Optimal. Leaf size=109 \[ \frac {2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4-b x\right )^{3/4}}{a x^3-b}\right )}{3 \sqrt [4]{a} b}+\frac {2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4-b x\right )^{3/4}}{a x^3-b}\right )}{3 \sqrt [4]{a} b} \]
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Rubi [A] time = 0.21, antiderivative size = 159, normalized size of antiderivative = 1.46, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2056, 466, 465, 377, 212, 206, 203} \begin {gather*} \frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4-b x}}+\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{a x^3-b} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3-b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 465
Rule 466
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (b+a x^3\right ) \sqrt [4]{-b x+a x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-b+a x^3} \left (b+a x^3\right )} \, dx}{\sqrt [4]{-b x+a x^4}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^{12}} \left (b+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^4}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{-b x+a x^4}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{b-2 a b x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{-b x+a x^4}}\\ &=\frac {\left (2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 b \sqrt [4]{-b x+a x^4}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 b \sqrt [4]{-b x+a x^4}}\\ &=\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{-b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{-b x+a x^4}}+\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{-b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{-b x+a x^4}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 76, normalized size = 0.70 \begin {gather*} \frac {4 x \sqrt [4]{1-\frac {a x^3}{b}} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 a x^3}{a x^3+b}\right )}{3 b \sqrt [4]{\frac {a x^3}{b}+1} \sqrt [4]{a x^4-b x}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.40, size = 109, normalized size = 1.00 \begin {gather*} \frac {2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{3 \sqrt [4]{a} b}+\frac {2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (-b x+a x^4\right )^{3/4}}{-b+a x^3}\right )}{3 \sqrt [4]{a} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 107.75, size = 430, normalized size = 3.94 \begin {gather*} -\frac {2}{3} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a x^{4} - b x\right )}^{\frac {3}{4}} a b^{3} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + {\left (2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {a x^{4} - b x} a b x \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (3 \, a^{2} b^{3} x^{3} - a b^{4}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {\sqrt {\frac {1}{2}} b^{2} \sqrt {\frac {1}{a b^{4}}}}\right )}}{a x^{3} + b}\right ) + \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{3} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) - \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b x} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b x\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {1}{a b^{4}}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{3} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b x\right )}^{\frac {3}{4}}}{a x^{3} + b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 209, normalized size = 1.92 \begin {gather*} \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a b} + \frac {2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}} b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{3}}}\right )}{6 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{3}+b \right ) \left (a \,x^{4}-b x \right )^{\frac {1}{4}}}\, dx\]
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 \begin {gather*} \int \frac {1}{{\left (a x^{4} - b x\right )}^{\frac {1}{4}} {\left (a x^{3} + b\right )}}\,{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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a\,x^4-b\,x\right )}^{1/4}\,\left (a\,x^3+b\right )} \,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]{x \left (a x^{3} - b\right )} \left (a x^{3} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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