Optimal. Leaf size=193 \[ -\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{a} \]
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Rubi [A] time = 0.138051, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\log \left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt{2} a}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx &=-\frac{4 \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{a}\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac{2 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}\\ &=-\frac{\log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}+\frac{\log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}-\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}+\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac{\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac{\log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}-\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}+\frac{\log \left (1+\frac{\sqrt{1-a x}}{\sqrt{1+a x}}+\frac{\sqrt{2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt{2} a}\\ \end{align*}
Mathematica [C] time = 0.0077021, size = 42, normalized size = 0.22 \[ -\frac{2 \sqrt [4]{2} (1-a x)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-a x)\right )}{3 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [4]{-ax+1}}} \left ( ax+1 \right ) ^{-{\frac{3}{4}}}}\, 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 (a x + 1\right )}^{\frac{3}{4}}{\left (-a x + 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.5946, size = 1235, normalized size = 6.4 \begin{align*} 2 \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a^{3} \frac{1}{a^{4}}^{\frac{3}{4}} - \sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a \frac{1}{a^{4}}^{\frac{1}{4}} +{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} - \sqrt{a x + 1} \sqrt{-a x + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} + a x - 1}{a x - 1}\right ) + 2 \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a^{3} \frac{1}{a^{4}}^{\frac{3}{4}} - \sqrt{2}{\left (a^{4} x - a^{3}\right )} \sqrt{-\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a \frac{1}{a^{4}}^{\frac{1}{4}} -{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} + \sqrt{a x + 1} \sqrt{-a x + 1}}{a x - 1}} \frac{1}{a^{4}}^{\frac{3}{4}} - a x + 1}{a x - 1}\right ) - \frac{1}{2} \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a \frac{1}{a^{4}}^{\frac{1}{4}} +{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} - \sqrt{a x + 1} \sqrt{-a x + 1}}{a x - 1}\right ) + \frac{1}{2} \, \sqrt{2} \frac{1}{a^{4}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2}{\left (a x + 1\right )}^{\frac{1}{4}}{\left (-a x + 1\right )}^{\frac{3}{4}} a \frac{1}{a^{4}}^{\frac{1}{4}} -{\left (a^{3} x - a^{2}\right )} \sqrt{\frac{1}{a^{4}}} + \sqrt{a x + 1} \sqrt{-a x + 1}}{a x - 1}\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]{- a x + 1} \left (a x + 1\right )^{\frac{3}{4}}}\, 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 (a x + 1\right )}^{\frac{3}{4}}{\left (-a x + 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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