Optimal. Leaf size=83 \[ \frac {15}{8} a b x \sqrt {a+b x^2}+\frac {5}{4} b x \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{x}+\frac {15}{8} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {283, 201, 223,
212} \begin {gather*} \frac {15}{8} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}+\frac {5}{4} b x \left (a+b x^2\right )^{3/2}+\frac {15}{8} a b x \sqrt {a+b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 283
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^2} \, dx &=-\frac {\left (a+b x^2\right )^{5/2}}{x}+(5 b) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac {5}{4} b x \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{x}+\frac {1}{4} (15 a b) \int \sqrt {a+b x^2} \, dx\\ &=\frac {15}{8} a b x \sqrt {a+b x^2}+\frac {5}{4} b x \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{x}+\frac {1}{8} \left (15 a^2 b\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {15}{8} a b x \sqrt {a+b x^2}+\frac {5}{4} b x \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{x}+\frac {1}{8} \left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {15}{8} a b x \sqrt {a+b x^2}+\frac {5}{4} b x \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{5/2}}{x}+\frac {15}{8} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 73, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-8 a^2+9 a b x^2+2 b^2 x^4\right )}{8 x}-\frac {15}{8} a^2 \sqrt {b} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 92, normalized size = 1.11
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 b^{2} x^{4}-9 a b \,x^{2}+8 a^{2}\right )}{8 x}+\frac {15 a^{2} \sqrt {b}\, \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{8}\) | \(61\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{a x}+\frac {6 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{a}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.43, size = 59, normalized size = 0.71 \begin {gather*} \frac {5}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b x + \frac {15}{8} \, \sqrt {b x^{2} + a} a b x + \frac {15}{8} \, a^{2} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 140, normalized size = 1.69 \begin {gather*} \left [\frac {15 \, a^{2} \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, b^{2} x^{4} + 9 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt {b x^{2} + a}}{16 \, x}, -\frac {15 \, a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b^{2} x^{4} + 9 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt {b x^{2} + a}}{8 \, x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.33, size = 117, normalized size = 1.41 \begin {gather*} - \frac {a^{\frac {5}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {3}{2}} b x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {11 \sqrt {a} b^{2} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8} + \frac {b^{3} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 87, normalized size = 1.05 \begin {gather*} -\frac {15}{16} \, a^{2} \sqrt {b} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, a^{3} \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} + \frac {1}{8} \, {\left (2 \, b^{2} x^{2} + 9 \, a b\right )} \sqrt {b x^{2} + a} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.05, size = 40, normalized size = 0.48 \begin {gather*} -\frac {{\left (b\,x^2+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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