Optimal. Leaf size=279 \[ -\frac{\log \left (-\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{b x+1}}+\frac{\sqrt{b} \sqrt{1-a x}}{\sqrt{b x+1}}+\sqrt{a}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\log \left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{b x+1}}+\frac{\sqrt{b} \sqrt{1-a x}}{\sqrt{b x+1}}+\sqrt{a}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{b x+1}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{b x+1}}+1\right )}{\sqrt [4]{a} b^{3/4}} \]
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Rubi [A] time = 0.303988, antiderivative size = 279, 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} \[ -\frac{\log \left (-\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{b x+1}}+\frac{\sqrt{b} \sqrt{1-a x}}{\sqrt{b x+1}}+\sqrt{a}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\log \left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{b x+1}}+\frac{\sqrt{b} \sqrt{1-a x}}{\sqrt{b x+1}}+\sqrt{a}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{b x+1}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{b x+1}}+1\right )}{\sqrt [4]{a} b^{3/4}} \]
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+b x)^{3/4}} \, dx &=-\frac{4 \operatorname{Subst}\left (\int \frac{x^2}{\left (1+\frac{b}{a}-\frac{b x^4}{a}\right )^{3/4}} \, dx,x,\sqrt [4]{1-a x}\right )}{a}\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{x^2}{1+\frac{b x^4}{a}} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{a}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{1+\frac{b x^4}{a}} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{a \sqrt{b}}-\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{1+\frac{b x^4}{a}} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{a \sqrt{b}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{b}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{\sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}\\ &=-\frac{\log \left (\sqrt{a}+\frac{\sqrt{b} \sqrt{1-a x}}{\sqrt{1+b x}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\log \left (\sqrt{a}+\frac{\sqrt{b} \sqrt{1-a x}}{\sqrt{1+b x}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}-\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}+\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}\\ &=\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\log \left (\sqrt{a}+\frac{\sqrt{b} \sqrt{1-a x}}{\sqrt{1+b x}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\log \left (\sqrt{a}+\frac{\sqrt{b} \sqrt{1-a x}}{\sqrt{1+b x}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt{2} \sqrt [4]{a} b^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0353814, size = 65, normalized size = 0.23 \[ -\frac{4 (1-a x)^{3/4} \left (\frac{a b x+a}{a+b}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b-a b x}{a+b}\right )}{3 a (b x+1)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [4]{-ax+1}}} \left ( bx+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{1}{4}}{\left (b x + 1\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56453, size = 586, normalized size = 2.1 \begin{align*} -4 \, \left (-\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (-a x + 1\right )}^{\frac{3}{4}}{\left (b x + 1\right )}^{\frac{1}{4}} a b^{2} \left (-\frac{1}{a b^{3}}\right )^{\frac{3}{4}} -{\left (a^{2} b^{2} x - a b^{2}\right )} \sqrt{\frac{{\left (a b^{2} x - b^{2}\right )} \sqrt{-\frac{1}{a b^{3}}} - \sqrt{-a x + 1} \sqrt{b x + 1}}{a x - 1}} \left (-\frac{1}{a b^{3}}\right )^{\frac{3}{4}}}{a x - 1}\right ) - \left (-\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (a b x - b\right )} \left (-\frac{1}{a b^{3}}\right )^{\frac{1}{4}} +{\left (-a x + 1\right )}^{\frac{3}{4}}{\left (b x + 1\right )}^{\frac{1}{4}}}{a x - 1}\right ) + \left (-\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (a b x - b\right )} \left (-\frac{1}{a b^{3}}\right )^{\frac{1}{4}} -{\left (-a x + 1\right )}^{\frac{3}{4}}{\left (b x + 1\right )}^{\frac{1}{4}}}{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 (b 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{1}{4}}{\left (b x + 1\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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