Optimal. Leaf size=299 \[ \frac{2 a d (f x)^{3/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{3}{4};-\frac{3}{2},-\frac{3}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 a e (f x)^{7/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{7}{4};-\frac{3}{2},-\frac{3}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{7 f^3 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.354368, antiderivative size = 299, 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} \[ \frac{2 a d (f x)^{3/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{3}{4};-\frac{3}{2},-\frac{3}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 a e (f x)^{7/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{7}{4};-\frac{3}{2},-\frac{3}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{7 f^3 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 1335
Rule 1141
Rule 510
Rubi steps
\begin{align*} \int \sqrt{f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\int \left (d \sqrt{f x} \left (a+b x^2+c x^4\right )^{3/2}+\frac{e (f x)^{5/2} \left (a+b x^2+c x^4\right )^{3/2}}{f^2}\right ) \, dx\\ &=d \int \sqrt{f x} \left (a+b x^2+c x^4\right )^{3/2} \, dx+\frac{e \int (f x)^{5/2} \left (a+b x^2+c x^4\right )^{3/2} \, dx}{f^2}\\ &=\frac{\left (a d \sqrt{a+b x^2+c x^4}\right ) \int \sqrt{f 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{\left (a e \sqrt{a+b x^2+c x^4}\right ) \int (f x)^{5/2} \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}{f^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}}}}\\ &=\frac{2 a d (f 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 f \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 e (f x)^{7/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{7}{4};-\frac{3}{2},-\frac{3}{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 f^3 \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 [A] time = 6.19388, size = 490, normalized size = 1.64 \[ \frac{2 x \sqrt{f x} \left (12 x^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) \left (420 a^2 c^2 e-309 a b^2 c e+684 a b c^2 d-95 b^3 c d+45 b^4 e\right )+28 a \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) \left (-156 a b c e+836 a c^2 d-57 b^2 c d+27 b^3 e\right )+7 \left (a+b x^2+c x^4\right ) \left (b c \left (624 a e+7 c x^2 \left (323 d+231 e x^2\right )\right )+c^2 \left (a \left (3971 d+2415 e x^2\right )+77 c x^4 \left (19 d+15 e x^2\right )\right )+12 b^2 c \left (19 d+7 e x^2\right )-108 b^3 e\right )\right )}{153615 c^2 \sqrt{a+b x^2+c x^4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \sqrt{fx} \left ( e{x}^{2}+d \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}\, 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{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )} \sqrt{f 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 ({\left (c e x^{6} +{\left (c d + b e\right )} x^{4} +{\left (b d + a e\right )} x^{2} + a d\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{f 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 \sqrt{f x} \left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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