Optimal. Leaf size=96 \[ \frac {5 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^3}-\frac {5 b \sqrt {a+\frac {b}{x}} x^2}{12 a^2}+\frac {\sqrt {a+\frac {b}{x}} x^3}{3 a}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{7/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 44, 65,
214} \begin {gather*} -\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{7/2}}+\frac {5 b^2 x \sqrt {a+\frac {b}{x}}}{8 a^3}-\frac {5 b x^2 \sqrt {a+\frac {b}{x}}}{12 a^2}+\frac {x^3 \sqrt {a+\frac {b}{x}}}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+\frac {b}{x}}} \, dx &=-\text {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x}} x^3}{3 a}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{6 a}\\ &=-\frac {5 b \sqrt {a+\frac {b}{x}} x^2}{12 a^2}+\frac {\sqrt {a+\frac {b}{x}} x^3}{3 a}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{8 a^2}\\ &=\frac {5 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^3}-\frac {5 b \sqrt {a+\frac {b}{x}} x^2}{12 a^2}+\frac {\sqrt {a+\frac {b}{x}} x^3}{3 a}+\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{16 a^3}\\ &=\frac {5 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^3}-\frac {5 b \sqrt {a+\frac {b}{x}} x^2}{12 a^2}+\frac {\sqrt {a+\frac {b}{x}} x^3}{3 a}+\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{8 a^3}\\ &=\frac {5 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^3}-\frac {5 b \sqrt {a+\frac {b}{x}} x^2}{12 a^2}+\frac {\sqrt {a+\frac {b}{x}} x^3}{3 a}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 69, normalized size = 0.72 \begin {gather*} \frac {\sqrt {a+\frac {b}{x}} x \left (15 b^2-10 a b x+8 a^2 x^2\right )}{24 a^3}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{7/2}} \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. \(163\) vs.
\(2(76)=152\).
time = 0.04, size = 164, normalized size = 1.71
method | result | size |
risch | \(\frac {\left (8 a^{2} x^{2}-10 a b x +15 b^{2}\right ) \left (a x +b \right )}{24 a^{3} \sqrt {\frac {a x +b}{x}}}-\frac {5 b^{3} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {x \left (a x +b \right )}}{16 a^{\frac {7}{2}} x \sqrt {\frac {a x +b}{x}}}\) | \(97\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}}-36 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b x +48 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{2}-18 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{2}-24 a \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{3}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3}\right )}{48 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 137, normalized size = 1.43 \begin {gather*} \frac {5 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {7}{2}}} + \frac {15 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{3} - 40 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a b^{3} + 33 \, \sqrt {a + \frac {b}{x}} a^{2} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} a^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} a^{4} + 3 \, {\left (a + \frac {b}{x}\right )} a^{5} - a^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 151, normalized size = 1.57 \begin {gather*} \left [\frac {15 \, \sqrt {a} b^{3} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{3} x^{3} - 10 \, a^{2} b x^{2} + 15 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, a^{4}}, \frac {15 \, \sqrt {-a} b^{3} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (8 \, a^{3} x^{3} - 10 \, a^{2} b x^{2} + 15 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, a^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.40, size = 128, normalized size = 1.33 \begin {gather*} \frac {x^{\frac {7}{2}}}{3 \sqrt {b} \sqrt {\frac {a x}{b} + 1}} - \frac {\sqrt {b} x^{\frac {5}{2}}}{12 a \sqrt {\frac {a x}{b} + 1}} + \frac {5 b^{\frac {3}{2}} x^{\frac {3}{2}}}{24 a^{2} \sqrt {\frac {a x}{b} + 1}} + \frac {5 b^{\frac {5}{2}} \sqrt {x}}{8 a^{3} \sqrt {\frac {a x}{b} + 1}} - \frac {5 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{8 a^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 106, normalized size = 1.10 \begin {gather*} \frac {1}{24} \, \sqrt {a x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, x}{a \mathrm {sgn}\left (x\right )} - \frac {5 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {15 \, b^{2}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} - \frac {5 \, b^{3} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {7}{2}}} + \frac {5 \, b^{3} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{16 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.40, size = 77, normalized size = 0.80 \begin {gather*} \frac {11\,x^3\,\sqrt {a+\frac {b}{x}}}{8\,a}-\frac {5\,x^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3\,a^2}+\frac {5\,x^3\,{\left (a+\frac {b}{x}\right )}^{5/2}}{8\,a^3}+\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{8\,a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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