Optimal. Leaf size=153 \[ -\frac {3 a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}{\sqrt {a x^2-b}-\sqrt {b}}\right )}{4 \sqrt {2} b^{7/4}}+\frac {3 a \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^2-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt {2} b^{7/4}}+\frac {\sqrt [4]{a x^2-b}}{2 b x^2} \]
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Rubi [A] time = 0.22, antiderivative size = 228, normalized size of antiderivative = 1.49, 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, 51, 63, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {3 a \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {a x^2-b}+\sqrt {b}\right )}{8 \sqrt {2} b^{7/4}}+\frac {3 a \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}+\sqrt {a x^2-b}+\sqrt {b}\right )}{8 \sqrt {2} b^{7/4}}-\frac {3 a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{7/4}}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^2-b}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{7/4}}+\frac {\sqrt [4]{a x^2-b}}{2 b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 204
Rule 211
Rule 266
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (-b+a x^2\right )^{3/4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (-b+a x)^{3/4}} \, dx,x,x^2\right )\\ &=\frac {\sqrt [4]{-b+a x^2}}{2 b x^2}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{x (-b+a x)^{3/4}} \, dx,x,x^2\right )}{8 b}\\ &=\frac {\sqrt [4]{-b+a x^2}}{2 b x^2}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{2 b}\\ &=\frac {\sqrt [4]{-b+a x^2}}{2 b x^2}+\frac {3 \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 )}{4 b^{3/2}}+\frac {3 \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 )}{4 b^{3/2}}\\ &=\frac {\sqrt [4]{-b+a x^2}}{2 b x^2}-\frac {(3 a) \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 )}{8 \sqrt {2} b^{7/4}}-\frac {(3 a) \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 )}{8 \sqrt {2} b^{7/4}}+\frac {(3 a) \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 )}{8 b^{3/2}}+\frac {(3 a) \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 )}{8 b^{3/2}}\\ &=\frac {\sqrt [4]{-b+a x^2}}{2 b x^2}-\frac {3 a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{8 \sqrt {2} b^{7/4}}+\frac {3 a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{8 \sqrt {2} b^{7/4}}+\frac {(3 a) \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 )}{4 \sqrt {2} b^{7/4}}-\frac {(3 a) \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 )}{4 \sqrt {2} b^{7/4}}\\ &=\frac {\sqrt [4]{-b+a x^2}}{2 b x^2}-\frac {3 a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{7/4}}+\frac {3 a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^2}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{7/4}}-\frac {3 a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{8 \sqrt {2} b^{7/4}}+\frac {3 a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}+\sqrt {-b+a x^2}\right )}{8 \sqrt {2} b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.25 \begin {gather*} \frac {2 a \sqrt [4]{a x^2-b} \, _2F_1\left (\frac {1}{4},2;\frac {5}{4};1-\frac {a x^2}{b}\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 152, normalized size = 0.99 \begin {gather*} \frac {\sqrt [4]{-b+a x^2}}{2 b x^2}+\frac {3 a \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 )}{4 \sqrt {2} b^{7/4}}+\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {b}+\sqrt {-b+a x^2}}\right )}{4 \sqrt {2} b^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 212, normalized size = 1.39 \begin {gather*} \frac {12 \, b x^{2} \left (-\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{2} - b\right )}^{\frac {1}{4}} a b^{5} \left (-\frac {a^{4}}{b^{7}}\right )^{\frac {3}{4}} - \sqrt {b^{4} \sqrt {-\frac {a^{4}}{b^{7}}} + \sqrt {a x^{2} - b} a^{2}} b^{5} \left (-\frac {a^{4}}{b^{7}}\right )^{\frac {3}{4}}}{a^{4}}\right ) + 3 \, b x^{2} \left (-\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (3 \, b^{2} \left (-\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} a\right ) - 3 \, b x^{2} \left (-\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-3 \, b^{2} \left (-\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} a\right ) + 4 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{8 \, b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 199, normalized size = 1.30 \begin {gather*} \frac {\frac {6 \, \sqrt {2} a^{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 {7}{4}}} + \frac {6 \, \sqrt {2} a^{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 {7}{4}}} + \frac {3 \, \sqrt {2} a^{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 {7}{4}}} - \frac {3 \, \sqrt {2} a^{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 {7}{4}}} + \frac {8 \, {\left (a x^{2} - b\right )}^{\frac {1}{4}} a}{b x^{2}}}{16 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (a \,x^{2}-b \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 202, normalized size = 1.32 \begin {gather*} \frac {{\left (a x^{2} - b\right )}^{\frac {1}{4}} a}{2 \, {\left ({\left (a x^{2} - b\right )} b + b^{2}\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} a \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 {3}{4}}} + \frac {2 \, \sqrt {2} a \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 {3}{4}}} + \frac {\sqrt {2} a \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 {3}{4}}} - \frac {\sqrt {2} a \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 {3}{4}}}\right )}}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 72, normalized size = 0.47 \begin {gather*} \frac {{\left (a\,x^2-b\right )}^{1/4}}{2\,b\,x^2}+\frac {3\,a\,\mathrm {atan}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{4\,{\left (-b\right )}^{7/4}}+\frac {3\,a\,\mathrm {atanh}\left (\frac {{\left (a\,x^2-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{4\,{\left (-b\right )}^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.15, size = 42, normalized size = 0.27 \begin {gather*} - \frac {\Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{2}}} \right )}}{2 a^{\frac {3}{4}} x^{\frac {7}{2}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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