Optimal. Leaf size=75 \[ -\frac {\left (a x^2+b\right )^{3/4}}{2 x^2}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt [4]{a x^2+b}}{\sqrt [4]{b}}\right )}{4 \sqrt [4]{b}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt [4]{a x^2+b}}{\sqrt [4]{b}}\right )}{4 \sqrt [4]{b}} \]
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Rubi [A] time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {266, 47, 63, 298, 203, 206} \begin {gather*} -\frac {\left (a x^2+b\right )^{3/4}}{2 x^2}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt [4]{a x^2+b}}{\sqrt [4]{b}}\right )}{4 \sqrt [4]{b}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt [4]{a x^2+b}}{\sqrt [4]{b}}\right )}{4 \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rubi steps
\begin {align*} \int \frac {\left (b+a x^2\right )^{3/4}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+a x)^{3/4}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (b+a x^2\right )^{3/4}}{2 x^2}+\frac {1}{8} (3 a) \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^2\right )\\ &=-\frac {\left (b+a x^2\right )^{3/4}}{2 x^2}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^2}\right )\\ &=-\frac {\left (b+a x^2\right )^{3/4}}{2 x^2}-\frac {1}{4} (3 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^2}\right )+\frac {1}{4} (3 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^2}\right )\\ &=-\frac {\left (b+a x^2\right )^{3/4}}{2 x^2}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^2}}{\sqrt [4]{b}}\right )}{4 \sqrt [4]{b}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^2}}{\sqrt [4]{b}}\right )}{4 \sqrt [4]{b}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.49 \begin {gather*} \frac {2 a \left (a x^2+b\right )^{7/4} \, _2F_1\left (\frac {7}{4},2;\frac {11}{4};\frac {a x^2}{b}+1\right )}{7 b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 75, normalized size = 1.00 \begin {gather*} -\frac {\left (a x^2+b\right )^{3/4}}{2 x^2}+\frac {3 a \tan ^{-1}\left (\frac {\sqrt [4]{a x^2+b}}{\sqrt [4]{b}}\right )}{4 \sqrt [4]{b}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt [4]{a x^2+b}}{\sqrt [4]{b}}\right )}{4 \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 185, normalized size = 2.47 \begin {gather*} -\frac {12 \, \left (\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{2} \arctan \left (-\frac {\left (\frac {a^{4}}{b}\right )^{\frac {1}{4}} {\left (a x^{2} + b\right )}^{\frac {1}{4}} a^{3} - \sqrt {\sqrt {a x^{2} + b} a^{6} + \sqrt {\frac {a^{4}}{b}} a^{4} b} \left (\frac {a^{4}}{b}\right )^{\frac {1}{4}}}{a^{4}}\right ) + 3 \, \left (\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{2} \log \left (27 \, {\left (a x^{2} + b\right )}^{\frac {1}{4}} a^{3} + 27 \, \left (\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 3 \, \left (\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{2} \log \left (27 \, {\left (a x^{2} + b\right )}^{\frac {1}{4}} a^{3} - 27 \, \left (\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) + 4 \, {\left (a x^{2} + b\right )}^{\frac {3}{4}}}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 209, normalized size = 2.79 \begin {gather*} \frac {\frac {6 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{2} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {1}{4}}} + \frac {6 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{2} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {1}{4}}} + \frac {3 \, \sqrt {2} a^{2} \left (-b\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (a x^{2} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{2} + b} + \sqrt {-b}\right )}{b} + \frac {3 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{2} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{2} + b} + \sqrt {-b}\right )}{\left (-b\right )^{\frac {1}{4}}} - \frac {8 \, {\left (a x^{2} + b\right )}^{\frac {3}{4}} a}{x^{2}}}{16 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{2}+b \right )^{\frac {3}{4}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 74, normalized size = 0.99 \begin {gather*} \frac {3}{8} \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{2} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (a x^{2} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{2} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}}\right )} - \frac {{\left (a x^{2} + b\right )}^{\frac {3}{4}}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 55, normalized size = 0.73 \begin {gather*} \frac {3\,a\,\mathrm {atan}\left (\frac {{\left (a\,x^2+b\right )}^{1/4}}{b^{1/4}}\right )}{4\,b^{1/4}}-\frac {{\left (a\,x^2+b\right )}^{3/4}}{2\,x^2}-\frac {3\,a\,\mathrm {atanh}\left (\frac {{\left (a\,x^2+b\right )}^{1/4}}{b^{1/4}}\right )}{4\,b^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.18, size = 42, normalized size = 0.56 \begin {gather*} - \frac {a^{\frac {3}{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{2}}} \right )}}{2 \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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