Optimal. Leaf size=54 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2}{a^2 \sqrt {a+b x}}+\frac {2}{3 a (a+b x)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ \frac {2}{a^2 \sqrt {a+b x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2}{3 a (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^{5/2}} \, dx &=\frac {2}{3 a (a+b x)^{3/2}}+\frac {\int \frac {1}{x (a+b x)^{3/2}} \, dx}{a}\\ &=\frac {2}{3 a (a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {a+b x}}+\frac {\int \frac {1}{x \sqrt {a+b x}} \, dx}{a^2}\\ &=\frac {2}{3 a (a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {a+b x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a^2 b}\\ &=\frac {2}{3 a (a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {a+b x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 32, normalized size = 0.59 \[ \frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b x}{a}+1\right )}{3 a (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 177, normalized size = 3.28 \[ \left [\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, a b x + 4 \, a^{2}\right )} \sqrt {b x + a}}{3 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b x + 4 \, a^{2}\right )} \sqrt {b x + a}\right )}}{3 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 45, normalized size = 0.83 \[ \frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, {\left (3 \, b x + 4 \, a\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.80 \[ \frac {2}{3 \left (b x +a \right )^{\frac {3}{2}} a}-\frac {2 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{\sqrt {b x +a}\, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 53, normalized size = 0.98 \[ \frac {\log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x + 4 \, a\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 42, normalized size = 0.78 \[ \frac {\frac {2\,\left (a+b\,x\right )}{a^2}+\frac {2}{3\,a}}{{\left (a+b\,x\right )}^{3/2}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.99, size = 697, normalized size = 12.91 \[ \frac {8 a^{7} \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {3 a^{7} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {6 a^{7} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {14 a^{6} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {9 a^{6} b x \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {18 a^{6} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {6 a^{5} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {9 a^{5} b^{2} x^{2} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {18 a^{5} b^{2} x^{2} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} + \frac {3 a^{4} b^{3} x^{3} \log {\left (\frac {b x}{a} \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} - \frac {6 a^{4} b^{3} x^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{3 a^{\frac {19}{2}} + 9 a^{\frac {17}{2}} b x + 9 a^{\frac {15}{2}} b^{2} x^{2} + 3 a^{\frac {13}{2}} b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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