Optimal. Leaf size=61 \[ -\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {283, 223, 212}
\begin {gather*} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 283
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{x^4} \, dx &=-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b \int \frac {\sqrt {a+b x^2}}{x^2} \, dx\\ &=-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b^2 \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b^2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 57, normalized size = 0.93 \begin {gather*} \frac {\left (-a-4 b x^2\right ) \sqrt {a+b x^2}}{3 x^3}-b^{3/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs.
\(2(49)=98\).
time = 0.04, size = 100, normalized size = 1.64
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (4 b \,x^{2}+a \right )}{3 x^{3}}+b^{\frac {3}{2}} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )\) | \(44\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 66, normalized size = 1.08 \begin {gather*} \frac {\sqrt {b x^{2} + a} b^{2} x}{a} + b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{3 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 112, normalized size = 1.84 \begin {gather*} \left [\frac {3 \, b^{\frac {3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (4 \, b x^{2} + a\right )} \sqrt {b x^{2} + a}}{6 \, x^{3}}, -\frac {3 \, \sqrt {-b} b x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (4 \, b x^{2} + a\right )} \sqrt {b x^{2} + a}}{3 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.14, size = 78, normalized size = 1.28 \begin {gather*} - \frac {a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} - \frac {b^{\frac {3}{2}} \log {\left (\frac {a}{b x^{2}} \right )}}{2} + b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x^{2}} + 1} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs.
\(2 (49) = 98\).
time = 0.97, size = 114, normalized size = 1.87 \begin {gather*} -\frac {1}{2} \, b^{\frac {3}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {4 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {3}{2}} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {3}{2}} + 2 \, a^{3} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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