Optimal. Leaf size=80 \[ -\frac {5}{2} a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {\left (a+b x^2\right )^{5/2}}{2 x^2}+\frac {5}{6} b \left (a+b x^2\right )^{3/2}+\frac {5}{2} a b \sqrt {a+b x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 80, 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} \[ -\frac {5}{2} a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {\left (a+b x^2\right )^{5/2}}{2 x^2}+\frac {5}{6} b \left (a+b x^2\right )^{3/2}+\frac {5}{2} a b \sqrt {a+b x^2} \]
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^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{5/2}}{2 x^2}+\frac {1}{4} (5 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {5}{6} b \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{2 x^2}+\frac {1}{4} (5 a b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=\frac {5}{2} a b \sqrt {a+b x^2}+\frac {5}{6} b \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{2 x^2}+\frac {1}{4} \left (5 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {5}{2} a b \sqrt {a+b x^2}+\frac {5}{6} b \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{2 x^2}+\frac {1}{2} \left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=\frac {5}{2} a b \sqrt {a+b x^2}+\frac {5}{6} b \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{2 x^2}-\frac {5}{2} a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.46 \[ \frac {b \left (a+b x^2\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {b x^2}{a}+1\right )}{7 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 142, normalized size = 1.78 \[ \left [\frac {15 \, a^{\frac {3}{2}} b x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, b^{2} x^{4} + 14 \, a b x^{2} - 3 \, a^{2}\right )} \sqrt {b x^{2} + a}}{12 \, x^{2}}, \frac {15 \, \sqrt {-a} a b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, b^{2} x^{4} + 14 \, a b x^{2} - 3 \, a^{2}\right )} \sqrt {b x^{2} + a}}{6 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 82, normalized size = 1.02 \[ \frac {\frac {15 \, a^{2} b^{2} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} + 12 \, \sqrt {b x^{2} + a} a b^{2} - \frac {3 \, \sqrt {b x^{2} + a} a^{2} b}{x^{2}}}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 88, normalized size = 1.10 \[ -\frac {5 a^{\frac {3}{2}} b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2}+\frac {5 \sqrt {b \,x^{2}+a}\, a b}{2}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}{6}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} b}{2 a}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 76, normalized size = 0.95 \[ -\frac {5}{2} \, a^{\frac {3}{2}} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {5}{6} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b}{2 \, a} + \frac {5}{2} \, \sqrt {b x^{2} + a} a b - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.95, size = 66, normalized size = 0.82 \[ \frac {b\,{\left (b\,x^2+a\right )}^{3/2}}{3}-\frac {a^2\,\sqrt {b\,x^2+a}}{2\,x^2}+2\,a\,b\,\sqrt {b\,x^2+a}+\frac {a^{3/2}\,b\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.22, size = 112, normalized size = 1.40 \[ - \frac {a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}}}{2 x^{2}} + \frac {7 a^{\frac {3}{2}} b \sqrt {1 + \frac {b x^{2}}{a}}}{3} + \frac {5 a^{\frac {3}{2}} b \log {\left (\frac {b x^{2}}{a} \right )}}{4} - \frac {5 a^{\frac {3}{2}} b \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2} + \frac {\sqrt {a} b^{2} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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