Optimal. Leaf size=86 \[ \frac {15}{8} b^2 \sqrt {a+b x^2}-\frac {15}{8} \sqrt {a} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {5 b \left (a+b x^2\right )^{3/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{5/2}}{4 x^4} \]
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Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 50, 63, 208} \begin {gather*} \frac {15}{8} b^2 \sqrt {a+b x^2}-\frac {15}{8} \sqrt {a} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {\left (a+b x^2\right )^{5/2}}{4 x^4}-\frac {5 b \left (a+b x^2\right )^{3/2}}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{5/2}}{4 x^4}+\frac {1}{8} (5 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {5 b \left (a+b x^2\right )^{3/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{5/2}}{4 x^4}+\frac {1}{16} \left (15 b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=\frac {15}{8} b^2 \sqrt {a+b x^2}-\frac {5 b \left (a+b x^2\right )^{3/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{5/2}}{4 x^4}+\frac {1}{16} \left (15 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {15}{8} b^2 \sqrt {a+b x^2}-\frac {5 b \left (a+b x^2\right )^{3/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{5/2}}{4 x^4}+\frac {1}{8} (15 a b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=\frac {15}{8} b^2 \sqrt {a+b x^2}-\frac {5 b \left (a+b x^2\right )^{3/2}}{8 x^2}-\frac {\left (a+b x^2\right )^{5/2}}{4 x^4}-\frac {15}{8} \sqrt {a} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.45 \begin {gather*} -\frac {b^2 \left (a+b x^2\right )^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};\frac {b x^2}{a}+1\right )}{7 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 70, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-2 a^2-9 a b x^2+8 b^2 x^4\right )}{8 x^4}-\frac {15}{8} \sqrt {a} b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 145, normalized size = 1.69 \begin {gather*} \left [\frac {15 \, \sqrt {a} b^{2} x^{4} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (8 \, b^{2} x^{4} - 9 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{16 \, x^{4}}, \frac {15 \, \sqrt {-a} b^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (8 \, b^{2} x^{4} - 9 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {b x^{2} + a}}{8 \, x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 88, normalized size = 1.02 \begin {gather*} \frac {\frac {15 \, a b^{3} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 8 \, \sqrt {b x^{2} + a} b^{3} - \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{3} - 7 \, \sqrt {b x^{2} + a} a^{2} b^{3}}{b^{2} x^{4}}}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 116, normalized size = 1.35 \begin {gather*} -\frac {15 \sqrt {a}\, b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8}+\frac {15 \sqrt {b \,x^{2}+a}\, b^{2}}{8}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}{8 a}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}{8 a^{2}}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}{8 a^{2} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 104, normalized size = 1.21 \begin {gather*} -\frac {15}{8} \, \sqrt {a} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {15}{8} \, \sqrt {b x^{2} + a} b^{2} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}}{8 \, a^{2}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}{8 \, a} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.01, size = 71, normalized size = 0.83 \begin {gather*} b^2\,\sqrt {b\,x^2+a}-\frac {9\,a\,{\left (b\,x^2+a\right )}^{3/2}}{8\,x^4}+\frac {7\,a^2\,\sqrt {b\,x^2+a}}{8\,x^4}+\frac {\sqrt {a}\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,15{}\mathrm {i}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.73, size = 117, normalized size = 1.36 \begin {gather*} - \frac {15 \sqrt {a} b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8} - \frac {a^{3}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 a^{2} \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {a b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{\frac {5}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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