Optimal. Leaf size=287 \[ \frac{2 d x^2 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{a x+b x^3+c x^5}}+\frac{2 e x^4 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{7 \sqrt{a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.399178, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1954, 1335, 1141, 510} \[ \frac{2 d x^2 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{a x+b x^3+c x^5}}+\frac{2 e x^4 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{7 \sqrt{a x+b x^3+c x^5}} \]
Antiderivative was successfully verified.
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Rule 1954
Rule 1335
Rule 1141
Rule 510
Rubi steps
\begin{align*} \int \frac{x \left (d+e x^2\right )}{\sqrt{a x+b x^3+c x^5}} \, dx &=\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{\sqrt{x} \left (d+e x^2\right )}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \left (\frac{d \sqrt{x}}{\sqrt{a+b x^2+c x^4}}+\frac{e x^{5/2}}{\sqrt{a+b x^2+c x^4}}\right ) \, dx}{\sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (d \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a x+b x^3+c x^5}}+\frac{\left (e \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{x^{5/2}}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (d \sqrt{x} \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{\sqrt{x}}{\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 x+b x^3+c x^5}}+\frac{\left (e \sqrt{x} \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{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}}}} \, dx}{\sqrt{a x+b x^3+c x^5}}\\ &=\frac{2 d x^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{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}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{a x+b x^3+c x^5}}+\frac{2 e x^4 \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{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}{b+\sqrt{b^2-4 a c}}\right )}{7 \sqrt{a x+b x^3+c x^5}}\\ \end{align*}
Mathematica [A] time = 5.13468, size = 239, normalized size = 0.83 \[ \frac{2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \left (7 d x^2 F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+3 e x^4 F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )}{21 \sqrt{x \left (a+b x^2+c x^4\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{x \left ( e{x}^{2}+d \right ){\frac{1}{\sqrt{c{x}^{5}+b{x}^{3}+ax}}}}\, 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 )} x}{\sqrt{c x^{5} + b x^{3} + a 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{\sqrt{c x^{5} + b x^{3} + a x}{\left (e x^{2} + d\right )}}{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{x \left (d + e x^{2}\right )}{\sqrt{x \left (a + b x^{2} + c x^{4}\right )}}\, 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 )} x}{\sqrt{c x^{5} + b x^{3} + a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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