Optimal. Leaf size=62 \[ -\frac {3 b \sqrt {a+b x}}{4 x}-\frac {(a+b x)^{3/2}}{2 x^2}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
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Rubi [A]
time = 0.01, 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 = {43, 65, 214}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
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) \text {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 = 0.09, size = 53, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {a+b x} (2 a+5 b x)}{4 x^2}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 3.49, size = 61, normalized size = 0.98 \begin {gather*} -\frac {a \sqrt {b} \sqrt {1+\frac {a}{b x}}}{2 x^{\frac {3}{2}}}-\frac {3 b^2 \text {ArcSinh}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ]}{4 \sqrt {a}}-\frac {5 b^{\frac {3}{2}} \sqrt {1+\frac {a}{b x}}}{4 \sqrt {x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 52, normalized size = 0.84
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (5 b x +2 a \right )}{4 x^{2}}-\frac {3 b^{2} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{4 \sqrt {a}}\) | \(42\) |
derivativedivides | \(2 b^{2} \left (-\frac {\frac {5 \left (b x +a \right )^{\frac {3}{2}}}{8}-\frac {3 a \sqrt {b x +a}}{8}}{b^{2} x^{2}}-\frac {3 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}\right )\) | \(52\) |
default | \(2 b^{2} \left (-\frac {\frac {5 \left (b x +a \right )^{\frac {3}{2}}}{8}-\frac {3 a \sqrt {b x +a}}{8}}{b^{2} x^{2}}-\frac {3 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}\right )\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 86, normalized size = 1.39 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 124, normalized size = 2.00 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.60, size = 76, normalized size = 1.23 \begin {gather*} - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 86, normalized size = 1.39 \begin {gather*} \frac {-\frac {5 \sqrt {a+b x} \left (a+b x\right ) b^{3}-3 \sqrt {a+b x} b^{3} a}{4 \left (a+b x-a\right )^{2}}+\frac {3 b^{3} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {-a}}\right )}{4 \sqrt {-a}}}{b} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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