Optimal. Leaf size=139 \[ \frac {b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}{\sqrt {a x^2-b}-\sqrt {b}}\right )}{\sqrt {2}}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^2-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt {2}}+\frac {2}{3} \left (a x^2-b\right )^{3/4} \]
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Rubi [A] time = 0.24, antiderivative size = 213, normalized size of antiderivative = 1.53, number of steps used = 12, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {266, 50, 63, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {a x^2-b}+\sqrt {b}\right )}{2 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {a x^2-b}+\sqrt {b}\right )}{2 \sqrt {2}}+\frac {b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2}}+\frac {2}{3} \left (a x^2-b\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 204
Rule 266
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^2\right )^{3/4}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-b+a x)^{3/4}}{x} \, dx,x,x^2\right )\\ &=\frac {2}{3} \left (-b+a x^2\right )^{3/4}-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^2\right )\\ &=\frac {2}{3} \left (-b+a x^2\right )^{3/4}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a}\\ &=\frac {2}{3} \left (-b+a x^2\right )^{3/4}+\frac {b \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a}-\frac {b \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{a}\\ &=\frac {2}{3} \left (-b+a x^2\right )^{3/4}-\frac {b^{3/4} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \sqrt {2}}-\frac {b^{3/4} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 \sqrt {2}}-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^2}\right )\\ &=\frac {2}{3} \left (-b+a x^2\right )^{3/4}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2}}-\frac {b^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2}}+\frac {b^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2}}\\ &=\frac {2}{3} \left (-b+a x^2\right )^{3/4}+\frac {b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{\sqrt {2}}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 79, normalized size = 0.57 \begin {gather*} \frac {2}{3} \left (a x^2-b\right )^{3/4}+(-b)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{-b}}\right )-(-b)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{-b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 139, normalized size = 1.00 \begin {gather*} \frac {2}{3} \left (-b+a x^2\right )^{3/4}-\frac {b^{3/4} \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {b}+\sqrt {-b+a x^2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 159, normalized size = 1.14 \begin {gather*} 2 \, \left (-b^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-b^{3}\right )^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{2} - \sqrt {\sqrt {a x^{2} - b} b^{4} - \sqrt {-b^{3}} b^{3}} \left (-b^{3}\right )^{\frac {1}{4}}}{b^{3}}\right ) - \frac {1}{2} \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{2} + \left (-b^{3}\right )^{\frac {3}{4}}\right ) + \frac {1}{2} \, \left (-b^{3}\right )^{\frac {1}{4}} \log \left ({\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{2} - \left (-b^{3}\right )^{\frac {3}{4}}\right ) + \frac {2}{3} \, {\left (a x^{2} - b\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 175, normalized size = 1.26 \begin {gather*} -\frac {1}{2} \, \sqrt {2} b^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \sqrt {2} b^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right ) + \frac {1}{4} \, \sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) - \frac {1}{4} \, \sqrt {2} b^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right ) + \frac {2}{3} \, {\left (a x^{2} - b\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}-b \right )^{\frac {3}{4}}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 178, normalized size = 1.28 \begin {gather*} -\frac {1}{4} \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{2} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{2} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} b + \frac {2}{3} \, {\left (a x^{2} - b\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 63, normalized size = 0.45 \begin {gather*} \frac {2\,{\left (a\,x^2-b\right )}^{3/4}}{3}+{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )-{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.07, size = 48, normalized size = 0.35 \begin {gather*} - \frac {a^{\frac {3}{4}} x^{\frac {3}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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