Optimal. Leaf size=62 \[ -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {(a+b x)^{3/2}}{2 x^2}-\frac {3 b \sqrt {a+b x}}{4 x} \]
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Rubi [A] time = 0.02, antiderivative size = 62, 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 = {47, 63, 208} \[ -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {(a+b x)^{3/2}}{2 x^2}-\frac {3 b \sqrt {a+b x}}{4 x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{x^3} \, dx &=-\frac {(a+b x)^{3/2}}{2 x^2}+\frac {1}{4} (3 b) \int \frac {\sqrt {a+b x}}{x^2} \, dx\\ &=-\frac {3 b \sqrt {a+b x}}{4 x}-\frac {(a+b x)^{3/2}}{2 x^2}+\frac {1}{8} \left (3 b^2\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=-\frac {3 b \sqrt {a+b x}}{4 x}-\frac {(a+b x)^{3/2}}{2 x^2}+\frac {1}{4} (3 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )\\ &=-\frac {3 b \sqrt {a+b x}}{4 x}-\frac {(a+b x)^{3/2}}{2 x^2}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 68, normalized size = 1.10 \[ -\frac {2 a^2+3 b^2 x^2 \sqrt {\frac {b x}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x}{a}+1}\right )+7 a b x+5 b^2 x^2}{4 x^2 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 124, normalized size = 2.00 \[ \left [\frac {3 \, \sqrt {a} b^{2} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (5 \, a b x + 2 \, a^{2}\right )} \sqrt {b x + a}}{8 \, a x^{2}}, \frac {3 \, \sqrt {-a} b^{2} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (5 \, a b x + 2 \, a^{2}\right )} \sqrt {b x + a}}{4 \, a x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 64, normalized size = 1.03 \[ \frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3} - 3 \, \sqrt {b x + a} a b^{3}}{b^{2} x^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 0.82 \[ 2 \left (-\frac {3 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}+\frac {\frac {3 \sqrt {b x +a}\, a}{8}-\frac {5 \left (b x +a \right )^{\frac {3}{2}}}{8}}{b^{2} x^{2}}\right ) b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 86, normalized size = 1.39 \[ \frac {3 \, b^{2} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{8 \, \sqrt {a}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {b x + a} a b^{2}}{4 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 46, normalized size = 0.74 \[ \frac {3\,a\,\sqrt {a+b\,x}}{4\,x^2}-\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{4\,\sqrt {a}}-\frac {5\,{\left (a+b\,x\right )}^{3/2}}{4\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.28, size = 76, normalized size = 1.23 \[ - \frac {a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{2 x^{\frac {3}{2}}} - \frac {5 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{4 \sqrt {x}} - \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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