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 = 0.30, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {63, 331, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
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*}
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Mathematica [C] time = 0.04, size = 65, normalized size = 0.23 \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 176, normalized size = 0.63 \begin {gather*} \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt [4]{1-a x} \left (\frac {\sqrt [4]{a} \sqrt {b x+1}}{\sqrt {2} \sqrt [4]{b} \sqrt {1-a x}}-\frac {\sqrt [4]{b}}{\sqrt {2} \sqrt [4]{a}}\right )}{\sqrt [4]{b x+1}}\right )}{\sqrt [4]{a} b^{3/4}}+\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b x+1}}{\sqrt [4]{1-a x} \left (\frac {\sqrt {a} \sqrt {b x+1}}{\sqrt {1-a x}}+\sqrt {b}\right )}\right )}{\sqrt [4]{a} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 247, normalized size = 0.89 \begin {gather*} -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 {gather*}
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 \begin {gather*} \int \frac {1}{{\left (-a x + 1\right )}^{\frac {1}{4}} {\left (b x + 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-a x +1\right )^{\frac {1}{4}} \left (b x +1\right )^{\frac {3}{4}}}\, dx \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (-a x + 1\right )}^{\frac {1}{4}} {\left (b x + 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (1-a\,x\right )}^{1/4}\,{\left (b\,x+1\right )}^{3/4}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt [4]{- a x + 1} \left (b x + 1\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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