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 = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 228
Rule 619
Rule 622
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*}
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Mathematica [C] time = 0.01, 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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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (c x^{2} + b x\right )}^{\frac {1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 36, normalized size = 0.62 \[ \frac {4\,x\,{\left (\frac {c\,x}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ -\frac {c\,x}{b}\right )}{3\,{\left (c\,x^2+b\,x\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [4]{b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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