Optimal. Leaf size=146 \[ \frac{2 a \sqrt{d x} \sqrt{a+b x^2+c x^4} F_1\left (\frac{1}{4};-\frac{3}{2},-\frac{3}{2};\frac{5}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.127411, antiderivative size = 146, normalized size of antiderivative = 1., 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} \[ \frac{2 a \sqrt{d x} \sqrt{a+b x^2+c x^4} F_1\left (\frac{1}{4};-\frac{3}{2},-\frac{3}{2};\frac{5}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 1141
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{\sqrt{d x}} \, dx &=\frac{\left (a \sqrt{a+b x^2+c x^4}\right ) \int \frac{\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}}{\sqrt{d x}} \, 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 \sqrt{d x} \sqrt{a+b x^2+c x^4} F_1\left (\frac{1}{4};-\frac{3}{2},-\frac{3}{2};\frac{5}{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 )}{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*}
Mathematica [B] time = 0.732367, size = 415, normalized size = 2.84 \[ \frac{2 x \left (5 \left (51 a^2 c+4 a b^2+76 a b c x^2+66 a c^2 x^4+29 b^2 c x^4+4 b^3 x^2+40 b c^2 x^6+15 c^3 x^8\right )-4 b x^2 \left (3 b^2-28 a c\right ) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-20 a \left (b^2-36 a c\right ) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )}{975 c \sqrt{d x} \sqrt{a+b x^2+c x^4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.289, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{dx}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{\sqrt{d x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{d x}}{d x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{\sqrt{d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{\sqrt{d x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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