Optimal. Leaf size=148 \[ \frac {2 a (d x)^{3/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {3}{4};-\frac {3}{2},-\frac {3}{2};\frac {7}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 d \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.13, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1141, 510} \[ \frac {2 a (d x)^{3/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {3}{4};-\frac {3}{2},-\frac {3}{2};\frac {7}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 d \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1141
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a+b x^2+c x^4}\right ) \int \sqrt {d x} \left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{3/2} \, dx}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}\\ &=\frac {2 a (d x)^{3/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {3}{4};-\frac {3}{2},-\frac {3}{2};\frac {7}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 d \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}}\\ \end {align*}
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Mathematica [B] time = 0.71, size = 417, normalized size = 2.82 \[ \frac {2 x \sqrt {d x} \left (7 \left (209 a^2 c+12 a b^2+328 a b c x^2+286 a c^2 x^4+12 b^3 x^2+131 b^2 c x^4+196 b c^2 x^6+77 c^3 x^8\right )+12 b x^2 \left (36 a c-5 b^2\right ) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {7}{4};\frac {1}{2},\frac {1}{2};\frac {11}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right )-28 a \left (3 b^2-44 a c\right ) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {3}{4};\frac {1}{2},\frac {1}{2};\frac {7}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right )\right )}{8085 c \sqrt {a+b x^2+c x^4}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \sqrt {d x}\, \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x}\,{d x} \]
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 \[ \int \sqrt {d\,x}\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \]
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 \[ \int \sqrt {d x} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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