Optimal. Leaf size=69 \[ -\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}} \]
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Rubi [A] time = 0.118679, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {466, 511, 510} \[ -\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 466
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{c+\frac{d x^8}{e^4}}}{x^2 \left (a+\frac{b x^8}{e^4}\right )} \, dx,x,\sqrt{e x}\right )}{e}\\ &=\frac{\left (2 \sqrt{c+d x^4}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{d x^8}{c e^4}}}{x^2 \left (a+\frac{b x^8}{e^4}\right )} \, dx,x,\sqrt{e x}\right )}{e \sqrt{1+\frac{d x^4}{c}}}\\ &=-\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{1+\frac{d x^4}{c}}}\\ \end{align*}
Mathematica [B] time = 0.113718, size = 143, normalized size = 2.07 \[ \frac{x \left (14 b d x^8 \sqrt{\frac{d x^4}{c}+1} F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-10 x^4 \sqrt{\frac{d x^4}{c}+1} (b c-4 a d) F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-70 a \left (c+d x^4\right )\right )}{35 a^2 (e x)^{3/2} \sqrt{c+d x^4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{4}+a}\sqrt{d{x}^{4}+c} \left ( ex \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 \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac{3}{2}}}\,{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{d x^{4} + c} \sqrt{e x}}{b e^{2} x^{6} + a e^{2} x^{2}}, 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{\sqrt{c + d x^{4}}}{\left (e x\right )^{\frac{3}{2}} \left (a + b 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{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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