \frac {{\left (5 \, B b^{3} - 12 \, B a b c - 6 \, A b^{2} c + 8 \, A a c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{32 \, c^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 286, normalized size = 1.87 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, B \,x^{4}}{6 c}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, A \,x^{2}}{4 c}-\frac {5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, B b \,x^{2}}{24 c^{2}}-\frac {A a \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}+\frac {3 A \,b^{2} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}+\frac {3 B a b \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {5}{2}}}-\frac {5 B \,b^{3} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {7}{2}}}-\frac {3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, A b}{8 c^{2}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, B a}{3 c^{2}}+\frac {5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, B \,b^{2}}{16 c^{3}} \]
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 \[ \int \frac {x^5\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
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 \[ \int \frac {x^{5} \left (A + B x^{2}\right )}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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