Optimal. Leaf size=58 \[ \frac{\sqrt{2} b \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{c \sqrt [4]{b x+c x^2}} \]
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Rubi [A] time = 0.0226108, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {622, 619, 228} \[ \frac{\sqrt{2} b \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{c \sqrt [4]{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 622
Rule 619
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{b x+c x^2}} \, dx &=\frac{\sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} \int \frac{1}{\sqrt [4]{-\frac{c x}{b}-\frac{c^2 x^2}{b^2}}} \, dx}{\sqrt [4]{b x+c x^2}}\\ &=-\frac{\left (b^2 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{b^2 x^2}{c^2}}} \, dx,x,-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right )}{\sqrt{2} c^2 \sqrt [4]{b x+c x^2}}\\ &=\frac{\sqrt{2} b \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (1+\frac{2 c x}{b}\right )\right |2\right )}{c \sqrt [4]{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0098401, size = 45, normalized size = 0.78 \[ \frac{4 x \sqrt [4]{\frac{c x}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{c x}{b}\right )}{3 \sqrt [4]{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.608, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [4]{c{x}^{2}+bx}}}\, 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{1}{{\left (c x^{2} + b x\right )}^{\frac{1}{4}}}\,{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{1}{{\left (c x^{2} + b x\right )}^{\frac{1}{4}}}, 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{1}{\sqrt [4]{b x + c x^{2}}}\, 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{1}{{\left (c x^{2} + b x\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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