Optimal. Leaf size=68 \[ -\frac {\sqrt {a+b x}}{2 a x^2}+\frac {3 b \sqrt {a+b x}}{4 a^2 x}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 68, 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 = {44, 65, 214}
\begin {gather*} -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {3 b \sqrt {a+b x}}{4 a^2 x}-\frac {\sqrt {a+b x}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a+b x}} \, dx &=-\frac {\sqrt {a+b x}}{2 a x^2}-\frac {(3 b) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{4 a}\\ &=-\frac {\sqrt {a+b x}}{2 a x^2}+\frac {3 b \sqrt {a+b x}}{4 a^2 x}+\frac {\left (3 b^2\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{8 a^2}\\ &=-\frac {\sqrt {a+b x}}{2 a x^2}+\frac {3 b \sqrt {a+b x}}{4 a^2 x}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{4 a^2}\\ &=-\frac {\sqrt {a+b x}}{2 a x^2}+\frac {3 b \sqrt {a+b x}}{4 a^2 x}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 56, normalized size = 0.82 \begin {gather*} \frac {\sqrt {a+b x} (-2 a+3 b x)}{4 a^2 x^2}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 4.73, size = 102, normalized size = 1.50 \begin {gather*} \frac {\frac {-2 a^{\frac {11}{2}} x \left (a+b x\right )}{b}+a^{\frac {9}{2}} x^2 \left (a+b x\right )+3 a^{\frac {7}{2}} b x^3 \left (a+b x\right )-3 a^3 b^{\frac {5}{2}} x^{\frac {9}{2}} \text {ArcSinh}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ] \left (\frac {a+b x}{b x}\right )^{\frac {3}{2}}}{4 a^{\frac {11}{2}} \sqrt {b} x^{\frac {9}{2}} \left (\frac {a+b x}{b x}\right )^{\frac {3}{2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 66, normalized size = 0.97
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-3 b x +2 a \right )}{4 a^{2} x^{2}}-\frac {3 b^{2} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}}}\) | \(45\) |
derivativedivides | \(2 b^{2} \left (-\frac {\sqrt {b x +a}}{4 a \,b^{2} x^{2}}-\frac {3 \left (-\frac {\sqrt {b x +a}}{2 a b x}+\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )\) | \(66\) |
default | \(2 b^{2} \left (-\frac {\sqrt {b x +a}}{4 a \,b^{2} x^{2}}-\frac {3 \left (-\frac {\sqrt {b x +a}}{2 a b x}+\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 92, normalized size = 1.35 \begin {gather*} \frac {3 \, b^{2} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{8 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {b x + a} a b^{2}}{4 \, {\left ({\left (b x + a\right )}^{2} a^{2} - 2 \, {\left (b x + a\right )} a^{3} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 123, normalized size = 1.81 \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 (3 \, a b x - 2 \, a^{2}\right )} \sqrt {b x + a}}{8 \, a^{3} x^{2}}, \frac {3 \, \sqrt {-a} b^{2} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b x - 2 \, a^{2}\right )} \sqrt {b x + a}}{4 \, a^{3} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.70, size = 102, normalized size = 1.50 \begin {gather*} - \frac {1}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {\sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {3 b^{\frac {3}{2}}}{4 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 97, normalized size = 1.43 \begin {gather*} \frac {2 \left (-\frac {-3 \sqrt {a+b x} \left (a+b x\right ) b^{3}+5 \sqrt {a+b x} a b^{3}}{8 a^{2} \left (a+b x-a\right )^{2}}+\frac {3 b^{3} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {-a}}\right )}{4 a^{2}\cdot 2 \sqrt {-a}}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 51, normalized size = 0.75 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{3/2}}{4\,a^2\,x^2}-\frac {5\,\sqrt {a+b\,x}}{4\,a\,x^2}-\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{4\,a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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