Optimal. Leaf size=297 \[ \frac{2 d (f x)^{5/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{9/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{9}{4};\frac{1}{2},\frac{1}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 f^3 \sqrt{a+b x^2+c x^4}} \]
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Rubi [A] time = 0.328178, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1335, 1141, 510} \[ \frac{2 d (f x)^{5/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{9/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{9}{4};\frac{1}{2},\frac{1}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 f^3 \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1335
Rule 1141
Rule 510
Rubi steps
\begin{align*} \int \frac{(f x)^{3/2} \left (d+e x^2\right )}{\sqrt{a+b x^2+c x^4}} \, dx &=\int \left (\frac{d (f x)^{3/2}}{\sqrt{a+b x^2+c x^4}}+\frac{e (f x)^{7/2}}{f^2 \sqrt{a+b x^2+c x^4}}\right ) \, dx\\ &=d \int \frac{(f x)^{3/2}}{\sqrt{a+b x^2+c x^4}} \, dx+\frac{e \int \frac{(f x)^{7/2}}{\sqrt{a+b x^2+c x^4}} \, dx}{f^2}\\ &=\frac{\left (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}}}\right ) \int \frac{(f x)^{3/2}}{\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}}}} \, dx}{\sqrt{a+b x^2+c x^4}}+\frac{\left (e \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}}}\right ) \int \frac{(f x)^{7/2}}{\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}}}} \, dx}{f^2 \sqrt{a+b x^2+c x^4}}\\ &=\frac{2 d (f x)^{5/2} \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}}} 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}{b+\sqrt{b^2-4 a c}}\right )}{5 f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{9/2} \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}}} F_1\left (\frac{9}{4};\frac{1}{2},\frac{1}{2};\frac{13}{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 )}{9 f^3 \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.62466, size = 354, normalized size = 1.19 \[ \frac{2 f \sqrt{f x} \left (x^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} (5 c d-3 b e) F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-5 a e \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+5 e \left (a+b x^2+c x^4\right )\right )}{25 c \sqrt{a+b x^2+c x^4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{(e{x}^{2}+d) \left ( fx \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}\, 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 (e x^{2} + d\right )} \left (f x\right )^{\frac{3}{2}}}{\sqrt{c x^{4} + b x^{2} + a}}\,{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 (e f x^{3} + d f x\right )} \sqrt{f x}}{\sqrt{c x^{4} + b x^{2} + a}}, 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 (f x\right )^{\frac{3}{2}} \left (d + e x^{2}\right )}{\sqrt{a + b x^{2} + c x^{4}}}\, 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 (e x^{2} + d\right )} \left (f x\right )^{\frac{3}{2}}}{\sqrt{c x^{4} + b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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